Real time PI-backstepping induction machine drive with efficiency optimization

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Real time PI-backstepping induction machine drive with efficiency optimization Fethi Farhani n, Chiheb Ben Regaya, Abderrahmen Zaafouri, Abdelkader Chaari Industrial Systems Engineering and Renewable Energies Research Laboratory (LISIER), Higher National Engineering School of Tunis (ENSIT), University of Tunis, 5 Av. Taha Hussein, BP 56, 1008 Tunis, Tunisia

art ic l e i nf o

a b s t r a c t

Article history: Received 18 March 2016 Received in revised form 16 April 2017 Accepted 1 July 2017

This paper describes a robust and efficient speed control of a three phase induction machine (IM) subjected to load disturbances. First, a Multiple-Input Multiple-Output (MIMO) PI-Backstepping controller is proposed for a robust and highly accurate tracking of the mechanical speed and rotor flux. Asymptotic stability of the control scheme is proven by LYAPUNOV Stability Theory. Second, an active online optimization algorithm is used to optimize the efficiency of the drive system. The efficiency improvement approach consists of adjusting the rotor flux with respect to the load torque in order to minimize total losses in the IM. A dSPACE DS1104 R&D board is used to implement the proposed solution. The experimental results released on 3 kW squirrel cage IM, show that the reference speed as well as the rotor flux are rapidly achieved with a fast transient response and without overshoot. A good load disturbances rejection response and IM parameters variation are fairly handled. The improvement of drive system efficiency reaches up to 180% at light load. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Field Oriented Control Efficiency Induction machine MIMO PI-backstepping Optimization

1. Introduction The economic reasons and environmental friendliness are taken into primary consideration in the design of electric drive systems. Almost all of IMs drive strategies, based on the Field Oriented Control (FOC) introduced initially by BLASCHKE in 1972. This control technique considers that the magnetic flux reference is constant and closes to its rated level ( λ rnm) [1,2] for high dynamic performance regardless of the operating points. Such procedure results in an unsatisfactory energy efficiency when the IM is under-loaded [3]. Many investigations have shown that almost 45% of IMs drive 40% of their rated load [4]. To overcome this drawback, the flux must be auto-adjusted online in respect of the load torque to achieve a minimum power loss [5,6]. In fact, the efficiency optimization algorithm tracks the unique optimal flux leakage value for each operating point and apply it to controlled IM. In the literature, many approaches are proposed to optimize the drive system efficiency [4–10], these solutions are basically divided into two categories: Model Based Optimization (MBO) and Search Algorithm based Optimization (SAO). Main drawbacks of the MBO are the number of arithmetic operations involved in the solution of the loss model and its high sensitivity to parameters variations [11]. The SAO are characterized by their parameter n

Corresponding author. E-mail address: [email protected] (F. Farhani).

insensitivity. However, it has some serious disadvantages such as a high torque ripples [10] and very slow convergence to optimal operating point compared to MBO. In these cases, the search space is very large and results in a more time to seeking the optimal operating conditions. The combination of this two methods is so called Hybrid Algorithm based Optimization (HAO) which tries to get benefit from their advantages. A complete overview of existing optimization methods is found in [7]. Based on a mathematical model of IM, the MBO can rapidly converge to the optimal flux. J. Rivera et al. propose on their work [9] a loss model based on the resistive and core loss under the assumption that all IM parameters are invariant and the core loss can be emulated by a constant resistance. However, on the one hand, the stator and rotor resistances may vary up 50% and 100% respectively [12]. On the other hand, the core-loss considerably depends on the flux level, stator frequency and inverter switching frequency in practices [6,13]. Indeed, without a real-time parameters adaptation mechanisms, the optimization algorithm can underestimate the optimal flux, can also lead to destabilize the drive system and cause the “motor stalling” issue specially in presence of sudden change in load torque. Even with IM parameters tracking algorithms, it is very difficult to identify the IM parameters simultaneously and accurately in the full operating region [14]. To overcome the parameters variation problems and its impact on the stability of the drive system, this paper proposes a cooperative two-step efficiency optimization. Firstly, by using a simplified model losses, the controller quickly achieves a first

http://dx.doi.org/10.1016/j.isatra.2017.07.003 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Nomenclature

vsd , vsq isd , isq λ rd , λ rq Ce , Cr Ptopt , Ptot

: : : : :

d and q components of stator voltages; d and q components of stator currents; d and q components of rotor flux; Electromagnetic and load torques; Optimal and total power losses;

approximation of optimal flux and so called “suboptimal flux”. This step, reduces the possible solutions space without disestablishing the drive system. Concerning the second step, from the near-optimal operating point, a search algorithm achieves a smooth and robust convergence to the optimal flux. As mentioned early, the efficiency optimization consists of decreasing the flux leakage value in the IM to its acceptable minimum level. As a matter of fact, the success of the efficiency optimization highly depends on the accuracy and the robustness of the flux regulation. Indeed, accurate estimation and robust flux controller guarantees the maximization of the energy saving. In addition, it avoids the motor stalling caused by low flux level. In this respective, the paper refers to Direct Field Oriented Control (DFOC) scheme to track the flux reference generated by the hybrid optimization algorithm. Recently, various control schemes have been proposed and tested in different control loops of DFOC, such as, Backstepping [15], sliding mode [9,16–20], feedback linearization [21–23], LPV approach [24]. However, conventional proportional/integral (PI) regulators still commonly used in industry due to its simplicity and ease of implementation [25]. Nevertheless, without parameters adaptation, the desired performances response (i.e accuracy, response times, smooth running…) are not obtained, especially in presence of sudden change in load torque and external disturbances [26]. To overcome its sensitivity to parameters variation, an online parameter tracking algorithm is used to estimate the corresponding parameters in the FOC [27]. However, this solution increases the complexity of the controller and the difficulty of a real-time implementation. Based on the scheme proposed by Benzineb et al [28] in stationary reference frame, this paper retains the conventional PI correctors and combines it with a Backstepping stage in rotating reference frame. This approach aims to benefit of simplicity of PI controller and the robustness of the Backstepping technique. Indeed, the PI controller achieves a high accuracy tracking of direct and quadrature currents references. Backstepping stage simultaneously guarantees, on the one hand a perfect tracking of rotor speed and flux references, and the decoupling between their dynamic on other hand. The stability of proposed controller is proven by LYAPUNOV Stability Theory.

: Actual, rated and optimal yields; η , η0 , ηopt : Stator, rotor and mutual inductances; Ls , L r , M : Stator and rotor resistances; Rs , R r p : Number of pole pairs; J : The inertia of IM and load; : Rotor and rotating frame angular velocity; ωr , ωs 2 M σ = 1 − L L : Total linkage coefficient. r s

⎧ di ⎪ Ld ⎪ dt ⎪ diLq ⎪ ⎪ dt ⎨ ⎪ dλ r ⎪ dt ⎪ ⎪ dω r ⎪ dt ⎩

(

)

(

)

= fid iLd , iLq, λ r , ωr + hd ( vsd )

( )

= fiq iLd , iLq, λ r , ωr + hq vsq = fλ ( iLd , λ r ) = M

(

Rr R iLd − r λ r Lr Lr

)

= fω iLq, λ r , ωr , Cr =

C − fω r pM λ r iLq − r JL r J

(1)

with

⎧ ⎪ f iLd , iLq, λ r , ωr ⎪ id ⎪ ⎪ Rr λ rd ⎪ +pωr iLq + ε Lr ⎪ ⎪ ⎪ fiq iLd , iLq, λ r , ωr ⎪ ⎨ ⎪ pω r ⎪ −pωr iLd − λ rd ε ⎪ ⎪ ⎪ h (v ) ⎪ d sd ⎪ ⎪h v q sq ⎪ ⎩

(

)

(

)

( )

⎛1 − σ R ⎞ 1 r =−⎜ + Rsc ⎟iLd σL s ⎠ ⎝ σ Lr +

2 R r iLq M L r λ rd

⎛1 − σ R ⎞ 1 r =−⎜ + Rsc ⎟iLq σL s ⎠ ⎝ σ Lr −

R r iLqiLd M λ rd Lr

=

Rcv vsd = μvsd σL s

=

Rcv vsq = μvsq σL s

(2)

where Rc is the core loss resistance, Rsc =

ε=

σLsL r . M

RsRc , Rs + Rc

Rcv =

Rc Rs + Rc

and

T

The load current iL = ⎡⎣ iLd iLq⎤⎦ is the difference between

( )

the stator current ( is) and the current i f consumed by Rc (Fig. 1) and given as

⎛R +R ⎞ v c iL = ⎜ s ⎟is − s Rc ⎝ Rc ⎠

(3)

The electromagnetic torque is described as follows:

⎛ 3 M⎞ C e = p⎜ ⎟λ r irq ⎝ 2L r ⎠

(4)

Stator copper, rotor copper, and core losses dominate the overall IM power losses and can be defined as:

Ptot = Pc + Pf

(5)

2. Model of IM in oriented (d, q) reference frame The FOC theory achieves the decoupling between flux and torque dynamics. This technique involves aligning the controlled flux space vector with d-axis of the rotating reference frame → → λ r = λ rd . In fact, the nonlinear state equation of IM can be

(

)

written by the following equation:

Fig. 1. Equivalent circuit of IM in ( d, q) reference frame ( lfs , lfr are stator and rotor leakage inductances receptively).

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

(

Pf =

)

(

2 2 2 2 Pc = Rs isd + isq + Rr ird + irq

where

(

Rc i2fd

+

i2fq

)

(copper

losses)

and

) (iron loss).

3. IM efficiency improvement strategy IM reaches optimal efficiency in its rated operating point. However, more than the half of used IMs are underloaded [29]. In these cases, the power losses minimization is a major gain. In this paper, a new algorithm of power loss minimization is developed based on tow cooperative steps. The main objective of the first one, is to allow a quick and first approximation of the optimal flux λ ropt . A simplified losses model of IM is enough to achieve this goal. However, the flux level generated by the MBO can be lower than the optimal one (the acceptable minimal flux level) due to the high uncertainty of IM model. In this case, the flux level applied to the controller causes the “motor stalling” issue. Hence, at the end of the first step of optimization processes, the output of the MBO will be multiplied by a scalar design coefficient γ great than the unit ( γ > 1) and so-called sub-optimal rotor flux λ rsopt . Second step of the optimization processes consists of a global power loss optimization. For this purpose, a second algorithm based on Simulated Annealing Method (SAM) is developed. The search algorithm measures the input power ( Pin), and then, iteratively decreases the flux level from its sub-optimal value, until the minimum of input power is reached.

(

)

(

)

Remark 1. In practice, the success of the motor efficiency optimization algorithm and the potential of energy saving depend highly to the quality and stability of the generated line current (stability, distortion…). In the presented work, the main power supply is considered as an ideal or regulated DC source (for more details the reader can refer to [30,31]). The modulation technique, the switching frequency and the inverter topology are optimized in order to limit their impact on the motor iron losses. 3.1. Step 1: Computing the suboptimal rotor flux From (4) and (5), the simplified power loss model can be expressed in terms of iLd , iLq and the operating conditions by the following equation:

⎛ 2L C ⎞2 2 r e ⎟ Ptot = RdiLd + Rq⎜ ⎝ 3pM2 iLd ⎠ M ω2 Rc + Rr s dP condition ditot = Ld

where Rd = Rs + The

2

(6)

and Rq = Rs + Rsc .

0 corresponding to minimum of power

loss defined by (6). Hence, the rotor flux current producer ( iLd) can be expressed as follows: 4 iLd =

⎞2 R q ⎛ 2L r ⎜ C⎟ 2 e Rd ⎝ 3pM ⎠

3

IM losses model and motor parameters variation ranges. On the one hand, a large coefficient γ consolidates the stability of the drive system and ensures the relaxation of the search algorithm which will be called in the second step of optimization cycle. On the other hand, this increases dramatically the convergence time to the global optimal flux. Therefore, the design coefficient will be chosen as γ ∈ ⎡⎣ 1.1 1.25 ⎤⎦. 3.2. Step 2: Seeking of the optimal rotor flux The second step of the efficiency optimization is enabled on the steady state and insured by the SAO technique. Indeed, a SAM algorithm adjusts finely the rotor flux in order to reduce the electric power losses caused by flux current component producer. The solution space is bounded between the suboptimal rotor flux λ rsopt and the acceptable minimal rotor flux ( λ rmin) corresponds to

(

)

the no-load operating point

(λ

ropt

∈ ⎡⎣ λ rmin λ rsopt ⎤⎦ . The optimal

)

rotor flux trajectory, at each iteration ( k ), is given by the following expression:

λ ropt (k ) = δλ ropt ( k − 1) + ( 1 − δ )λ rmin

(10)

Hence, the optimization algorithm decreases the level of the rotor flux from its initial value λ rsopt at each iteration. The convergence speed to the optimal flux is handled by the rate coefficient δ ∈ ⎤⎦ 0 1 ⎡⎣ . Indeed, the choice of δ is critical to the optimizing process and the stability of drive system. A very small δ causes high ripple torque since the rotor flux is lowered too quickly. However, if δ is near the unit ( δ ≈ 1), the flux level is decreased too slowly. Indeed, the optimization algorithm takes more time to seeking the optimal flux and can get trapped at a local optimum and can yield to a very poor efficiency. To ensure the success of the optimization process, the rate coefficient will be bounded by δmax and δmin δ ∈ ⎤⎦ δmin δmax ⎡⎣ . Where The upper and the lower limits of δ correspond respectively to acceptable ripple torque and the power measurement noise ( Pnos) [32]. These constraints ensure the convergence of the optimization algorithm but its relaxation time and its accuracy depends on the expertise of the drive design engineer (Fig. 2, dashed lines). To obtain the best solution for a quick and accurate convergence to optimal operating point (Fig. 2, solid line), for each iteration k, the rate coefficient δ will be increased smoothly from its lower limit towards its upper limit according to the current variation rate of input power. Fig. 3 shows the variation rate curves of input power versus the changing applied to the rotor flux level ( Δλ r ) for four different operating points simultaneously. The zone with negatives variation rates correspond to the minimization of input power. This zone can be divided in three zones, namely, high (red zone), medium (blue zone) and slow variation rate (green zone). Based on this repartition, the rate coefficient δ will be adjusted

(

(

)

)

(7)

By ignoring the dynamic components, substituting λ r from (1) into (7) yields:

λr =

4

dPtot =0 diLd

Rq Rd

2L r Ce 3p (8)

The suboptimal of the rotor flux can be expressed as follows:

λ rsopt = γ 4

Rq Rd

2L r Ce 3p

(9)

where γ is a scalar design coefficient depends on the accuracy of

Fig. 2. Impact of the rate coefficient ( δ ) on the optimal rotor flux trajectory (noload operating point).

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

Fig. 3. Input power variation rates versus rotor flux variation (Initial rotor flux level is 0.75Wb ).

Fig. 4. Block diagram of DFOC scheme based IM drive.

4.1. Control objectives and problem formulation accordingly by the following adaptive law:

(11)

The block diagram of electrical drive with efficiency optimization is shown in Fig. 4. The control errors of speed ( eω ), rotor flux ( eλ ), load current direct component ( eid ) and load current quad-

ν is a positive damping

rature component eiq may be described in ( d, q) reference frame

⎛ P (k ) − P ( k − 1) ⎞ in in ⎟⎟ δ = δmax + ν⎜⎜ ⎝ Pin( k − 1)Δλ r (k ) ⎠ where

Δλ r (k ) = λ ropt (k ) − λ ropt ( k − 1),

coefficient related to variation rate of input power

( ) and is ΔPin Δλr

defined by the following three tests:

if

ΔPin Δλ r

>

−P1

then ν = ν1

if

−P2

<

ΔPin Δλ r

< − P1 then ν = ν2

if

ΔPin Δλ r

<

−P2

then ν = ν3

( )

as follows:

⎧e = ω* − ω ω r r ⎨ ⎪ * − λ rd ⎩ eλ = λ rd

(13)

⎧ * − iLd = iLd ⎪ eid ⎨ ⎪ * − iLq ⎩ eiq = iLq

(14)

⎪

(12)

P1 and P2 are two levels of rate power saving (Fig. 3). ν1, ν2 and ν3 are three design scalars ensuring the growth of δ from its lower limit to its upper one. Finally, The optimization algorithm relaxes for one of the following conditions:

 λropt ≤ λrmin  ΔPin ≤ Pnos with ΔPin = Pin(k ) − Pin( k + 1) The first and the second condition correspond to no-load operation and optimal operating point respectively. Remark 2. The power measurement noise can be approximated by the gap between the largest and lowest input power measures of a given operating point, in this paper it is approximated by 50 W .

4. DFOC based on PI backstepping technique The key idea of the proposed scheme in ( d, q) reference frame is using only two PI regulators instead of four PI regulators in the case of DFOC classical approach [33] to improve the dynamic performance (i.e accuracy, raising time, overshoot…) of the drive system. Indeed, as mentioned previously, the inner regulation loops ( iLd and iLq regulation loops) are carried out by an MIMO PI regulator. Unlike the inner regulation loops, the outer regulation loops (speed and flux regulation loops) are ensured by a non-linear Backstepping controller. The absence of the PI speed and flux controllers ensures a faster dynamic response.

where the superscript “n” denotes the reference signals. We define two auxiliary functions and their first derivative as follows:

⎧⎧ ⎪⎪ϖ ⎪⎪ ⎪⎪ ⎪ ⎨ ϖ̇ ⎪⎪ ⎪ ⎪⎪ ⎨⎪ ⎪⎩ ⎪⎧ ⎪⎪ψ ⎪⎪ ⎪⎨ ⎪ ⎪ ψ̇ ⎪ ⎪ ⎩⎩

=−

pM λ r iLq JL r

=−

pM ⎡ ̇ λ r iLq + λ r fiq iLd , iLq, λ r , ωr JL r ⎣

( (

)

⎤ +hq vsq ⎦

( ))

=−M

Rr iLd Lr

=−M

Rr ⎡ ⎤ ⎣ f iLd , iLq, λ r , ωr + hd ( vsd )⎦ L r id

(

)

(15)

The dynamics of speed and flux regulation errors can be rewritten as the following differential equations:

⎧ C fω r +ϖ ⎪ eω̇ = ω̇ r* + r + ⎪ J J ⎨ R ⎪ ̇ eλ = λ ṙ * + r λ r + ψ ⎪ Lr ⎩

(16)

Eq. (16) highlights that the rotor velocity as well as the rotor flux control errors can be directly controlled respectively by ϖ and ψ . These two functions are themselves generated by both direct and quadrature stator voltage components ( vsd and vsq ). We can therefore conclude that the speed and flux control errors can be mitigated by the two auxiliary functions hd and hq . To solve the flux and speed tracking problems, a MIMO PI is used to generate * * and vsq * and tracks load currents references ( iLd the control laws vsd * and iLq ) as:

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

⎡ v* ⎤ sd vs*(t ) = ⎢ ⎥ = KpE(t ) + Ki * ⎥⎦ ⎢⎣ vsq

∫0

t

5

E(τ )dτ (17)

T

Where E = ⎡⎣ eid eiq ⎤⎦ is the control error vector of the load current iL . Kp and Ki are two matrices of appropriate dimensions that define the proportional and integral gains respectively. 4.2. Control lows synthesis To overcome the non-linearity of the IM as well as the parameters uncertainties and unknown load torque disturbances, the outer regulations loops are made by the Backstepping methodology. The following theorem defines the control laws T i* = ⎡⎣ i* i* ⎤⎦ ensuring the tracking of the IM speed and rotor L

Ld

Lq

Fig. 5. Experimental test bench.

flux references.

Theorem 3. Consider the dynamic system (16). For positives scalairs C where Cmax is the maximum load torque, the Kω and Kλ and Kcr > max J control errors are exponentially stable if the following condition is verified:

must be tuned to achieve the desired behavior and dynamic performances of the drive system.

⎧ ⎛ ⎞ ̇* + Rr λ r⎟ * = L r ⎜ Kλeλ + λ rd ⎪ iLd MR r ⎝ Lr ⎠ ⎪ ⎨ JL r ⎛ ⎪ fω r ⎞ * = ⎜ Kωeω + ω̇ r* − sig ( eω )Kcr + ⎟ ⎪ iLq pMλ r ⎝ J ⎠ ⎩

5. Experimental results and discussion

(18) □

Proof. The proof is given in Appendix A.

* and vsq * are provided by the The control laws of stator voltage vsd following theorem: Theorem 4. Considering the IM dynamic model (1)–(2) and the outer loop defined by the Theorem 3, the convergence of iL components to their references is guaranteed through MIMO PI controller whose proportional Kp and integral ( Ki) gains are defined as follows:

( )

⎡ r 0⎤ d ⎥ = η−1R Kp = η−1⎢ ⎢⎣ 0 rq ⎥⎦

(19)

⎡ s 0⎤ d ⎥ = η−1S Ki = η−1⎢ ⎢⎣ 0 sq ⎥⎦

(20)

with η =

Rcv I, σL s

R is a matrix with strictly positive real coefficients and S

is a positive definite matrix which satisfies the following condition

⎡s κ 0 ⎤ d d ⎥ > S¯ such as S=⎢ ⎢⎣ 0 sqκq ⎥⎦ κd(q)

Table 1 IM Nominal Parameters.

⎡ s¯ 0 ⎤ d ⎥ = ed(q) ∫ ed(q)(τ )dτ and S¯ = ⎢ 0 ⎢⎣ 0 s¯q ⎥⎦ t

with ⎧ Mλ rd eλeid + id*̇ − fid iLd , iLq, λ rd , ωr eid ⎪ s¯d >p JL r ⎪ ⎨ Rr ⎪ *̇ ⎪ s¯q >M L eω eiq + iq − fiq iLd , iLq, λ rd , ωr eiq ⎩ r

(

(

(

(

Proof. The proof is given in Appendix B.

To validate the proposed control scheme and evaluate its performances as well as its impact on the efficiency of the drive system, an experimental test bench shown in Fig. 5 has been built. The controller with the hybrid efficiency optimization algorithm has been implemented using a dSPACE DS1104 R&D board with TMS320F240 DSP. A 3 kW squirrel cage IM is used in this experimental investigation, which is coupled to a powder brake. The IM nameplate data and parameters are listed in Table 1. The IM is fed by a Semikron three-phase inverter with opto-isolation and gate driver circuit SKHI22A. The overall structure of the test bench is shown in Fig. 6. Two tests are made, without and with efficiency optimization. The first test, without efficiency optimization, verifies the performances of the PI-Backstepping controller. In the second test, the optimization algorithm is enabled to evaluate the electric energy saving and to verify the robustness of the proposed solution facing a sudden change in load torque. Furthermore, all these tests take into account the presence of parametric uncertainties (20%) in order to verify the robustness and the accuracy of proposed solution facing to parameter mismatch. The experimental test results will be analyzed with quantitative and qualitative methods. The quantitative analysis shows the static and dynamic behavior of the tested system (raising time, accuracy…etc.). However, the qualitative analysis shows the ability of the controller to respect the accuracy requirements. To this end, two performance metrics Cp

))

))

□

Remark 5. It is important to highlight that the Theorem 4 provides a sufficient condition on the choice of matrix S S > S¯ for the stability of the drive system. However, an excessive integral gain yields closed-loop instability. In practice, based on the design tradeoff between rapidity and stability, the MIMO PI controller

(

)

Symbol

Quantity

Unit of measure

Rs Rr Rc Ls Lr M σ J f ωr Pn Cmax p

Stator resistance

2.3

[Ω]

Rotor resistance Equivalent iron core-loss

1.8 92

[Ω] [Ω]

Stator inductance

261

[ mH ]

Rotor inductance Mutual inductance Leakage factor Moment of inertia Friction coefficient Rated motor Speed Rated power

261 245 0.134 0.22 0.001 149 3

[ mH ] [ mH ] – [Kg m2] – [rad s 1]

Rated load torque

20

Number of pole pairs

2

[ kW ] [ Nm ] –

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

Fig. 6. Overall structure of the test-bench.

Table 2 Evaluation board. Cp, Cpk

Tracking accuracy

0 − 0.67 0.67 − 1 1 − 1.33 1.33 − 1.67 1.67 − 2 >2

Very poor Poor Poor to moderate Moderate Good Excellent

(Process potential Capability) and Cpk (The capability index, which accounts for process centering) will be used. These indices are mostly used to evaluate the capability and the performance of industrial process through a statistical measurements. This method can be used, also, for comparison with another competitor. Cp is defined as the ratio of the allowable distribution over the actual distribution. The Cpk index indicate the process's ability to meet its reference within specification limits. Indeed, these indices should always be greater than 1.67 for a satisfactory accuracy. Capability studies can be conducted with the MATLABs function “capability”. We refer the reader to [34] for more details. The Table 2 shows the evaluation board.

Fig. 7. Speed response for reference with slow dynamic profile.

Remark 6. Several wave references for the rotor flux and the motor speed were completed to evaluate the performances of the proposed controller. However, for briefness, this paper presents only the most important results. Fig. 8. Speed response for reference: High dynamic profile.

5.1. System performance without efficiency optimization A speed control test and a flux control test will be made with the proposed PI-Backstepping controller and a conventional PI [26]. Their performances will be analyzed and compared. 5.1.1. Speed control test Fig. 7 illustrates the speed reference and these measured. The load torque is set to 15 Nm. A 5 Nm step was applied at 22.5 s and removed at 27.5 s . Unlike the PI controller, the proposed controller

present a higher robustness to load perturbation. Fig. 8 shows the motor speed responses to the rapid dynamic reference profile. Compared to conventional drive, the PI-Backstepping has a short rise time without overshoot. Fig. 9 shows the distribution of speed measurements. The drive system presents an excellent capability to maintain the speed control error less than 5%( Cp|BS = 3.22 ≫ 1.67) and a very smooth response. Therefore, the distribution of measured speed is limited on 33% of the speed

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

7

Fig. 9. Global distribution of the normalized mechanical speed.

Fig. 12. Load torque profile: rapid dynamic.

Fig. 10. Rotor flux response: High dynamic profile.

Fig. 13. Rotor flux reference.

Fig. 11. Distribution of rotor flux: Slow dynamic profile.

Fig. 14. Electric power losses ( Ptnm : Total power losses when λ r = λ rnm ).

tolerance range. The capability of the conventional controller is very poor ( Cp|PI = 0.88) due to the overshoot. 5.1.2. Rotor flux control test In this test, the rotor velocity is maintained constant as well as the load torque is fixed to 5Nm. Fig. 10 shows the rotor flux responses to the rapid dynamic reference profile. Both controllers release a high accurate reference tracking. However, the PI-Backstepping controller improves the setting time and presents the better capability ( Cp|BS = 5.06⋙Cp|PI = 0.93) to respect the specification requirements (Fig. 11) due to the overshoot illustrated by using the classical scheme. Fig. 15. IM efficiency.

5.2. Experimental results with efficiency optimization In this section, we discuss the ability of the proposed control approach to improving the efficiency of drive system and maintaining its static and dynamic performances. Indeed, the IM is subjected to high dynamic load torque profile shown in Fig. 12. The rotor speed reference is set to 120 rad/s . A 30 rad/s step was

applied at 35 s and removed at 40 s . In this round of evaluation, the efficiency performance of the proposed drive is compared to conventional FOC (without efficiency optimization). Fig. 13 shows the rotor flux reference generated by the HAO as well as the rated rotor flux. The optimization routines are activated at t = 5 s . Fig. 14 shows the calculated total power losses and which are

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

1 2 1 eω + eλ2 2 2

(A.1)

V̇ = eω eω̇ + eλeλ̇

(A.2)

V=

From Eqs. (1) and (A.2), one gets:

V̇ = V1 + V2 + V3

(A.3)

where

⎧ 2 2 ⎪ V1 = − Kωeω + Kλeλ ≤ 0 ⎪ ⎛ ⎞ ⎪ V = e ⎜ K e + ω̇ * + Cr + fωr + ϖ ⎟ r ω ω ω ⎨ 2 J J ⎝ ⎠ ⎪ ⎛ ⎞ ⎪ Rr λr + ψ ⎟ ⎪ V3 = eλ⎜ Kλeλ + λ ṙ * + Lr ⎝ ⎠ ⎩

(

Fig. 16. Measured rotor velocity and its reference.

significantly reduced after enabling the HAO algorithm (up to 70% at light loads). The power saving decrease with increasing the load torque. The IM efficiency is shown in Fig. 15. Whatever the load torque, the efficiency is kept at its maximum. Compared to the conventional FOC, the improvement in the efficiency up to 180% at light load. The rotor velocity closes perfectly to the reference at different operating points (Fig. 16). In addition, the controller ensures the same drive behavior without power saving and a good rejection of exterior disturbance caused by the sudden variation in load torque. The optimization block acts on the input power without degrading the dynamic performance of the drive system even for sudden change in load and avoid the motor stalling issue. The maximal convergence time is 2 s when the load is near the rated one.

6. Conclusion In this paper, an efficient and robust DFOC for IM variable speed control is described and implemented. The proposed scheme uses a hybrid optimization algorithm in order to maximize the drive system efficiency. The optimization cycle is achieved in two steps. The first one, computing a sub-optimal rotor flux via the dynamic model of IM taking into account the iron loss by using a shunt resistance. From the sub-optimal rotor flux, the second algorithm, based on the simulated annealing technique, tracks the optimal one regardless of the IM parameters. This strategy, overcomes the main drawbacks of conventional on-line efficiency optimization (motor stalling, ripple torque, sensitivity to parameters variations…) and maximize the electric energy saving. As a result, a smooth and fast minimization of the drive system input power is achieved. The success of the proposed scheme requires a highperformance vector control. Indeed, a PI-Backstepping controller is developed in order to control the motor speed and the rotor flux. The presented controller is robust to parameter variations and external torque disturbances. The stability and robustness of the closed-loop have been proven through LYAPUNOV stability theory. Extensive experimental results of the proposed design which are made on 3 kW IM show a great improvement in drive efficiency (up to 180% at light load) with a high convergence rate. A capability analysis of experimental results proves the effectiveness of the references tracking. Indeed, the capability of the speed control loop is 3.22 (two times larger than industrial standards), the rotor flux control loop has also a very high capability 5.06.

Appendix A. Proof of theorem 3 Let us define the candidate LYAPUNOV function V and its first time derivative V̇ as:

)

(A.4)

with Kω and Kλ are the positive design gains that determine the dynamic of closed loop. To ensure the convergence of control errors ( eω and eλ ) to zero, the first time derivative of the LYAPUNOV function candidate (A.3) must be negative definite. This condition is sufficiently satisfied when V2 + V3 = 0. However, to guarantee the tracking stability in the presence of an unknown external disturbance ( Cr ), V2 + V3 should be always negative definite regardless of the load torque. * and iLq * that stabilize the tracking errors Indeed, the control laws iLd are defined by the following Eq. (A.5) where Kcr >

Cmax . J

⎧ ⎛ ⎞ ̇ * + R r λ rd⎟ * = L r ⎜ Kλeλ + λ rd ⎪ iLd MR r ⎝ Lr ⎠ ⎪ ⎨ ⎞ JL r ⎛ ⎪ fω r * = − sig ( eω )Kcr ⎟ ⎜ Kωeω + ω̇ r* + ⎪ iLq pMλ rd ⎝ ⎠ J ⎩

(A.5)

Appendix B. Proof of theorem 4 Let us recall the current control errors as follows:

⎧ * − iLd = iLd ⎪ eid ⎨ ⎪ * − iLq ⎩ eiq = iLq

(B.1)

Consider the following LYAPUNOV function candidate V4 as:

V4 = V +

1 T EE 2

(B.2)

Substituting from (A.1) to (14), the simplified first time derivative of the augmented LYAPUNOV function V4̇ becomes:

(

V4̇ = − Kωeω2 + Kλeλ2

)

⎞ ⎛ ⎡ Mλ ⎤ rd ⎟ ⎜⎢ p eλ⎥ ⎟̇ ⎜ ⎢ JL r ⎥ + ET ⎜ ⎢ + E ⎟ ⎥ R r ⎟ ⎜⎢ M e ⎥ ω ⎟ ⎜⎢ L r ⎦⎥ ⎠ ⎝⎣

(B.3)

T * vsq * ⎤⎦ by the one Replace the voltages references vector ⎡⎣ vsd generated by the PI controller, the iL control errors dynamics yields:

⎤ ⎡ *̇ ⎛ ⎢ id − fid iLd , iLq, λ rd , ωr ⎥ Ė = ⎢ ⎥ − ϱ⎜⎝ KpE + Ki * ̇ ⎢⎣ iq − fiq iLd , iLq, λ rd , ωr ⎥⎦

( (

with ϱ =

Rcv I. σL s

) )

∫0

t

⎞ E(τ )dτ ⎟ ⎠

(B.4)

The simplified form of the time derivative of V4

becomes:

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i

F. Farhani et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

(

)

V4̇ = − Kωeω2 + Kλeλ2 − ET RE + ϑ(t )

(B.5)

with

R = ϱKp

(B.6)

⎛ ⎡ Mλ ⎤ r ⎜⎢ p eλ⎥ ⎡ * ⎤ JL r ⎥ ⎢ iḋ − fid iLd , iLq, λ r , ωr ⎥ T⎜ ⎢ ϑ(t ) = E ⎜ ⎢ ⎥ ⎥+⎢ * ⎜ ⎢ M R r e ⎥ ⎢⎣ iq̇ − fiq iLd , iLq, λ r , ωr ⎥⎦ ⎜⎢ L ω ⎥ ⎦ ⎝⎣ r

( (

−S

∫0

t

) )

⎞ E(τ )dτ ⎟ ⎠

(B.7)

where

⎡ s 0⎤ d ⎥ S = ϱKi = ⎢ ⎢⎣ 0 sq ⎥⎦

(B.8)

Such as R is a positive definite matrix, the condition ( ϑ(t ) < 0) is sufficient to ensure that V4̇ < 0. To guarantee that ϑ(t ) < 0, we seek sd and sq such that:

⎧ Mλ r eλeid + id*̇ − fid ( *) eid − sdeid ⎪p ⎪ JL r ⎨ ⎪ Rr *̇ ⎪ M L eω eiq + iq − fiq ( *) eiq − sqeiq ⎩ r

( (

) )

∫0

∫0

t

t

eid (τ )dτ < 0

eiq(τ )dτ

<0

(B.9)

t

as eid(q) ∫ eid(q)(τ )dτ ≥ 0 ∀ t ≥ 0, we impose on the pair sd and sq the 0 following condition:

⎧ ⎪ ⎪ sdeid ⎨ ⎪ se ⎪ ⎩ q iq

∫0 ∫0

t

t

eid (τ )dτ > s¯d eiq(τ )dτ > s¯q

(B.10)

with

⎧ Mλ r eλeid + ⎪p ⎪ JL r ⎨ ⎪ Rr ⎪ M L eω eiq + ⎩ r

( i *̇ − f ( i , i , λ , ω ))e ( i *̇ − f ( i , i , λ , ω ))e d

q

id

iq

Ld

Ld

Lq

Lq

r

r

r

r

id

iq

< s¯d < s¯q

We choose sd and sq sufficiently large to maintain ϑ(t ) always negative. From Eqs. (B.6) and (B.8), the MIMO PI controller gains are defined as follows:

⎧ −1 ⎪ Kp = ϱ R ⎨ −1 ⎪ ⎩ Ki = ϱ S

(B.11)

References [1] Zhang X. Sensorless induction motor drive using indirect vector controller and sliding-mode observer for electric vehicles. IEEE Trans Veh Technol 2013;62 (7):3010–8. [2] El Fadili A, Giri F, El Magri A, Lajouad R, Chaoui FZ. Towards a global control strategy for induction motor: speed regulation, flux optimization and power factor correction. Int J Electr Power Energy Syst 2012;43(1):230–44. [3] Navneet K, Thanga Raj C, Srivastavab SP. Adaptive control schemes for improving dynamic performance of efficiency-optimized induction motor drives. ISA Trans 2015;57:301–10. [4] Farhani F, Zaafouri A, Chaari A. Advances in efficiency optimisation control of electrical servo drive. Int J Renew Energy Technol 2014;5(2):125–40. [5] Stumper J-F, Dotlinger A, Kennel R. Loss minimization of induction machines

9

in dynamic operation. IEEE Trans Energy Convers 2013;28(3):726–35. [6] Qu Z, Ranta M, Hinkkanen M, Luomi J. Loss-minimizing flux level control of induction motor drives. IEEE Trans Ind Appl 2012;48(3):952–61. [7] Bazzi AM, Krein PT. Review of methods for real-time loss minimization in induction machines. IEEE Trans Ind Appl 2010;46(6):2319–28. [8] Hajian M, Arab Markadeh GR, Soltani J, Hoseinnia S. Energy optimized slidingmode control of sensorless induction motor drives. Energy Convers Manag 2009;50(9):2296–306. [9] Rivera Dominguez J, Mora-Soto C, Ortega-Cisneros S, Raygoza Panduro JJ, Loukianov AG. Copper and core loss minimization for induction motors using high-order sliding-mode control. IEEE Trans Ind Electron 2012;59(7):2877–89. [10] Uddin MN, Nam SW. Development of a nonlinear and model-based online loss minimization control of an IM drive. IEEE Trans Energy Convers 2008;23 (4):1015–24. [11] Cabezas Rebolledo AA, Valenzuela MA. Expected savings using loss-minimizing flux on IM drives; Part I: optimum flux and power savings for minimum losses. IEEE Trans Ind Appl 2015;51(2):1408–16. [12] Kenne G, Nguimfack Ndongmo JD, Kuate Fochie R, Fotsin BH. An online simplified nonlinear controller for transient stabilization enhancement of DFIG in multi-machine power systems. IEEE Trans Autom Contr 2015;60(9):2464–9. [13] Lee J-J, Kim Y-K, Nam H, Ha K-H, Hong J-P, Hwang D-H. Loss distribution of three-phase induction motor fed by pulsewidth-modulated inverter. IEEE Trans Magn 2004;40(2):762–5. [14] Chen B, Yao W, Chen F, Lu Z. Parameter sensitivity in sensorless induction motor drives with the adaptive full-order observer. IEEE Trans Ind Electron 2015;62(7):4307–18. [15] Abderrahmen Z, Chiheb BR, Hechmi BA, Abdelkader C. DSP-based adaptive backstepping using the tracking errors for high-performance sensorless speed control of induction motor drive. ISA Trans 2016;60:333–47. [16] Liutanakul P, Pierfederici S, Meibody-Tabar F. Application of SMC With I/O feedback linearization to the control of the cascade controlled-rectifier/inverter-motor drive system with small dc-link capacitor. IEEE Trans Power Electron 2008;23(5):2489–99. [17] Wu L, Gao H, Wang C. Quasi sliding mode control of differential linear repetitive processes with unknown input disturbance. IEEE Trans Ind Electron 2011;58(7):3059–68. [18] Wu L, Zheng WX, Gao H. Dissipativity-based sliding mode control of switched stochastic systems. IEEE Trans Autom Contr 2013;58(3):785–91. [19] Di Gennaro S, Rivera Dominguez J, Meza MA. Sensorless high order sliding mode control of induction motors with core loss. IEEE Trans Ind Electron 2014;61(6):2678–89. [20] Barambones O, Alkorta P. Position control of the induction motor using an adaptive sliding-mode controller and observers. IEEE Trans Ind Electron 2014;61(12):6556–65. [21] Chiasson J. A new approach to dynamic feedback linearization control of an induction motor. IEEE Trans Autom Contr 1998;43(3):391–7. [22] Wang W-J, Wang C-C. A new composite adaptive speed controller for induction motor based on feedback linearization. IEEE Trans Energy Convers 1998;13(1):1–6. [23] Boukas TK, Habetler TG. High-performance induction motor speed control using exact feedback linearization with state and state derivative feedback. IEEE Trans Power Electron 2004;19(4):1022–8. [24] Farhani F, Ben Regaya C, Zaafouri A, Chaari A. A quasi linear parameter varying approach to robust control of an Induction machine. In: Proceedings of the 10th international multi-conference syst. signals devices, SSD 2013; 2013. p. 1–6, http://dx.doi.org/10.1109/SSD.2013.6564053. [25] Masiala M, Vafakhah B, Salmon J, Knight AM. Fuzzy self-tuning speed control of an indirect field-oriented control induction motor drive. IEEE Trans Ind Appl 2008;44(6):1732–40. [26] Comanescu M, Xu L, Batzel TD. Decoupled current control of sensorless induction-motor drives by integral sliding mode. IEEE Trans Ind Electron 2008;55(11):3836–45. [27] Ben Regaya C, Zaafouri A, Chaari A. Electric drive control with rotor resistance and rotor speed observers based on fuzzy logic. Math Probl Eng 2014;2014:1–9. [28] Benzineb O, Salhi H, Tadjine M, Salah Boucherit M, Benbouzid M. A PI/backstepping approach for induction motor drives robust control. Int Rev Electr Eng 2010;5(2):426–32. [29] Hasanah RN. A contribution to energy saving in induction motors. Lausanne: École polytechnique fédérale de Lausanne EPFL; 2005 Jun. Report No.:3255 (2005), http://dx.doi.org/10.5075/epfl-thesis-3255. [30] Liu J, Laghrouche S, Wack M. Observer-based higher order sliding mode control of power factor in three-phase AC/DC converter for hybrid electric vehicle applications. Int J Control 2014;87(6):1117–30. [31] Liu J, Laghrouche S, Harmouche M, Wack M. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Eng Pract 2014;30:124–31. [32] Buchholz F-I, Kessel W, Melchert F. Noise power measurements and measurement uncertainties. IEEE Trans Instrum Meas 1992;41(4):476–81. [33] Reed DM, Hofmann HF. Direct field-oriented control of an induction machine using an adaptive rotor resistance estimator. In: Proceedings of IEEE energy conversion congress and exposition (ECCE); 2010, p. 1158–65, http://dx.doi. org/10.1109/ECCE.2010.5617841. [34] Pyzdek T, Keller P. The six sigma handbook. Fourth Edition. New York: McGraw-Hill Professional; 2014.

Please cite this article as: Farhani F, et al. Real time PI-backstepping induction machine drive with efficiency optimization. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.003i