energies Article
Dynamic Simulations of Adaptive Design Approaches to Control the Speed of an Induction Machine Considering Parameter Uncertainties and External Perturbations Kamran Zeb 1,2 ID , Waqar U. Din 1 ID , Muhammad Adil Khan 3 , Ayesha Khan 4 , Umair Younas 5 , Tiago Davi Curi Busarello 6 ID and Hee Je Kim 1, * 1 2 3 4 5 6
*
School of Electrical Engineering, Pusan National University, San 30, ChangJeon 2 Dong, Pusandaehak-ro 63 beon-gil 2, Geumjeong-gu, Busan 46241, Korea;
[email protected] (K.Z.);
[email protected] (W.U.D.) School of Electrical Engineering and Computer Science, National University of Sciences and Technology, Islamabad 44000, Pakistan Department of Electrical and Computer Engineering, Air University Islamabad 44000, Pakistan;
[email protected] Department of Electrical Engineering University of Management and Technology, Sialkot 51040, Pakistan;
[email protected] Department of Electrical Engineering, Selçuk University Konya 42100, Turkey;
[email protected] Department of Engineering, Federal University of Santa Catarina Blumenau, Rua João Pessoa 2750-89036-256, Brazil;
[email protected] Correspondence:
[email protected]; Tel.: +82-10-3462-1990
Received: 3 August 2018; Accepted: 3 September 2018; Published: 5 September 2018
Abstract: Recently, the Indirect Field Oriented Control (IFOC) scheme for Induction Motors (IM) has gained wide acceptance in high performance applications. The IFOC has remarkable characteristics of decoupling torque and flux along with an easy hardware implementation. However, the detuning limits the performance of drives due to uncertainties of parameters. Conventionally, the use of a Proportional Integral Differential (PID) controller has been very frequent in variable speed drive applications. However, it does not allow for the operation of an IM in a wide range of speeds. In order to tackle these problems, optimal, robust, and adaptive control algorithms are mostly in use. The work presented in this paper is based on new optimal, robust, and adaptive control strategies, including an Adaptive Proportional Integral (PI) controller, sliding mode control, Fuzzy Logic (FL) control based on Steepest Descent (SD), Levenberg-Marquardt (LM) algorithms, and Hybrid Control (HC) or adaptive sliding mode controller to overcome the deficiency of conventional control strategies. The main theme is to design a robust control scheme having faster dynamic response, reliable operation for parameter uncertainties and speed variation, and maximized torque and efficiency of the IM. The test bench of the IM control has three main parts: IM model, Inverter Model, and control structure. The IM is modelled in synchronous frame using dq modelling while the Space Vector Pulse Width Modulation (SVPWM) technique is used for modulation of the inverter. Our proposed controllers are critically analyzed and compared with the PI controller considering different conditions: parameter uncertainties, speed variation, load disturbances, and under electrical faults. In addition, the results validate the effectiveness of the designed controllers and are then related to former works. Keywords: induction motor; indirect field oriented control; PI controller; adaptive PI controller; fuzzy logic controller; sliding mode controller; adaptive sliding mode controller; space vector pulse width modulation
Energies 2018, 11, 2339; doi:10.3390/en11092339
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1. Introduction Induction Machines (IM) have a wide use in industries for different processes due to their high performance. Since their discovery, the IM has been considered the predominant actuator in constant speed maneuvers. The advantageous characteristics of IM are: (a) low cost, (b) high efficiency, (c) small inertia, (d) inherent self-starting, (e) absence of collector brooms systems, (f) less maintenance, and (g) simpler design [1]. Conversely, the IM is insufficient to provide flexible speed operation due to its non-linear, complex, and multivariable mathematical model [2]. The aforementioned limitations can be resolved by designing smart and sophisticated Vector Control (VC) and scalar control strategies [3]. Remarkable dynamic response is mandatory for the load disturbances and parameter uncertainties of electric drives to respond to speed and torque control loops. In scalar control, constant torque and flux response is achieved by simple a V/f or Volts/Hertz constant rule with the sluggish dynamic response. In the greater part of advanced, industrial electrical-drive control-applications, the standardized technique to control squirrel cage induction motors is based on the VC principle to accomplish the best dynamic performance [3,4]. In VC or Field-Oriented Control (FOC), like a separately excited DC motor, the flux and torque are individually controlled [5]. The addition of VC to IM lead to IM popularity by allowing variable speed operation and motion control drives [6]. The speed, torque, and position of the IM are conventionally controlled by a fixed gain Proportional Integral Differential (PID) control strategy that is load, speed, and parameter variation disturbance-sensitive. Adaptive control law is one of the solutions to solve these problems [7]. Recently, specialists have centered their effort on mitigating the effect of uncertainties and disturbances using various control approaches. The PID control scheme is widely utilized in industries because of the ease of design, low cost, and simple control structure [8]. However, this scheme is unable to provide perfect control for uncertainty, disturbances, and highly non-linear systems. In addition, these control schemes lack the knowledge of uncertainty and disturbances. However, non-linear control is used to tackle the parameter variation problem and enhance the performance of Indirect Field Oriented Control (IFOC) IM. These control methods include adaptive and robust control [9,10], variable structure control [11], Sliding Mode Control (SMC) [12,13], Fuzzy Logic Control (FLC) [14,15], Neural Network (NN) [16,17], Hybrid Controller (HC) [18], a genetic algorithm based on supervisory control [19], Model Reference Systems (MRAS) control [20], and optimal control [21] strategies. The above-mentioned non-linear control techniques improved control performance of the IM. Though, these linear and non-linear control strategies lack analysis of load disturbances, parameter uncertainties due to inductances and temperature, and speed variation in comparison with various optimal and adaptive control strategies. Moreover, electrical fault perturbation investigation, e.g., double phase, single phase, overvoltage, and undervoltage, has not been incorporated in previous work done on IFOC IM drives [22–25]. This paper is based on the design of the following different optimal control strategies: (a) FLC based on Steepest Descent (SD) and Levenberg-Marquardt (LM) algorithms (b) adaptive Proportional Integral (PI) control (c) SMC, and (d) HC or adaptive SMC for compensation of parameter uncertainties, load disturbances, electrical faults perturbations, and speed variation of IFOC IM. The Adaptive PI control scheme incorporates the promising features of conventional PI and Fuzzy Logic. The Fuzzy Inference System (FIS) is used to update proportional k p and integral k i gains in accordance with sudden load disturbances, parameter uncertainties, and speed variations. The proposed control design provides better and faster dynamic response with a zero Steady State Error (SSE) [22]. The presence of parameter uncertainties, load disturbance, and temperature variation makes the accomplishment of an exact model challenging. To tackle such deficiencies, FLC is implemented in the IM drives, as FLC does not require an exact mathematical model as it adds human logical thinking to the control scheme [14]. The FLC uses a linguistic approach for developing a control algorithm for the Multiple Input Multiple Output (MIMO) system by mapping the output and the input correlation based on human proficiency. The global non-linearities that are mostly problematic to the model are controlled by Fuzzy logics that makes FLC tolerable to parameter uncertainties, robust, and more
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accurate [15]. The valuable features of the proposed SMC are as follows: (a) it is insensitive to external load disturbance and robust to system uncertainties and parameter variation, (b) it offers a stable control system with faster dynamic response, (c) the software/hardware implementation of SMC is easy, and (d) the non-linear systems that are uncontrollable with linear controllers are handled by SMC [12,13]. In our designed HC, the better of the two non-linear control approaches, PI-fuzzy and SMC, are incorporated. PI-fuzzy reduces chattering during steady state operation. The transient state error is minimized by SMC, providing system stability and fast dynamic response [23]. In light of the above specified issues, the main contributions of our paper are:
•
•
• •
•
•
Our proposed control schemes are based on: (a) Fuzzy PI, (b) Fuzzy based on Levenberg Marquardt (LM) and Steepest Descent techniques, (c) Sliding Mode (SM), and (d) HC based on fuzzy PI and sliding mode principles for an IFOC IM drive. Superior Space Vector Pulse Width Modulation (SVPWM) technique-based inverter is designed. The dominant features of the SVPWM are: (a) low switching losses, (b) lower ripples, (c) simple digital implementation, (d) constant switching frequency, and (f) maximum DC-bus voltage utilization. The performance of the detuning effect of the IFOC caused by Rotor Resistance (RR) deviation at 200%, 150%, and 120% of the rated values are analyzed for proposed optimal control schemes. Electrical faults perturbations, e.g., double phasing, single phasing, overvoltage, and undervoltage, are scrutinized along with load disturbances in order to verify robustness and fault tolerant capability of the IFOC IM drive. Comparative analyses of the various proposed optimal control strategies for load disturbances concerning undershoot, overshoot, rise time, settling time, and fast response with traditionally tuned PI controller are also performed. Speed variation is also discussed, described, and analyzed to satisfy the requirement of variable speed drives.
The rest of paper is organized as follows: the modeling of IM in a synchronous frame is described in Section 2, Section 3 presents the IFOC schemes for the IM, and the design of different proposed controlled strategies is presented in Section 4. Section 5 shows the assessment and comparison of different control strategies and finally Section 6 concludes the paper and suggests future directions. 2. Modelling of IM in a Synchronous Reference Frame The mathematical model of a three phase Y-connected squirrel cage IM in direct quadrature (dq)-synchronous rotating reference frame is specified by the subsequent equations [24]. 2.1. Stator Model The stator dq-axis voltages with corresponding flux linkages and currents are described as: vsd = Rs isd − ωd λsq + Lls
d d i + Lm (isd + ird ) dt sd dt
(1)
vsq = Rs isq + ωd λsd + Lls
d d isq + Lm isq + irq dt dt
(2)
where Rs and Rr are stator and rotor resistances, respectively. vsd and vsq are synchronous frame and stator (dq-axis) voltages, respectively. λsd and λsq are synchronous frame and stator (dq-axis) fluxes, respectively. isd and isq synchronous frame and stator (dq-axis) currents, respectively. ωd is synchronous electrical speed and Lm is mutual inductance. 2.2. Rotor Model The rotor dq-axis voltages with corresponding flux linkages and currents are stated below
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Energies 2018, 11, x FOR PEER v REVIEW = rd
Rr ird − ωdA λrq + Llr
d d i + Lm (isd + ird ) dt rd dt
4 of 25(3)
|{z}
𝑑𝑑 𝑑𝑑 𝑖𝑖𝑟𝑟𝑟𝑟 + 𝐿𝐿𝑚𝑚 (𝑖𝑖𝑠𝑠𝑠𝑠 + 𝑖𝑖𝑟𝑟𝑟𝑟 ) 𝑑𝑑𝑑𝑑d 𝑑𝑑𝑑𝑑 d 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 vrq 𝑡𝑡𝑡𝑡 0= Rr irq + ωdA λrd + Llr irq + Lm isq + irq dt dt 𝑑𝑑 𝑑𝑑 |{z} 𝑣𝑣� = 𝑅𝑅𝑟𝑟 𝑖𝑖𝑟𝑟𝑟𝑟 +𝜔𝜔𝑑𝑑𝑑𝑑 𝜆𝜆𝑟𝑟𝑟𝑟 + 𝐿𝐿𝑙𝑙𝑙𝑙 𝑖𝑖𝑟𝑟𝑟𝑟 + 𝐿𝐿𝑚𝑚 �𝑖𝑖𝑠𝑠𝑠𝑠 + 𝑖𝑖𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟 equal to 0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 equal to 0
𝑣𝑣� 𝑟𝑟𝑟𝑟
= 𝑅𝑅𝑟𝑟 𝑖𝑖𝑟𝑟𝑟𝑟 −𝜔𝜔𝑑𝑑𝑑𝑑 𝜆𝜆𝑟𝑟𝑟𝑟 + 𝐿𝐿𝑙𝑙𝑙𝑙
(3)
(4) (4)
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑡𝑡𝑡𝑡 0
In the above equations, all parameters are the same as previously stated, only subscript “s” is In the above equations, all parameters are the same as previously stated, only subscript “s” is replaced with subscript “r” dA isisangular replaced with subscript “r”for forrotor rotorand andω𝜔𝜔 angularslip slipspeed. speed. 𝑑𝑑𝑑𝑑 The stator and rotor model Equations (1)–(4), are combined todescribe describethe thedqdqequivalent equivalent circuits The stator and rotor model Equations (1)–(4), are combined to circuits forfor thethe d-axis and q-axis d-axis and q-axisininFigure Figure1.1.
→
Lls +
isd
+
d λsd dt
Vsd
d λrd dt
Lm
−
− Rs
+
isq
Vsq
−
Rr
ird
d-axis Lls
+
+ d λsq dt
−
d λrq dt
Lm
−
q-axis
V rd
−
ωdA λrd
Llr
+
equat to 0
−
ωd λsd
→
ωdA λrq
Llr
Rr
→
+
ωd λsq
→
Rs
+
irq V rq
equat to 0
−
Figure 1. The dq equivalent circuits of the induction motor (IM). Figure 1. The dq equivalent circuits of the induction motor (IM).
2.3. Rotor Electromagnetic Torque
2.3. Rotor Electromagnetic The instantaneous Torque electromagnetic torque 𝑇𝑇𝑒𝑒𝑒𝑒 acting on the d-axis and the q-axis of the rotor can beThe written by superposition as follows: torque Tem acting on the d-axis and the q-axis of the rotor can instantaneous electromagnetic
be written by superposition as follows: 𝑇𝑇𝑒𝑒𝑒𝑒 = 𝑇𝑇𝑑𝑑,𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑇𝑇𝑞𝑞,𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
(5)
Equation (5), with correspondingTinductance and currents, is formulated as: =T + dq-axis T em
d, rotor
(5)
q, rotor
𝑝𝑝 𝑇𝑇𝑒𝑒𝑒𝑒 = 𝐿𝐿𝑚𝑚 �𝑖𝑖𝑠𝑠𝑠𝑠 𝑖𝑖𝑟𝑟𝑟𝑟 − 𝑖𝑖𝑠𝑠𝑠𝑠 𝑖𝑖𝑟𝑟𝑟𝑟 � 2 Equation (5), with corresponding inductance and dq-axis currents, is formulated as:
(6)
p Lm isq ird − isd irq (6) 2 Subtraction of load torque 𝑇𝑇𝐿𝐿 from electromagnetic torque 𝑇𝑇𝑒𝑒𝑒𝑒 acting on the moment of inertia 2.4.𝐽𝐽 Electrodynamics of IM 𝑒𝑒𝑒𝑒 determines the acceleration as presented in Equation (7a). In addition, Equation (7a) also demonstrates mechanical of the motor [5,24]. Whereas, Equation (7b) describes the actual Subtraction the of load torque Tpart L from electromagnetic torque Tem acting on the moment of inertia Jeq mechanical speed of the rotor in radians second (7a). as: In addition, Equation (7a) also demonstrates determines the acceleration as presented inper Equation 𝑝𝑝 the mechanical part of the motor [5,24]. 𝑇𝑇Whereas, Equation the actual mechanical speed 𝐿𝐿 �𝑖𝑖 𝑖𝑖 (7b) − 𝑖𝑖describes 𝑑𝑑 𝑠𝑠𝑠𝑠 𝑖𝑖𝑟𝑟𝑟𝑟 � − 𝑇𝑇𝐿𝐿 𝑒𝑒𝑒𝑒 − 𝑇𝑇𝐿𝐿 2 𝑚𝑚 𝑠𝑠𝑠𝑠 𝑟𝑟𝑟𝑟 (7a) 𝜔𝜔 = = of the rotor in radians per second as: 𝑑𝑑𝑑𝑑 𝑚𝑚𝑒𝑒𝑒𝑒ℎ 𝐽𝐽𝐽𝐽𝐽𝐽 𝐽𝐽𝑒𝑒𝑒𝑒 p d Tem − TL m isq ird − isd irq − TL 22L𝜔𝜔 (7b)(7a) ⁄ 𝜔𝜔 = 𝑝𝑝 ω = 𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚ℎ dt mech Jeq Jeq 2.4. Electrodynamics of IM
3. Field Oriented Control Schemes
Tem =
ωm = p/2ωmech
(7b)
The VC scheme has the remarkable advantage of decoupling the flux and torque. It makes AC drives equivalent to DC drives with superior dynamic response in high-performance applications. In
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3. Field Oriented Control Schemes The VC scheme has the remarkable advantage of decoupling the flux and torque. It makes AC drives equivalent to DC drives with superior dynamic response in high-performance applications. In the VC strategy, the position of the rotor flux linkages λr is assumed at an angle θ f from the stationary reference frame. Where θ f is the field angle that is used to calculate currents of 5the dq-axis Energies 2018, 11, x FOR PEER REVIEW of 25 in a synchronous reference frame via three phase stator currents through transformation. The stator the yields VC strategy, theflux position of the the electromagnetic rotor flux linkages 𝜆𝜆𝑟𝑟 is Tassumed at an angle 𝜃𝜃𝑓𝑓 from the current the rotor λr and torque in Figure 2. Moreover, e , as presented stationary frame. Where the fieldalong angle that used to calculate currents of the dqin Figure 2 thereference stator current vector 𝜃𝜃is resolved λr , ishorizontally generates field producing 𝑓𝑓 is axis in ai synchronous reference frametorque via three phase stator currents through transformation. The component , and vertically provides controlling component i . Controlling only the field T f stator current yields the rotor flux 𝜆𝜆 and the electromagnetic torque 𝑇𝑇 , as presented in Figure 2. 𝑟𝑟 𝑒𝑒 current controls the rotor flux linkages just like a separately-excited DC machine. In the DC motor, Moreover, in Figure 2 the stator current vector is resolved along 𝜆𝜆𝑟𝑟 , horizontally generates field the field current controls the field flux with no impact of armature current. The flux is controlled by producing component 𝑖𝑖𝑓𝑓 , and vertically provides torque controlling component 𝑖𝑖 𝑇𝑇 . Controlling the d component of the stator current based on the equation: only the field current controls the rotor flux linkages just like a separately-excited DC machine. In the DC motor, the field current controls the field flux with no impact of armature current. The flux is = f λ (8a) r , i f on the equation: controlled by the d component of the statorisdcurrent based
� (8a) 𝑠𝑠𝑠𝑠 = 𝑓𝑓�𝜆𝜆𝑟𝑟i , 𝑖𝑖𝑓𝑓at Similarly, controlling torque generating𝑖𝑖current constant rotor flux linkages, independently T Similarly, controlling currenttorque 𝑖𝑖 𝑇𝑇 atis controlled constant rotor flux linkages, controls electromagnetic torque,torque just likegenerating electromagnetic by armature current in a independently controls electromagnetic torque, separately excited DC machine [5], described as: just like electromagnetic torque is controlled by armature current in a separately excited DC machine [5], described as:
T𝑇𝑇e𝑒𝑒 ∝∝ λ𝜆𝜆r𝑟𝑟i𝑖𝑖T𝑇𝑇 ∝ ∝ 𝑖𝑖i𝑓𝑓f𝑖𝑖i𝑇𝑇T
(8b)
(8b)
is
vs φ θT
vsds
vsqe vsqs
vsde
θf
θr
θ sl
isqe = iT
isqs
λesq = λr
isde = i f
isds
Synchronous Reference Frame (q-d, e)
Rotor Reference Frame (q-d, r)
Stator Reference Frame (q-d, s)
Figure 2. Indirect Field Oriented Controller Phasor Diagram [24,25].
Figure 2. Indirect Field Oriented Controller Phasor Diagram [24,25].
The instantaneous rotor flux locus, 𝜃𝜃𝑓𝑓 , is defined as:
The instantaneous rotor flux locus, θ f , is defined as: 𝜃𝜃𝑓𝑓 = 𝜃𝜃𝑟𝑟 + 𝜃𝜃𝑠𝑠𝑠𝑠
where 𝜃𝜃𝑟𝑟 is the rotor is angle and 𝜃𝜃𝑠𝑠𝑠𝑠 is theθ fslip = angle. θr + θslThe 𝜃𝜃𝑓𝑓 is formulated as:
= �(𝜃𝜃 + 𝜃𝜃𝑠𝑠𝑠𝑠 )𝑑𝑑𝑑𝑑The = �θ𝜔𝜔𝑠𝑠is 𝑑𝑑𝑑𝑑formulated as: where θr is the rotor is angle and θsl is𝜃𝜃𝑓𝑓the slip𝑟𝑟 angle. f
(8c)
(8c) (8d)
Z Z 3.1. Field Oriented Control Scheme: AθTaxonomy + θsl )dt = ωs dt (θr Overview f =
Field oriented control scheme on the basis of field angle acquisition is classified into two subcategories. If the field angle 𝜃𝜃𝑓𝑓 is determined by using Hall sensors or terminal currents and voltages or flux-sensing windings, then it is recognized as a Direct Field Oriented Control (DFOC) scheme. When the field angle is calculated by utilizing the position of the rotor and partial estimation
(8d)
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3.1. Field Oriented Control Scheme: A Taxonomy Overview Field oriented control scheme on the basis of field angle acquisition is classified into two subcategories. If the field angle θ f is determined by using Hall sensors or terminal currents and Energies 2018, 11, x FOR PEER REVIEW 6 of 25 voltages or flux-sensing windings, then it is recognized as a Direct Field Oriented Control (DFOC) scheme. theparameters, field angle iswithout calculated by utilizing thevariables, position of thevoltages rotor and partial estimation with only When machine using additional e.g., and currents, it is with only machine parameters, without using additional variables, e.g., voltages and currents, it is called an IFOC scheme [24,25]. Figure 3 depicts the classification of Field Oriented Control Schemes called for an IFOC [24,25]. this Figure 3 depicts the classification Field Oriented Control Schemes (FOCS) IM [9].scheme Furthermore, research is based on estimatedof field angle. (FOCS) for IM [9]. Furthermore, this research is based on estimated field angle. Field Oriented Control Strategy
Direct Approach (Calculate θ f Directly)
Indirect Approach
Field Angle calculation from Rotor Angle and Estimated Slip Angle
Te*
λ
* r
Measured (with Machine Parameters)
θ sl
Phase Currents
++
θf
Line Voltages
Calculate Field Angle
Measured (with Machine Parameters)
λdr λqr
Magnitude and Angle Resolver
θf
λr
θr Figure 3. 3. Classification ofof the Field Oriented Control Figure Classification the Field Oriented ControlScheme Scheme[24,25]. [24,25].
3.2. 3.2.Implementation ImplementationofofIndirect IndirectField FieldOriented OrientedControl ControlScheme Scheme The Theindirect indirectFOCS FOCSisisadvantageous advantageousover overdirect directFOCS FOCSdue duetotoitsitsremarkable remarkablefeatures featureslike like(a)(a) reluctance Hall effect andand fluxflux sensors (b) reduced number of transducer and feedback loops, loops, and reluctancetoto Hall effect sensors (b) reduced number of transducer and feedback (c)and tranquil operation at zero speed and near zero speed [5]. The indirect FOCS is derived from (c) tranquil operation at zero speed and near zero speed [5]. The indirect FOCS is derived froma a dynamic simplicity, dynamicequation equationofofthe theIM IMininthe thesynchronous synchronousrevolving revolvingreference referenceframe. frame.For For simplicity,a acurrent current source-inverter is supposed that serves the stator phase current as an input and is neglected for source-inverter is supposed that serves the stator phase current as an input and is neglected for further further consideration. The IFOC scheme is derived in detail in [24]. In an indirect FOCS, the flux consideration. The IFOC scheme is derived in detail in [24]. In an indirect FOCS, the flux linkage 𝑑𝑑 linkage of winding rotor winding is aligned ond-axis, the d-axis, which gives 𝜆𝜆 λ,rq 𝜆𝜆= = 0, =[24] 0 [24] 𝑟𝑟𝑟𝑟 0, 𝑟𝑟𝑟𝑟 0 of rotor is aligned on the which gives λr =𝜆𝜆𝑟𝑟λ= andandd λrq𝜆𝜆= and rd , 𝑟𝑟𝑟𝑟 and modifies equation below: modifies thethe slipslip equation below:
dt 𝑑𝑑𝑑𝑑
Lmisq 𝐿𝐿 𝑚𝑚𝑖𝑖 ω = (9) 𝑠𝑠𝑠𝑠 𝜔𝜔𝑠𝑠𝑠𝑠sl = � Tr λ�r (9) 𝑇𝑇𝑟𝑟 𝜆𝜆𝑟𝑟 Equation (9) calculates slip speed. Where Tr =𝐿𝐿 Lrr is the rotor’s time constant. The rotor flux Equation (9) calculates slip speed. Where 𝑇𝑇𝑟𝑟 = 𝑟𝑟 Ris the rotor’s time constant. The rotor flux 𝑅𝑅𝑟𝑟 equation related to isd is presented as: equation related to 𝑖𝑖𝑠𝑠𝑠𝑠 is presented as: Lm λr (s) = 𝐿𝐿𝑚𝑚 i (s) (10) (𝑠𝑠) 𝜆𝜆𝑟𝑟 = (1 + sTr𝑖𝑖)𝑠𝑠𝑠𝑠sd(𝑠𝑠) (10) (1 + 𝑠𝑠𝑇𝑇𝑟𝑟 )
Equation (10), provides reliance isd and andisisknown knownas asthe thedynamics dynamicsequation equationofofthe the r on𝑖𝑖𝑠𝑠𝑠𝑠 Equation (10), provides thethe reliance of of 𝜆𝜆𝑟𝑟λon d-axisrotor rotor flux linkage. The generated rotor d-axis flux, which is acting on rotor the rotor d-axis flux linkage. The 𝑇𝑇𝑒𝑒𝑒𝑒Temis is generated byby thethe rotor d-axis flux, which is acting on the qq-axis winding, while the q-axis rotor flux is zero, established as: axis winding, while the q-axis rotor flux is zero, established as: 𝑝𝑝p 𝐿𝐿𝑚𝑚Lm 𝑇𝑇T𝑒𝑒𝑒𝑒 = 𝜆𝜆 � 𝑖𝑖𝑠𝑠𝑠𝑠isq� (11)(11) = λ 𝑟𝑟 em 2 r 2 𝐿𝐿𝑟𝑟Lr
Figure 4 depicts, the model of IM for field angle 𝜃𝜃𝑓𝑓 calculation, basically, this is the implementation of Equations (9)–(11). The flux linkage of the rotor 𝜆𝜆𝑟𝑟 is aligned with the d-axis. The rotor flux 𝜆𝜆𝑟𝑟 , field angle 𝜃𝜃𝑓𝑓 and electromagnetic torque 𝑇𝑇𝑒𝑒𝑒𝑒 can be obtained from 𝑖𝑖𝑠𝑠𝑠𝑠 , 𝑖𝑖𝑠𝑠𝑠𝑠 , and 𝜔𝜔𝑚𝑚𝑚𝑚𝑚𝑚ℎ as inputs.
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Figure 4 depicts, the model of IM for field angle θ f calculation, basically, this is the implementation of Equations (9)–(11). The flux linkage of the rotor λr is aligned with the d-axis. The rotor flux λr , field angle θ and electromagnetic torque Tem can be obtained from isd , isq , and ωmech as inputs. Energies 2018,f 11, x FOR PEER REVIEW 7 of 25 Energies 2018, 11, x FOR PEER REVIEW
i sd i sd
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λr λr
Lm 1 +LsT m r 1 + sT r
isq isq
Tr TDr N
Lm Lm
N
N/D
ωsl ωd Σ + ωsl ωd +
× λr×
ωmech ωmech
D N/D
λr λr
p Lm . Lrm 2p . L 2 Lr
λr
+Σ +
θf θf
1 s1 s
ωm ωm
Tem Tem
p 2p 2
Figure 4. The estimated model for 𝜃𝜃 calculation. Figure 4. The estimated model for 𝑓𝑓θ f calculation. Figure 4. The estimated model for 𝜃𝜃𝑓𝑓 calculation.
3.3. Proposed System Model of Indirect Field Oriented Control 3.3. Proposed System Model of Indirect Field Oriented Control 3.3. Proposed of Indirect FieldofOriented Control Figure 5 System depictsModel the block diagram the indirect VC scheme (VCS) or IFOC IM based on the Figure 5 depicts the block diagram of the indirect VC scheme (VCS) or IFOC IM based on the proposed controllers. Theblock machine driveofapplies indirect to(VCS) IM, while the IM switches Figure 5 depicts the diagram the indirect VC FOCS scheme or IFOC based of on the the proposed controllers. The machine drive applies indirect FOCS to IM, while the switches of the inverter inverter are operated by the SVPWM technique. The calculated values of 𝜆𝜆 , 𝑖𝑖 , and 𝑖𝑖 and the proposed controllers. The machine drive applies indirect FOCS to IM, while the switches of the 𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 aredesignated operated theof SVPWM The calculated values of(𝑣𝑣 λ∗ r , and isd i𝑖𝑖sq ,and the value 𝜃𝜃by used to generate reference voltages dq-axis currents inverter areby operated thetechnique. SVPWM technique. The calculated values of,𝑣𝑣and 𝜆𝜆∗ 𝑟𝑟), from 𝑖𝑖𝑠𝑠𝑠𝑠designated and the 𝑓𝑓 are 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 and ∗ ∗ ) for ∗ of θ ∗are used to generate reference voltages (v∗∗ ∗∗ value v∗sd ) 𝑣𝑣from dq-axis (ivoltages isq ∗ currents (𝑖𝑖designated for SVPWM. The reference values of 𝑣𝑣𝑎𝑎sq , and 𝑣𝑣𝑏𝑏∗ , and the𝑣𝑣three phase are sd and of 𝜃𝜃𝑓𝑓 are used to generate reference voltages (𝑣𝑣 and dq-axis currents 𝑐𝑐 𝑠𝑠𝑠𝑠for 𝑠𝑠𝑠𝑠 and f 𝑖𝑖𝑠𝑠𝑠𝑠 )value 𝑠𝑠𝑠𝑠 ) from ∗ for ∗ ∗ threeon SVPWM. values v∗a , v∗bvalues , and vof the phase voltages are measured using ∗using ∗ SVPWM c 𝑣𝑣𝑎𝑎∗ , is measured 𝜃𝜃𝑓𝑓 . The power electronics converter switching technique to (𝑖𝑖𝑠𝑠𝑠𝑠 andThe 𝑖𝑖𝑠𝑠𝑠𝑠 )reference for SVPWM. Theof reference 𝑣𝑣based , and 𝑣𝑣 for the three phase voltages are 𝑐𝑐 𝑏𝑏 θ f .deliver The power electronics converter is based on SVPWM switching technique to deliver real stator realusing stator voltages 𝑣𝑣𝑎𝑎 , 𝑣𝑣𝑏𝑏electronics , and 𝑣𝑣𝑐𝑐 [25]. measured 𝜃𝜃𝑓𝑓 . The power converter is based on SVPWM switching technique to voltages vc [25]. 𝑣𝑣𝑎𝑎 , 𝑣𝑣𝑏𝑏 , and 𝑣𝑣𝑐𝑐 [25]. a , vstator b , andvoltages delivervreal Flux Loop
Speed Loop Flux Loop
θ θf
Torque Speed Loop Loop Torque Loop
* sd * sd
i i
ωmech ωmech
Σ Σωmech
+ +
isd iisqsd isq
IVC Model for IVC Model f for
-
Design Controllers Design Controllers
ωmech
(measured) (measured)
isd isd* isd isd* + Σ
Σ
* + sq * sq+
i i
Σ isq Σ isq -
PI PI
PI
* vsd * vsd vsq** vsq
θf θf abc abc
dq dq
abc
ia*
*
abc
iib*a ii
dq
b
dq
va* vvb*a*
vvc** b vc*
** c
ic*
Motor
SV PWM SV PWM
Motor
PI
+
-
ωmech ωmech (measured) (measured)
d θ mech dtd θ mech Encoder dt
Figure 5. Proposed model of the Indirect Field Oriented Control (IFOC). Figure5.5.Proposed Proposedmodel modelof of the the Indirect Indirect Field Figure Field Oriented OrientedControl Control(IFOC). (IFOC).
Encoder
The SVPWM is designed from [24,25]. In the case of Sinusoidal Pulse Width Modulation (PWM) (𝑟𝑟𝑟𝑟𝑟𝑟) 𝑉𝑉𝐿𝐿𝐿𝐿,𝑚𝑚𝑚𝑚𝑚𝑚 𝑉𝑉𝐿𝐿𝐿𝐿,𝑚𝑚𝑚𝑚𝑚𝑚 utilizes 15% higher DC= 0.612𝑉𝑉 = of 0.707𝑉𝑉 The SVPWM is designed from [24,25]. In (𝑟𝑟𝑟𝑟𝑟𝑟) the case Sinusoidal Pulse Width Modulation (PWM) 𝑑𝑑 and SVPWM 𝑑𝑑 . The SVPWM ∗ The SVPWM isreference designed from [24,25]. In the case of Sinusoidal Pulse Width Modulation (PWM) bus voltage. The current 𝑖𝑖 is used to control the IM torque. The control structure is (𝑟𝑟𝑟𝑟𝑟𝑟) (𝑟𝑟𝑟𝑟𝑟𝑟) 𝑉𝑉𝐿𝐿𝐿𝐿,𝑚𝑚𝑚𝑚𝑚𝑚 and SVPWM 𝑉𝑉 . The SVPWM utilizes 15% higher DC= 0.612𝑉𝑉 = 0.707𝑉𝑉 𝑠𝑠𝑠𝑠 𝑑𝑑 𝐿𝐿𝐿𝐿,𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑 ∗inner VLL,max 0.612V SVPWM Vis 0.707V . the The SVPWM utilizes 15% higher (rms)of=two (rms ) = to composed cascaded The𝑖𝑖𝑠𝑠𝑠𝑠 loop is control torque while the outer loop is LL,max d andloops. bus voltage. The reference current used toused control the dIM torque. The control structure is ∗ is used to control the IM torque. The control structure is DC-bus voltage. The reference current i used to control the speed of the IM. The bandwidth of the inner loop is larger than the outer loop. composed of two cascaded loops. The inner loop is used to control the torque while the outer loop is sq The adaptive andof robust controllers are usedoftothe minimize theiserror between theouter reference usedadvance to control the speed the IM. The bandwidth inner loop larger than the loop. ∗ value and measured value provide a reference current 𝑖𝑖𝑠𝑠𝑠𝑠 to thethe internal The the PI controller The advance adaptive andand robust controllers are used to minimize error loop. between reference ∗ value and measured value and provide a reference current 𝑖𝑖𝑠𝑠𝑠𝑠 to the internal loop. The PI controller
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composed of two cascaded loops. The inner loop is used to control the torque while the outer loop is used to control the speed of the IM. The bandwidth of the inner loop is larger than the outer loop. The advance adaptive and robust controllers are used to minimize the error between the reference ∗ to the internal loop. The PI controller is value and measured value and provide a reference current isq used to minimize the error of the q-axis current isq . Similarly, the error of the d-axis current isd is also controlled by using a conventional PI controller [24]. 4. Optimal Speed Controllers: A Design Overview This section briefly describes the design and overview of different proposed controllers for indirect FOCS IM. 4.1. PI Control Scheme Design The baseline PI controller is comprehensively designed and tuned in [24]. The PI controller constants for the speed loop are calculated on the basis that the phase margin is 60◦ and the crossover open loop frequency is 25 rad/s. However, the torque loop has the same phase margin with 10 times the bandwidth of the speed loop. 4.2. Adaptive PI Control Scheme Design The PI controller has constant proportional and integral gains: k p and k i , respectively. The performance of the PI control scheme is enhanced by updating the gains, according to error e(t). The adaptation is achieved by employing Fuzzy Rules (FR). FR are simple If-THEN rules (or algorithms) with a condition and conclusion. FR are constructed to control the output variables as follows:
• • •
If the absolute error |e(t)| is zero, then k p is large and k i is zero. If the absolute error |e(t)| is small, then k p is large and k i is small. If the absolute error |e(t)| is large, then k p is large and k i is large.
In fuzzification and de-fuzzification the membership functions are required to convert crisp input into fuzzified output and vice versa. A Gaussian Membership Function (GMF) is employed in the fuzzy rules that utilizes two update parameters, i.e., standard deviation or variance σi and center ci as: 1 µ( x ) = exp − 2
xi − ci σi
2 ! (12)
Mathematically, a PI controller is described as: ∗ isq ( PI ) = k p e(t) + k i
Z
e(t)dt
(13)
∗ , and k and k are proportional and where the controller input is e(t), output of the controller is isq p i integral gains, respectively. Figure 6 demonstrates the adaptive fuzzy PI control scheme for the IFOC IM drive system. The error is minimized by updating PI gains using fuzzy rules. The update equation for the designed controller is presented in Equation (14). ∗ isq ( Fuzzy) = F1 K1 e(t) + F2 K2
Z
e(t)dt
(14)
where F1 and F2 are the output of the fuzzy controller for k p and k i and K1 and K2 are learning rate constants for k p and k i , respectively.
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Adaptive PI controller
F 1
× K 1 i
Error
+ +
U Absolute Fuzzy Logic Controller
F 2
× K 2
* Output
sq
1 s Integrator
Induction Motor
Figure6.6.Adaptive AdaptiveProportional ProportionalIntegral Integral(PI) (PI)controller controller[25]. [25]. Figure
4.3. Proposed FLC based on LM 4.3. Proposed FLC based on LM This segment describes the design of proposed FLC based on an LM algorithm. In the literature, This segment describes the design of proposed FLC based on an LM algorithm. In the literature, different control schemes are implemented for IFOC IM drives, each approach has its own favorable different control schemes are implemented for IFOC IM drives, each approach has its own favorable characteristics [26]. FLC based on LM is the most appropriate for the non-linear, inadequately characteristics [26]. FLC based on LM is the most appropriate for the non-linear, inadequately understood, and complex dynamic plants. FLC based on LM does not require an exact mathematical understood, and complex dynamic plants. FLC based on LM does not require an exact mathematical model and is known as a direct adaptive control procedure. IM can be optimally controlled by FLC model and is known as a direct adaptive control procedure. IM can be optimally controlled by FLC based on the LM technique [27]. The knowledge base modifier handles the uncertainty of a non-linear based on the LM technique [27]. The knowledge base modifier handles the uncertainty of a non-linear model, while the updating of the singleton Membership Function (MF) makes the controller adaptive. model, while the updating of the singleton Membership Function (MF) makes the controller adaptive. The selection of LM technique is based on the following discussion. Because of smaller step sizes, The selection of LM technique is based on the following discussion. Because of smaller step FLC based on SD is always convergent. The SD algorithm is slow but convergent [27,28]. The sizes, FLC based on SD is always convergent. The SD algorithm is slow but convergent [27,28]. probability of divergence increases in the Gauss-Newton (GN) procedure, although it minimizes the The probability of divergence increases in the Gauss-Newton (GN) procedure, although it minimizes error function faster than does the SD technique. At any time, if the Jacobian matrix inverse inclines the error function faster than does the SD technique. At any time, if the Jacobian matrix inverse inclines to infinity, the GN procedure fails. The LM technique is a hybrid method which is stable with the to infinity, the GN procedure fails. The LM technique is a hybrid method which is stable with the Jacobian matrix, uses the fastness of GN, and the convergence property of SD [27]. LM technique Jacobian matrix, uses the fastness of GN, and the convergence property of SD [27]. LM technique optimally solves least square minimization problems. The cost function of a control scheme is optimally solves least square minimization problems. The cost function of a control scheme is optimized optimized by the LM algorithm as [25,28]: by the LM algorithm as [25,28]: 1 1 ∑m𝑚𝑚 𝑟𝑟22 (𝑥𝑥) (15) f ( x𝑓𝑓(𝑥𝑥) (15) ) ==∑ 𝑗𝑗=1r 𝑖𝑖 ( x ) 2 2 j =1 i 𝑅𝑅𝑛𝑛 to 𝑅𝑅 . It is In Equation (15), 𝑥𝑥 = [𝑥𝑥 𝑥𝑥 𝑥𝑥 ⋯ 𝑥𝑥 ]𝑇𝑇 ∈ 𝑅𝑅𝑛𝑛×1 and 𝑟𝑟𝑖𝑖 is the function from In Equation (15), x = [ x1 x21 x32 · 3· · xn ] T𝑛𝑛 ∈ Rn×1 and ri is the function from Rn to R. It is assumed assumed that 𝑚𝑚 ≥ 𝑛𝑛 for the 𝑟𝑟 , where 𝑟𝑟 is called the residuals. The different parameters of the LM that m ≥ n for the ri , where ri is𝑖𝑖 called the𝑖𝑖 residuals. The different parameters of the LM algorithm is algorithm is updated by the following equation. updated by the following equation. (16) 𝑤𝑤𝑘𝑘+1 = 𝑤𝑤𝑘𝑘 −𝜆𝜆(𝐽𝐽𝑘𝑘𝑇𝑇 𝐽𝐽𝑘𝑘 + 𝜇𝜇𝜇𝜇)−1 𝐽𝐽𝑘𝑘 𝑒𝑒𝑘𝑘 −1 T λ Jacobean Jk Jk + µI matrix, Jk ek 𝑤𝑤 is the previous value, 𝜇𝜇 is(16) k +1 =𝐽𝐽wkis−the 𝑤𝑤 presents the updated w value, the 𝑘𝑘+1
𝑘𝑘
𝑘𝑘
combination coefficient, 𝐼𝐼 is the identity matrix, and 𝑒𝑒𝑘𝑘 is the error. In order to guarantee that the wk+1 presents the updated value, Jk is the Jacobean matrix, wk is the previous value, µ is the estimated Hessian Matrix (HM) is convertible, the LM technique familiarizes another approximation combination coefficient, I is the identity matrix, and ek is the error. In order to guarantee that the to HM as: estimated Hessian Matrix (HM) is convertible, the LM technique familiarizes another approximation 𝐻𝐻 ≅ (𝐽𝐽𝑘𝑘𝑇𝑇 𝐽𝐽𝑘𝑘 + 𝜇𝜇𝜇𝜇 )−1 to HM as: −1 H∼ = toJkT Jmake k + µI the algorithm stable and convergent [26]. The parameters 𝜇𝜇 and 𝜆𝜆 are used Furthermore, FLC integrates four individual parts: fuzzifier that converts crisp input into fuzzy The parameters µ and λ are used to make the algorithm stable and convergent [26]. Furthermore, output using a membership function, simple rule base of IF-THEN rules or algorithms that are fired FLC integrates four individual parts: fuzzifier that converts crisp input into fuzzy output using a to control output, inference engine that evaluates the rules and dictates which rules should be fired, membership function, simple rule base of IF-THEN rules or algorithms that are fired to control output, and de-fuzzifier that converts the inference conclusion into crisp or real output as presented in Figure 7. inference engine that evaluates the rules and dictates which rules should be fired, and de-fuzzifier that converts the inference conclusion into crisp or real output as presented in Figure 7.
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isd
IVC Model for
isd
Speed Loop Torque Loop
i
ωmech
-
(measured)
Fuzzy conclusions
x2
. . . xn
Inference Mechanism
Rule-base
-
Σ
+ Crisp outputs
y1
i
y2 Defuzzification
+
Σ
Crisp inputs
* sd
x1 Fuzzification
ωmech
Fuzzified inputs
* isd
. . .
* sq
Σ
+
PI
v
ic*
dq
θf
* vsd
PI
ia*
ib*
isq
θf
Flux Loop
abc
* sq
abc
dq
va* vb* vc*
Motor
SV PWM
-
isq
ym
FLC based On Levenberg Marquardt
ωmech (measured)
d θ mech dt
Encoder
Figure 7. Fuzzy based on on the the Levenberg-Marquardt Levenberg-Marquardt (LM) (LM) algorithm. algorithm. Figure 7. Fuzzy Logic Logic Control Control (FLC) (FLC) based
4.3.1. Controller Controller Output Output Equation Equation 4.3.1. The design design develops develops from from the the Cost Cost equation equation as: as: The 1 2 (17) 𝑒𝑒𝑚𝑚 = 1 h�𝑓𝑓𝑚𝑚 − 𝑦𝑦𝑟𝑟𝑟𝑟𝑟𝑟i�2 em = 2 f m − yre f (17) 2 The center of gravity approach is utilized to accomplish the de-fuzzification of output MF as The center gravity approach is utilized to accomplish the de-fuzzification of output MF as described in the of accompanying Equation. described in the accompanying Equation. �∑𝑅𝑅𝑖𝑖=1 𝑏𝑏𝑖𝑖 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗𝑚𝑚 , 𝑘𝑘�� 𝑓𝑓𝑚𝑚 = 𝑅𝑅 (18) R �∑ 𝜇𝜇 �𝑥𝑥 𝑚𝑚m, 𝑘𝑘�� ∑ i =𝑖𝑖=1 1 bi µ𝑖𝑖 i 𝑗𝑗x j , k (18) fm = m Equation (18), the FLC based on LM output, input MFs are updated according to output ∑iR=1 µand i xj , k MFs, as presented in Equation (18). Equation (18), the on LM output, andFunction input MFs(GMF) are updated according to output The updating of FLC the based Gaussian Membership is faster and works wellMFs, for as presentedfunctions, in Equation (18). continuous additionally, it update two parameters i.e., Center 𝑐𝑐𝑖𝑖 and Variance 𝜎𝜎1 . The updating of the Gaussian Membership Function (GMF) is faster and works well for continuous functions, additionally, it update two parameters i.e., Center ci and Variance σ1 . 4.3.2. Jacobian Calculation of cost Equation (17), w.r.t output MF 𝑏𝑏𝑗𝑗 is: 4.3.2.Derivative Jacobian Calculation
𝜕𝜕𝑒𝑒𝑚𝑚 𝜕𝜕𝜕𝜕 Derivative of cost Equation (17), w.r.t output j is: = �𝑓𝑓𝑚𝑚MF − 𝑦𝑦b𝑟𝑟𝑟𝑟𝑟𝑟 � (19) 𝜕𝜕𝑏𝑏𝑖𝑖 𝜕𝜕𝑏𝑏𝑖𝑖 ∂ε ∂em By substituting the estimations of GMF =we fobtain: (19) m − yre f ∂bi ∂bi 𝑅𝑅 𝑚𝑚 𝜕𝜕𝑒𝑒𝑚𝑚 𝜕𝜕 �∑𝑖𝑖=1 𝑏𝑏𝑖𝑖 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗 , 𝑘𝑘�� = 𝜀𝜀 we�obtain: � (20) By substituting the estimations𝜕𝜕𝑏𝑏 of GMF 𝜕𝜕𝑏𝑏𝑖𝑖 �∑𝑅𝑅𝑖𝑖=1 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗𝑚𝑚 , 𝑘𝑘�� 𝑖𝑖 R m , k𝑚𝑚 𝑖𝑖 2 i =1 bi µi 1x j 𝑥𝑥 ∂em ⎡ ∂𝑅𝑅 ∑ 𝑗𝑗 − 𝑐𝑐𝑗𝑗 ⎤ ∏ 2 � 𝑖𝑖 � �� =⎢�∑ ε 𝑖𝑖=1 𝑏𝑏𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒 �− (20) ⎥ 𝜎𝜎 R ∂bi m , k 𝑗𝑗 𝜕𝜕𝑒𝑒𝑚𝑚 ∂bi 𝜕𝜕 µ x ∑ i i = 1 ⎢ ⎥ j = 𝜀𝜀 (21) 𝑚𝑚 𝑖𝑖 2 𝜕𝜕𝑏𝑏𝑖𝑖 𝜕𝜕𝑏𝑏𝑖𝑖 ⎢ 1 𝑥𝑥𝑗𝑗 − 𝑐𝑐𝑗𝑗 !! ⎥ 𝑅𝑅 ⎢ �∑𝑖𝑖=1 ∏ 𝑒𝑒𝑒𝑒𝑒𝑒 �− � 𝑖𝑖 �2 �� ⎥ 2 xm𝜎𝜎𝑗𝑗−ci ⎣ R ⎦ j j 1 ∑i=1 bi ∏ exp − 2 σji ∂e ∂ m The Jacobian of each span, MF, variance, and center, is obtained by derivating(21) the = i.e., ε output m i 2 !! ∂b ∂b x − c i i cost equation. The Jacobian of output GMF 𝑏𝑏R𝑖𝑖 is given in Equations (19)–(21). Similarly, by derivating j j ∑i=1 ∏ exp − 12 σji the cost equation w. r. t. 𝑐𝑐𝑖𝑖 and 𝜎𝜎𝑖𝑖 , we individually demonstrate the Jacobian of the center and variance of the GMF.
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The Jacobian of each span, i.e., output MF, variance, and center, is obtained by derivating the cost equation. The Jacobian of output GMF bi is given in Equations (19)–(21). Similarly, by derivating the cost equation w. r. t. ci and σi , we individually demonstrate the Jacobian of the center and variance of the GMF. 4.3.3. Update Equation for Output Membership Function The output of the controller or the output MF is updated by Equation (22). The output MF is presented by variable bi . The output of the controller is updated regularly according to the output to minimize the error. bi ( k ) = bi ( k − 1 ) − λ ε
−1 T m, k µi x m , k µi x m µ x i j j ,k j ε ε + µI ε ∑iR=1 µi x m ∑iR=1 µi x m ∑iR=1 µi x m j ,k j ,k j ,k
(22)
4.3.4. Update Equation for Variance Equation (23) demonstrates the adaptation of variance σi or the spread of the MF. There is an inverse relation between variance and the magnitude of MF. The notch will be greater if the variance value is lower, and vice versa. h i −1 σi (k) = σi (k − 1) − λ AA T + µI Aε " " where A = ε
(∑iR=1 bi )− f m
∑iR=1 µi x m j ,k
#
(23)
# 2 ! i xm j −c j m µi x j , k . 3 σji
4.3.5. Update Equation for Center Based on the crisp input to the controller, the GMF center acquires different values, which are given by Equation (24). h i −1 ci (k) = ci (k − 1) − λ BB T + µI Bε (24) # " " # ! i xm (∑iR=1 bi )− f m j −c j m where B = ε R µi x j , k . 2 m ∑i=1 µi x j ,k
σji
4.4. Proposed Fuzzy Logic Controller Based on SD The Steepest Descent (SD) algorithm is also known as the Error Back-Propagation (EBP) or first order-order algorithm as it employs a first derivative to reduce error. The updated equation of FLC based on SD is given as: wk+1 = wk − αgk (25a) where gk is the gradient while α is the step size or learning rate constant. According to the EBP update rule, as illustrated in Figure 8, the equation that updates the center, variance, and output membership function are similar to the FLC based on LM with little modification. The asymptotic convergence principle is followed by the FLC based on SD technique for a training process. Around the solution, all the elements of the gradient vector will be very trivial. In addition, in comparison to the LM and GN techniques, the weight change is small. 4.4.1. Update Equation for Output Membership Function The derivative of the learning error function equation with respect to output MF b j results in the calculation of a gradient through the following subsequent equations:
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∂ε ∂em = f m − yre f ∂bi ∂bi
(25b)
By placing the estimations of GMF we get: R m, k b µ x ∑ i i i = 1 j ∂em ∂ = ε R ∂bi ∂bi m ∑ i =1 µ i x j , k Energies 2018, 11, x FOR PEER REVIEW
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(25c) 12 of 25
𝑚𝑚 𝑐𝑐𝑗𝑗𝑖𝑖ci 2 ⎤ ⎡�∑𝑅𝑅R 𝑏𝑏 ∏ 𝑒𝑒𝑒𝑒𝑒𝑒 �− 1 �1𝑥𝑥𝑗𝑗 x−mj − �j �� 𝑖𝑖b ∏ exp 2 𝑖𝑖=1 𝑖𝑖 − ∑ ⎢ ⎥ i =1 i 2 𝜎𝜎𝑗𝑗 σi 𝜕𝜕𝑒𝑒𝑚𝑚 𝜕𝜕 j ∂em = 𝜀𝜀 ∂ ⎢ ⎥ 2 !! = ε 𝑚𝑚 𝑖𝑖 𝜕𝜕𝑏𝑏 𝜕𝜕𝑏𝑏𝑖𝑖 ⎢ 𝑥𝑥 −m − 𝑐𝑐𝑗𝑗ci 2 ⎥ 1 𝑗𝑗 ∂bi 𝑖𝑖 ∂bi 𝑅𝑅 x ⎢ �∑𝑖𝑖=1 �j �� ⎥ j𝑖𝑖 R ∏ 𝑒𝑒𝑒𝑒𝑒𝑒 �− �1 ⎣ ∑i=1 ∏ exp 2− 2 𝜎𝜎𝑗𝑗 i ⎦
!!
(25d) (25d)
σj
The finalized law for updated output MF is given as:
The finalized law for updated output MF is given as:
𝑖𝑖 2
𝑚𝑚
⎤ ⎡�∑𝑅𝑅 𝑏𝑏 ∏ 𝑒𝑒𝑒𝑒𝑒𝑒 �− 1 �𝑥𝑥𝑗𝑗 − 𝑐𝑐𝑗𝑗 � �2�!! i 𝑖𝑖=1 𝑖𝑖 𝑖𝑖m ⎥ 𝜕𝜕 ⎢ ∑ R b ∏ exp 2− 1 𝜎𝜎x𝑗𝑗 j −c j ⎢ ⎥ 𝜀𝜀 i =1 i 2 𝑏𝑏𝑖𝑖 (𝑘𝑘) = 𝑏𝑏𝑖𝑖 (𝑘𝑘 − 1) − 𝛼𝛼 σji 2 𝑚𝑚 𝑖𝑖 ∂ ⎢ 𝜕𝜕𝑏𝑏 ⎥ ε 1 𝑥𝑥𝑗𝑗 − 𝑐𝑐𝑗𝑗 !! 𝑅𝑅 bi (k ) = bi (k − 1) − α 𝑖𝑖 ∏ ⎢ ⎥ �∑ 𝑒𝑒𝑒𝑒𝑒𝑒 �− � � � � 𝑖𝑖=1 ∂bi 2 1 𝜎𝜎x𝑗𝑗𝑖𝑖mj −cij 2 ⎦ R ⎣ ∑i=1 ∏ exp − 2 σi
(25e)
(25e)
j
4.4.2. Update Equation for Variance
4.4.2. Update Equation for Variance
The update law for variance is obtained by taking the derivative of Equation (25b) w. r. t. 𝜎𝜎𝑖𝑖 (𝑘𝑘) Theputting update law forvalues variance is obtained taking theinderivative of Equation (25b) w. r. t. σi (k ) and the final in Equation (25a)by as presented Equation (25f).
and putting the final values in Equation (25a) as presented in𝑚𝑚Equation (25f). 𝑖𝑖 2
�𝑥𝑥𝑗𝑗 − 𝑐𝑐𝑗𝑗 � (∑𝑅𝑅 𝑏𝑏 ) − 𝑓𝑓 𝛼𝛼�𝜀𝜀� 𝑅𝑅𝑖𝑖=1 𝑖𝑖 𝑚𝑚 𝑚𝑚 �� 𝜎𝜎𝑖𝑖 (𝑘𝑘) = 𝜎𝜎𝑖𝑖 (𝑘𝑘 − 1) − �𝜇𝜇2𝑖𝑖 �𝑥𝑥𝑗𝑗𝑚𝑚 , 𝑘𝑘�� 𝜀𝜀 𝑖𝑖 3 ∑R𝑖𝑖=1 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗 , 𝑘𝑘� m i �𝜎𝜎 � x𝑗𝑗j − c j ∑ i = 1 bi − f m
σi (k) = σi (k − 1) − αε 4.4.3. Update Equation for Center
∑iR=1
µi x m j ,k
3 σji
(25f)
m µi x j , k ε
(25f)
Similarly, the update relation for the center is acquired:
4.4.3. Update Equation for Center
(∑𝑅𝑅 𝑏𝑏 ) − 𝑓𝑓 ∑𝑖𝑖=1 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗 , 𝑘𝑘�
�𝑥𝑥 𝑚𝑚 − 𝑐𝑐𝑗𝑗𝑖𝑖 �
𝑗𝑗 𝑚𝑚 𝑖𝑖=1 𝑖𝑖 Similarly, the update for 𝑐𝑐𝑖𝑖 (𝑘𝑘) =relation 𝑐𝑐𝑖𝑖 (𝑘𝑘 − 1) − the 𝛼𝛼 �𝜀𝜀center � 𝑅𝑅 is acquired: �� 𝑚𝑚
2
�𝜎𝜎𝑗𝑗𝑖𝑖 �
� 𝜇𝜇𝑖𝑖 �𝑥𝑥𝑗𝑗𝑚𝑚 , 𝑘𝑘�� 𝜀𝜀
(25g)
R m i − f monthexSD −technique. c ∑i=FLC Figure 8 shows a block diagram of the The implementation of 1 bi based m j j control − 1) − αout µi xFLC , k based (25g) (k carried (k) = ci is ε in a closed-loop The ε on the SD j the proposedcischeme system. 2 R i µi x m , k ∑ σ i = 1 algorithm performs fuzzification of input, selects aj specific rule from the set of rules, and finally dej fuzzifies the error function to generate a control signal to the plant. FLC based On SD
Induction Motor
De-fuzzification
Reference Input
Fuzzification
Inference Mechanism
Output of Controller
Output
Rule-base
Figure 8. FLC based on the Steepest Descent (SD).
Figure 8. FLC based on the Steepest Descent (SD).
4.5. Sliding Mode Control (SMC) Strategy The SMC is a non-linear control scheme applicable to linear and non-linear systems. SMC has the remarkable features of robustness, accuracy, easy implementation, and tranquil tuning. SMC is insensitive to disturbance rejection and parameter variation [12,13,29,30]. The mechanical equation of the IM is represented as:
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Figure 8 shows a block diagram of the FLC based on the SD technique. The implementation of the proposed scheme is carried out in a closed-loop control system. The FLC based on the SD algorithm performs fuzzification of input, selects a specific rule from the set of rules, and finally de-fuzzifies the error function to generate a control signal to the plant. 4.5. Sliding Mode Control (SMC) Strategy The SMC is a non-linear control scheme applicable to linear and non-linear systems. SMC has the remarkable features of robustness, accuracy, easy implementation, and tranquil tuning. SMC is insensitive to disturbance rejection and parameter variation [12,13,29,30]. The mechanical equation of the IM is represented as: . Te = J ωm + Bωm + TL (26) The mechanical equation based on IFOC is equivalently described as: Te = k T isq
(27a)
where k T is electromagnetic torque constant and is represented as: kT =
3PLm λ 4Lr rd
Substituting Equation (27a) into Equation (26), we have: .
biqs = ωm + aωm + f
(27b)
where b = KT /J, a = B/J, and f = TL /J. Applying uncertainties ∆a, ∆b, and ∆ f in terms of a, b, and f, respectively, we have: ωm = (b + ∆b)isq − ( a + ∆a)ωm − ( f + ∆ f ) The error equation for speed is defined as: ∗ e ( t ) = ωm ( t ) − ωm (t)
(27c)
∗ is the reference rotor speed. Derivating Equation (27c), with respect to time provides: where ωm .
.
. ∗
e(t) = ω m (t) − ω m (t) = − ae(t) + u(t) + d(t) . ∗ ∗ (t) − f (t) − ω u(t) = bisq − aωm m
(28a)
and the uncertainties d(t) are: d(t) = ∆bisq − ∆aωm − ∆ f . The Sliding Surface (SS) S(t) with an integral component for the speed loop is defined as: S(t) = e(t) −
Zt
(k − a)e(τ )dτ
0
The SMC controller is designed to drive the error to the sliding surface as presented in Figure 9. At the SS error, the derivative and integral of error tend to zero. In SMC the designed control system trajectories are forced towards SS. The SMC consists of SS and control law designing. Moreover, the discontinuous control law results in chattering in the system that is overcome by adapting continuous control law, where k is the constant defining the slope of SS, and is equal to k < 0. . S(t) = S(t) = 0 at SS. The controlled system dynamic behaviour is represented as: .
e(t) − (k − a)e(t) = 0
(28b)
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e /∫ e
Sliding Surface S(t)=0
e Chattering Phenomena Reaching Phase
Figure 9. Graphical interpretation of the sliding mode controller. Figure 9. Graphical interpretation of the sliding mode controller.
4.6. Designed Hybrid Controller The variable structure controller is designed as: The proposed HC or Fuzzy Sliding Mode Controller incorporates two adaptive non-linear control techniques: (a) Sliding Mode uControl and((b) S) PI-fuzzy control. The advantageous (28c) (t) = ke(SMC) (t) − βsgn features of the SMC and PI-fuzzy control are combined in HC [31]. The SMC minimizes error in transient states,(28c), enlarging a system’s stability and is providing a fast-dynamic is (.) In Equation β is the switching gain and constrained to β ≥ |d(tresponse. SS and sgn )|. S is theChattering reduced by PI-fuzzy during steady state. Figure 10 demonstrates the designed HC scheme. The 𝑘𝑘 , 𝑝𝑝 is the sign function and is represented as: and 𝑘𝑘𝑖𝑖 values are updated using a sliding surface based on fuzzy rules. The SS control law is ( designed for the updated values from the fuzzy-PI1control i f S(scheme. t) > 0 The HC aim is to design a control Sgn S t = (29a) ( ( )) law that tracks reference speed [32]. Error and derivative error −1 i f S(of t) < 0 are always directed towards the sliding surface in SMC. SS 𝑆𝑆(𝑡𝑡) with a derivative component of the speed loop is defined as: ∗ command is obtained as: Substituting Equation (28c) into Equation 𝑆𝑆(𝑡𝑡) =(28a), 𝑒𝑒̇ (𝑡𝑡) +isq 𝜆𝜆𝜆𝜆(𝑡𝑡)
(30a)
i dependent of the system In Equation (30a), 𝜆𝜆 is a positive that is. ∗bandwidth 1 h arbitrary constant ∗ ∗ (29b) isq = ke − βsgn(s) + aωm (t) + ω m (t) + f and is defined as: b 4.6. Designed Hybrid Controller
𝜆𝜆𝜆𝜆(𝑡𝑡) = 𝐹𝐹1 𝐾𝐾1 𝑒𝑒(𝑡𝑡) + 𝐹𝐹2 𝐾𝐾2 � 𝑒𝑒(𝑡𝑡) 𝑑𝑑𝑑𝑑
(30b)
The proposed or Fuzzyusing Sliding Mode Controller incorporates adaptive non-linear The 𝜆𝜆𝜆𝜆(𝑡𝑡) HC is updated the adaptive PI controller. Thetwo discontinuous control control law techniques: (SMC) and The advantageous features introduced(a) in Sliding EquationMode (31a) Control causes oscillations in (b) the PI-fuzzy electrical control. system where PWM control signals of are the employed. SMC and PI-fuzzy control are combined in HC [31]. The SMC minimizes error in transient states, enlarging a system’s stability and providing a fast-dynamic response. Chattering is reduced ∗ 𝑖𝑖𝑠𝑠𝑠𝑠 = −𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈(𝑆𝑆) (31a) by PI-fuzzy during steady state. Figure 10 demonstrates the designed HC scheme. The k p , and k i where is a sufficiently large positive and values are𝑈𝑈updated using a sliding surfaceconstant based on fuzzy rules. The SS control law is designed for the updated values from the fuzzy-PI control scheme. 𝑈𝑈 The𝑖𝑖𝑖𝑖𝑖𝑖 HC>aim 0 is to design a control law that tracks 𝑆𝑆𝑆𝑆𝑆𝑆�𝑆𝑆(𝑡𝑡)� = � −𝑈𝑈 𝑖𝑖𝑖𝑖𝑖𝑖 < reference speed [32]. Error and derivative of error are always0directed towards the sliding surface in SMC. SS t) with a derivative component of thethe speed loop iscontrol definedlaw as: is used to implement a ToS(minimize the chattering phenomena, alternate continuous smooth approximation instead of the. discontinuous “sgn” function described as: S(t) = e(t) + λe(t) (30a) 𝑆𝑆 ∗ 𝑖𝑖𝑠𝑠𝑠𝑠 = −𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈(𝜎𝜎; 𝜀𝜀) = �−𝑈𝑈 � 𝜀𝜀 > 0 𝜀𝜀 ≈ 0 (31b) |𝑆𝑆| + 𝜀𝜀 that is bandwidth dependent of the system In Equation (30a), λ is a positive arbitrary constant
and is defined as:
λe(t) = F1 K1 e(t) + F2 K2
Z
e(t)dt
(30b)
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The λe(t) is updated using the adaptive PI controller. The discontinuous control law introduced in Equation (31a) causes oscillations in the electrical system where PWM control signals are employed. ∗ isq = −Usgn(S)
(31a)
where U is a sufficiently large positive constant and ( Sgn(S(t)) =
if S > 0 if S < 0
U −U
To minimize the chattering phenomena, the alternate control law is used to implement a continuous smooth approximation instead of the discontinuous “sgn” function described as: ∗ isq = −Usat(σ; ε) Energies 2018, 11, x FOR PEER REVIEW
Flux Loop Speed Loop
i
Torque Loop
* sd
i
* sd
ωmech
+
Σ
K1 + +
U
ωmech
-
Fuzzy Logic Controller
×
F2
K2
1 s
Integrator
d dt
(measured)
+ + Sliding Surface
Control Law
ε≈0
ε>0
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isd
* vsd
-
Σ
+
×
F1
Absolute
S = −U |S| + ε
* isq
Σ
+
PI
vsq*
PI
-
isq
Adaptive Sliding Mode Controller
Figure10. 10.Hybrid Hybridcontroller controllerdesign. design. Figure
Resultsand andDiscussions Discussions 5.5.Results Theperformance performanceof ofthe thedesigned designedcontrol controlstrategies, strategies,for forexample, example,PI, PI,adaptive adaptivePI, PI,FLC FLCbased basedon on The LM, fuzzy logic (FL) based on SD, SMC, and HC was validated and investigated by several simulation LM, fuzzy logic (FL) based on SD, SMC, and HC was validated and investigated by several simulation results.For Forthe theIFOC IFOCIM IMdrive drivesystem, system,the thesoftware softwareused usedfor forresults resultswas wasMATLAB/Simulink MATLAB/Simulink (R2017a, (R2017a, results. MathWorks, Natick, MA, USA). The designed system was subjected to different perturbations, MathWorks, Natick, MA, USA). The designed system was subjected to different perturbations, namely: namely: parameter uncertainties, speed variation, electrical and load disturbances. Initially, parameter uncertainties, speed variation, electrical faults, andfaults, load disturbances. Initially, the drive the drive was started from a halt condition; by sustaining the torque at zero, the rated flux was was started from a halt condition; by sustaining the torque at zero, the rated flux was attained. attained. The fundamental purpose of the simulations was twofold. First, to authenticate the The fundamental purpose of the simulations was twofold. First, to authenticate the exceptional exceptionaland robustness outcomes the designed Second, to relate the robustness outcomesand of the designedof control strategies.control Second,strategies. to relate the effectiveness of the effectiveness of the proposed control approaches. Table A1 summarizes the parameters of the IM, proposed control approaches. Table A1 summarizes the parameters of the IM, furthermore, the update furthermore, constants usedschemes by the designed control constants usedthe byupdate the designed control are elaborated inschemes Table A2.are elaborated in Table A2. 5.1.Rotor RotorResistance Resistanceand andLoad LoadDisturbances Disturbances 5.1. Thedisturbance disturbancerejection rejectioncapability capabilityfor forthe theproposed proposedcontrol controlschemes schemeswas wasverified verifiedby byapplying applying The full load loadtill tillt 𝑡𝑡== 11 s,s, and and for for the the rest restofofthe thetest testthe theload loadwas washalved. halved. Moreover, Moreover, to to verify verify the the aa full robustness of the design control strategies for parameter uncertainty, 𝑅𝑅 was varied from rated to robustness of the design control strategies for parameter uncertainty, Rr𝑟𝑟 was varied from rated to 120%,150%, 150%,and and200% 200%of ofthe therated ratedvalues. values. As As illustrated illustrated in in Figure Figure 11 11 at at rated rated R𝑅𝑅r𝑟𝑟, ,the thePI PIcontroller controller 120%, simulationresults resultsconfirm confirmslow slowconvergence, convergence,high hightransient transientoscillation, oscillation,and andSSE SSEfor forparameter parameterand and simulation loadvariation. variation. The The performance performance evaluation in load evaluation of ofthe theproposed proposedoptimal optimalcontrol controlschemes schemesisiselaborated elaborated Table demonstrated in in Figure Figure 12. 12. The Theproposed proposed in Table1.1.The Theresult resultofofthe theadaptive adaptivePI PI control control scheme scheme is is demonstrated controlscheme schemevalidates validatesless lesssusceptibility susceptibilityto toparameter parameteruncertainty uncertaintyand andsudden suddenload loaddisturbances, disturbances, control withaasmall smallSSE SSEand andapproximately approximatelynegligible negligibleoscillation. oscillation. with 210 ω
ref
ω at rated R
200
r
ω at 120% R
𝑟𝑟
simulation results confirm slow convergence, high transient oscillation, and SSE for parameter and load variation. The performance evaluation of the proposed optimal control schemes is elaborated in Table 1. The result of the adaptive PI control scheme is demonstrated in Figure 12. The proposed control scheme validates less susceptibility to parameter uncertainty and sudden load disturbances, Energies 2018, 11, 2339 16 of 25 with a small SSE and approximately negligible oscillation. 210 ω
ref
ω at rated R
200
r
ω at 120% R
ω (rad/sec)
ω at 150% R ω at 200% R
190
180
r r r
192 190 188
170
186
3
2.5
2
1.5 t (sec)
1
0.5
0
1.4
1.2
1 160
Figure Speed control indirect field oriented control using a PI controller. Figure 11.11. Speed control ofof indirect field oriented control IMIM using a PI controller. Table 1. Performance assessment of designed controllers. Control Strategies
OS
US
RT
FT
IAE
ISE
ITAE
PI
(1). (2). (3). (4).
6.746 6.770 6.781 10.14
(1). (2). (3). (4).
0.754 0.753 0.454 25.25
(1). (2). (3). (4).
37.82 31.88 33.59 35.27
(1). (2). (3). (4).
103.8 14.11 120.4 279.0
(1). (2). (3). (4).
0.997 3.295 1.425 1.864
(1). (2). (3). (4).
4.792 32.96 4.318 4.858
(1). (2). (3). (4).
1.114 1.312 1.232 1.515
Adaptive PI
(1). (2). (3). (4).
1.106 1.103 1.026 1.186
(1). (2). (3). (4).
1.554 1.654 1.754 1.854
(1). (2). (3). (4).
373.3 337.1 181.9 127.8
(1). (2). (3). (4).
2.760 4.044 3.490 3.405
(1). (2). (3). (4).
0.315 0.286 0.222 0.223
(1). (2). (3). (4).
0.170 0.151 0.120 0.118
(1). (2). (3). (4).
0.147 0.146 0.126 0.144
FL based on LM
(1). (2). (3). (4).
0.246 0.346 0.345 0.376
(1). (2). (3). (4).
0.554 0.204 0.203 0.199
(1). (2). (3). (4).
265.2 11.38 181.9 126.6
(1). (2). (3). (4).
2.656 4.238 3.490 3.731
(1). (2). (3). (4).
0.235 0.298 0.222 0.296
(1). (2). (3). (4).
0.134 1.352 0.120 0.197
(1). (2). (3). (4).
0.106 0.102 0.126 0.200
FL based on SD
(1). (2). (3). (4).
0.666 0.665 0.664 0.663
(1). (2). (3). (4).
0.001 0.079 0.194 0.424
(1). (2). (3). (4).
2.021 1.933 1.199 1.623
(1). (2). (3). (4).
6.335 4.060 243.5 253.5
(1). (2). (3). (4).
0.498 0.559 0.650 0.784
(1). (2). (3). (4).
0.125 0.150 0.193 0.257
(1). (2). (3). (4).
0.997 1.104 1.267 1.470
SM
(1). (2). (3). (4).
0.236 0.237 0.356 0.376
(1). (2). (3). (4).
0.224 0.301 0.363 0.365
(1). (2). (3). (4).
812.8 697.8 609.7 669.4
(1). (2). (3). (4).
0.010 808.0 0.001 0.001
(1). (2). (3). (4).
0.254 0.209 0.248 0.290
(1). (2). (3). (4).
0.056 0.021 0.022 0.030
(1). (2). (3). (4).
0.153 0.220 0.337 0.471
Hybrid
(1). (2). (2). (4).
0.036 0.037 0.038 0.040
(1). (2). (3). (4).
0.014 0.016 0.017 0.019
(1). (2). (3). (4).
12.08 0.001 0.006 0.008
(1). (2). (3). (4).
21.19 168.1 192.9 143.4
(1). (2). (3). (4).
0.010 0.021 0.010 0.009
(1). (2). (3). (4).
.0001 .0004 .0005 .0006
(1). (2). (3). (4).
0.007 0.019 0.008 0.009
OS: Overshoot, US: Undershoot, RT: Rise Time, FT: Fall Time, ISE: Integral Square Error, SD: Steepest Descent, IAE: Integral Absolute Error, LM: Levenberg-Marquardt, ITAE: Integral of Time Weighted Absolute Error, FL: Fuzzy Logic, SM: Sliding Mode. Note: (1) Rated Rr , (2) 120% of rated Rr , (3) 150% of rated Rr , and (4) 200% of rated Rr .
The performance of the FLC based on LM is evaluated in Figure 13. The transient oscillation, settling time, and SSE increase comparatively less than with parameter and load variation. The FLC based on the SD technique SSE increases from 0.21 rad/s to 0.34 rad/s with parameter and load variation from rated Rr to 200% Rr as illustrated in Figure 14. The convergence property of SMC is verified in Figure 15. The SSE is almost zero for rated Rr and increases to 0.08 rad/s for 200% of rated
SM
Energies 2018, 11, 2339
Hybrid
(1). 0.236 (2). 0.237 (3). 0.356 (4). 0.376 (1). 0.036 (2). 0.037 (2). 0.038 (4). 0.040
(1). 0.224 (2). 0.301 (3). 0.363 (4). 0.365 (1). 0.014 (2). 0.016 (3). 0.017 (4). 0.019
(1). 812.8 (2). 697.8 (3). 609.7 (4). 669.4 (1). 12.08 (2). 0.001 (3). 0.006 (4). 0.008
(1). 0.010 (2). 808.0 (3). 0.001 (4). 0.001 (1). 21.19 (2). 168.1 (3). 192.9 (4). 143.4
(1). 0.254 (2). 0.209 (3). 0.248 (4). 0.290 (1). 0.010 (2). 0.021 (3). 0.010 (4). 0.009
(1). 0.056 (2). 0.021 (3). 0.022 (4). 0.030 (1). .0001 (2). .0004 (3). .0005 (4). .0006
(1). 0.153 (2). 0.220 (3). 0.337 (4). 0.471 (1). 0.007 (2). 0.019 (3). 0.008 (4). 0.009
17 of 25
Overshoot, Undershoot, RT: Rise Time, FT: Fall Time, ISE: Integral Error, SD: Steepest Rr . Also, the OS: behavior of US: SMC is oscillatory initially and settles after Square t = 1.5 s. The HC strategy is Descent, IAE: Integral Absolute Error, LM: Levenberg-Marquardt, ITAE: Integral of Time Weighted investigated in Figure 16. The HC scheme results validate the insensitivity and robustness to parameter Absolute Error, FL: Fuzzy Logic, SM: Sliding Mode. Note: (1) Rated 𝑅𝑅𝑟𝑟 , (2) 120% of rated 𝑅𝑅𝑟𝑟 , (3) 150% and load variation fast(4)settling time𝑅𝑅and zero SSE. 200% of rated of ratedwith 𝑅𝑅𝑟𝑟 , and 𝑟𝑟 . ω
187
ref
ω at rated R
ω (rad/sec)
186.5
r
ω at 120% R
186
ω at 150% R
185.5
ω at 200% R
185 184.5 184
183.5 Energies Energies 2018, 2018, 11, 11, xx FOR FOR PEER PEER REVIEW REVIEW 183 0 0.5
186.4 186.2 186 185.8 185.6 185.4 185.2
1
1.05
1
1.5
r r r
17 17 of of 25 25
1.1 2
2.5
3
rated t (sec) initially rated 𝑅𝑅 𝑅𝑅𝑟𝑟𝑟𝑟.. Also, Also, the the behavior behavior of of SMC SMC is is oscillatory oscillatory initially and and settles settles after after 𝑡𝑡𝑡𝑡 = = 1.5 1.5 s. s. The The HC HC strategy is investigated in Figure 16. The HC scheme results validate the insensitivity and robustness strategy is investigated 16. The HC scheme results validate the adaptive insensitivity and robustness Figure in 12.Figure Speed control of indirect field oriented control IM using PI. Figure Speed control of indirect field oriented IM using adaptive PI. to and load variation with fast time zero to parameter parameter and12. load variation with fast settling settling time and andcontrol zero SSE. SSE.
(rad/sec) ωω(rad/sec)
The performance of the FLC based on LM is evaluated in Figure 13. The transient oscillation, settling time, and SSE increase comparatively less than with parameter and load variation. The FLC ωω 187 ref ref with parameter and load based on the SD187 technique SSE increases from 0.21 rad/s to 0.34 rad/s ωω at Rrr property of SMC at rated rated R variation from186.5 rated 𝑅𝑅𝑟𝑟 to 200% 𝑅𝑅𝑟𝑟 as illustrated in Figure 14. The convergence 186.5 R ωω at 120% Rr rad/s for 200% of at 120% is verified in Figure 15. The SSE is almost zero for rated 𝑅𝑅𝑟𝑟 and increases to 0.08 r 186 186
ωω at at 150% 150% R Rr
185.5 185.5
ωω at Rr at 200% 200% R
r r
185 185
186.4 186.4 186.2 186.2 186 186 185.8 185.8 185.6 185.6 185.4 185.4 185.2 185.2
184.5 184.5 184 184 183.5 183.5 183 1830 0
0.5 0.5
11
1.02 1.02 1.5 1.5 tt (sec) (sec)
11
1.04 1.04 22
33
2.5 2.5
Figure Speed control IFOC IM using LM. Figure 13. 13. Speed control ofof IFOC IM using Figure 13. Speed control of IFOC IM usingLM. LM. ωω ref
187.5 187.5
ref
ωω at Rr at rated rated R
187 187
r
ωω at at 120% 120% R Rr
r
(rad/sec) ωω(rad/sec)
186.5 186.5
ωω at at 150% 150% R Rr
r
ωω at at 200% 200% R Rr
186 186
r
185.5 185.5 185 185
185.8 185.8 185.6 185.6 185.4 185.4
184.5 184.5 184 184 183.5 183.5 00
185.2 185.2 11 0.5 0.5
11
1.02 1.04 1.06 1.06 1.08 1.08 1.02 1.04 1.5 22 1.5 tt (sec) (sec)
2.5 2.5
33
Figure 14. Speed control of IFOC IM using SD. Figure Speed control IFOC usingSD. SD. Figure 14. 14. Speed control of of IFOC IMIM using 187 187 186.5 186.5
ωω ref
ref
ωω at at rated rated R Rr
r
ωω at Rr at 120% 120% R
r
c) )
186 186
ωω at at 150% 150% R Rr
r
ωω at Rr at 200% 200% R
184.5
185.6
184
185.4 185.2
183.5
1.02 1.04 1.06 1.08 1.5 2 t (sec)
1 0
Energies 2018, 11, 2339
0.5
1
2.5
3
18 of 25
Figure 14. Speed control of IFOC IM using SD. ω
187 186.5
r
ω at 120% R ω at 150% R
186 ω (rad/sec)
ref
ω at rated R
ω at 200% R
185.5
r r r
185 185.6 184.5
185.4
184
185.2 185
183.5
1 0
0.5
1
1.02 1.04 1.06 1.5 t (sec)
2
2.5
3
15. REVIEW Speed control of IFOC IM using Sliding Mode Control (SMC). Energies 2018, 11, xFigure FOR PEER
Figure 15. Speed control of IFOC IM using Sliding Mode Control (SMC).
18 of 25
185.5 ω
ref
ω at rated R
185.4
r
ω at 120% R
ω (rad/sec)
ω at 150% R ω at 200% R
185.3
r r r
185.2 185.28 185.26
185.1
1 185
0
0.5
1
1.2
1.4
1.5 t (sec)
1.6 2
2.5
3
Figure 16. Speed control of IFOC IM using Hybrid Control (HC).
Figure 16. Speed control of IFOC IM using Hybrid Control (HC). 5.2. Electrical Faults with Load Variation
5.2. Electrical Faults with Load Variation
Electrical fault perturbation analysis evaluated stability and dynamic performance of the IFOC
IM for the designed control strategies. proposed system was various faults: of (a) the Single Electrical fault perturbation analysisThe evaluated stability andsubjected dynamictoperformance IFOC Phasing, (b) Double (c) Under Voltage, and (d) Over Voltage, asto demonstrated in Figures IM for the designed controlPhasing, strategies. The proposed system was subjected various faults: (a) Single respectively. In (c) single phasing, phase for 200 as msdemonstrated at 𝑡𝑡 = 1 s. Extra heat was Phasing,17–20 (b) Double Phasing, Under Voltage, and“a” (d)opened Over Voltage, in Figures 17–20 generated in the stator winding because of more current being drawn by phase “b” and “c”. PI respectively. In single phasing, phase “a” opened for 200 ms at t = 1 s. Extra heat was generated in the controller provided a large undershoot of 8.25 rad/s and overshoot of 14.25 rad/s. Further, the PI stator winding because of more current being drawn by phase “b” and “c”. PI controller provided a control scheme settled after 𝑡𝑡 = 1.8 s, resulting a high oscillation in response. The responses of the large undershoot 8.25 rad/s of SD, 14.25 rad/s. Further, thethe PI same control scheme settled adaptive PI,ofFLC based on and LM, overshoot FLC based on and SMC were almost with an average after t =settling 1.8 s, resulting a high oscillation in response. The responses of the adaptive PI, FLC based time of 𝑡𝑡 = 1.3 s, overshoot of 3.25 rad/s, and undershoot of 3.75 rad/s. The chattering on LM, phenomena FLC basedin onthe SD, and SMC in were the same withHC anstrategies average provided settling time of t = 1.3 s, SMC resulted largealmost oscillation. Moreover, an undershoot of 0.05 rad/s and and overshoot of 0.03 of rad/s settled 𝑡𝑡 = 1.3 s with negligible The overshoot of 3.25 rad/s, undershoot 3.75that rad/s. Theatchattering phenomena in oscillation. the SMC resulted was subjected to HC double phase fault at 𝑡𝑡 = an 1 sundershoot for 200 ms along with loadand variation. The of in large system oscillation. Moreover, strategies provided of 0.05 rad/s overshoot undershoot of 28.25 of 72.75 rad/s, settling time of 𝑡𝑡 = 2 was s, and large dips 0.03 rad/s that settled at trad/s, = 1.3overshoot s with negligible oscillation. The system subjected towere double noted for the speed response by the PI control strategy. An average undershoot of 7.25 rad/s, phase fault at t = 1 s for 200 ms along with load variation. The undershoot of 28.25 rad/s, overshoot of overshoot of 9.75 rad/s, and settling time of 𝑡𝑡 = 1.45 s along with chattering for adaptive PI, FLC 72.75 rad/s, settling time of t = 2 s, and large dips were noted for the speed response by the PI control based on LM, FLC based on SD, and SMC control schemes were found. Furthermore, an undershoot strategy.ofAn average undershoot of 7.25 rad/s, overshoot of 9.75 rad/s, andnegligible settling time of t =was 1.45 s 2.05 rad/s, overshoot of 1.15 rad/s, output settles after 𝑡𝑡 = 1.5 s, and chattering along with chattering for adaptive PI, FLC based on LM, FLC based on SD, and SMC control schemes recorded for the HC scheme. were found.The Furthermore, undershoot of 2.05control rad/s,strategies overshoot 1.15 rad/s, settles after fault tolerantan capability of the design wasofinvestigated foroutput overvoltage and under electrical fault perturbation. At for 𝑡𝑡 = the 1 s HC the three phase voltages were maximized and t = 1.5 s, and voltage negligible chattering was recorded scheme. minimized to 150% and 50% of the rated values for 200 ms. The PI control strategy shows undershoots of 13.75 rad/s and 9.25 rad/s, settling times of 𝑡𝑡 = 1.8 s and 𝑡𝑡 = 1.7 s, and overshoots of 28.75 rad/s and 13.75 rad/s, respectively, and bulky transient oscillation was noted for undervoltage and overvoltage. The dynamic behavior of adaptive PI, FLC based on LM, FLC based on SD, and SMC generated average settling times of 𝑡𝑡 = 1.62 s and 𝑡𝑡 = 1.58 s, overshoots of 2.75 rad/s and 4.75 rad/s, and undershoots of 2.25 rad/s and 5.25 rad/s, for overvoltage and undervoltage faults, respectively,
Energies 2018, 11, 2339
19 of 25
Energies 2018, 11, x FOR PEER REVIEW Energies Energies2018, 2018,11, 11,xxFOR FORPEER PEERREVIEW REVIEW
19 of 25 19 of of 25 25 19
220 220 220
ω ωref
ref
PI PI Adaptive PI Adaptive PI FLC based on LM FLC based on LM FLC based on SD FLC based on SD SMC SMC HC HC
210 210 210
(rad/sec) ω (rad/sec) (rad/sec) ωω
200 200 200 190 190 190 180 180 180 170 170 170 160 160 160 0 00
0.5 0.5 0.5
1 11
190 190 188 188 186 186 184 184 182 182 1 1 1.5 1 1.5 1.5 t (sec) (sec) tt (sec)
1.1 1.1 21.1 2 2
1.2 1.2 1.2 2.5 2.5 2.5
3 33
Figure17. 17. Single Single phasing. Figure phasing. Figure 17. 17. Single Single phasing. phasing. Figure 260 260 260
ω ωref ω ref
PIref PI PI Adaptive PI Adaptive PI Adaptive PIon LM FLC based FLC based on LM FLC based on SD LM FLC based on FLC based on SD FLC SMC based on SD SMC SMC HC HC HC
ωω(rad/sec) (rad/sec) ω(rad/sec)
240 240 240 220 220 220 200 200 200 180 180 180 160 160 160 0 0 0
0.5 0.5 0.5
1 1 1
190 190 190 185 185 185 180 180 1 180 1 1 1.5 1.5 1.5 t (sec) t (sec) t (sec)
2 2 2
1.1 1.1 1.1
1.2 1.2 1.2 2.5 2.5 2.5
3 3 3
Figure 18. Double phasing. Figure 18. Double phasing.
(rad/sec) ωω (rad/sec) ω(rad/sec)
Figure 18. 18. Double Double phasing. Figure phasing. 215 215 215 210 210 210 205 205 205 200 200 200 195 195 195 190 190 190 185 185 185 180 180 180 175 175 175 170 170 170 165 1650 165 0 0
ω ωref ω ref
PIref PI PI Adaptive PI Adaptive PI Adaptive PIon LM FLC based FLC based on LM FLC based on SD LM FLC based on SD SMC based on SD FLC SMC HC SMC HC HC 190 190 190 185 185 185
0.5 0.5 0.5
1 1 1
1 1 1 1.5 1.5 t (sec) 1.5 t (sec) t (sec)
1.1 1.1 1.1 2 2 2
Figure 19. Undervoltage. Figure 19. Undervoltage. Figure 19. Undervoltage. Undervoltage. Figure 19.
1.3 1.2 1.3 1.2 1.2 2.5 1.3 2.5 2.5
3 3 3
Energies 2018, 11, 2339 Energies 2018, 11, x FOR PEER REVIEW
20 of 25 20 of 25
ω
205 200 195 ω (rad/sec)
ref
PI Adaptive PI FLC based on LM FLC based on SD SMC HC
190 185 187 180
186
175
185
170 165
184 1 0
0.5
1
1.1
1.5 t (sec)
1.2 2
1.3 2.5
3
Figure 20. 20. Overvoltage. Figure Overvoltage.
5.3. Load Disturbances in Presence of Speed Variation The fault tolerant capability of the design control strategies was investigated for overvoltage and The robustness of fault the designed control validated for low speed under voltage electrical perturbation. At tschemes = 1 s thewere threesuccessfully phase voltages were maximized and variation and the results areofshown in Figure speed was reducedstrategy to half shows (92.62 undershoots rad/s) along minimized to 150% and 50% the rated values21. forThe 200 ms. The PI control with load disturbance 𝑡𝑡 = 1 s. Thetimes non-minimum of the PI controller produced an of 13.75 rad/s and 9.25 at rad/s, settling of t = 1.8 sphase and t behavior = 1.7 s, and overshoots of 28.75 rad/s and overshoot 7.06 rad/s and of 25.625 rad/s. The speed slowlyand after 𝑡𝑡 = 1.5 s. 13.75 rad/s,ofrespectively, and undershoot bulky transient oscillation was noted for settled undervoltage overvoltage. Furthermore, the simulation results confirm slow convergence, high transient oscillation, high The dynamic behavior of adaptive PI, FLC based on LM, FLC based on SD, and SMC generatedand average SSE of the PI control strategy. of adaptive PI, rad/s FLC based on LM, FLC based on SD, and settling times of t = 1.62 s andThe t = responses 1.58 s, overshoots of 2.75 and 4.75 rad/s, and undershoots of SMCrad/s wereand almost same settling and timeundervoltage of 𝑡𝑡 = 1.4 s,faults, average overshoot and of 3.05 rad/s, and 2.25 5.25the rad/s, forwith overvoltage respectively, comparatively undershoot of for 3.15 rad/s. Theand chattering phenomena in the SMC in large oscillations. less oscillation overvoltage undervoltage. The overvoltage andresulted undervoltage faults of the HC Moreover, HC strategies 0.04 that settled 𝑡𝑡 = 1.4 sand and0.05 0.08rad/s, rad/s scheme provides settlingprovided times of tan = overshoot 1.35 s andof t= 1.4rad/s s, overshoots of at 0.02 rad/s undershoot with negligible oscillation. The tracking error for the dynamic response of the speed loop and undershoots of 0.03 rad/s and 0.1 rad/s, respectively. is presented in Figure 22. Among the designed controllers, the PI control strategy showed sensitivity, 5.3. Disturbances in Presence of SpeedtoVariation largeLoad slow convergence, and oscillation load variation. However, the FLC based on LM algorithm, FLC The based on SD technique, SMC, and HC schemes (Adaptivewere Sliding Mode Controller (ASMC)) provided robustness of the designed control successfully validated for low speed allowable sensitivity, less oscillation, and fast convergence when compared to PI. variation and the results are shown in Figure 21. The speed was reduced to half (92.62 rad/s) along
ω (rad/sec)
with load disturbance at t = 1 s. The non-minimum phase behavior of the PI controller produced an overshoot of 7.06 settled slowly after t = 1.5 s. 180 rad/s and undershoot of 25.625 rad/s. The speed ω ref Furthermore, the simulation results confirm slow convergence, high transient oscillation, and high PI 160 strategy. The responses of adaptive PI, FLC based on LM, FLC based on SD, SSE of the PI control Adaptive PI and SMC were almost the same with settling time of t = 1.4 s, FLC average overshoot of 3.05 rad/s, based on LM 140 and undershoot of 3.15 rad/s. The chattering phenomena in the SMC resulted in large oscillations. FLC based on SD Moreover, HC strategies provided an overshoot of 0.04 rad/s that SMC settled at t = 1.4 s and 0.08 rad/s HC undershoot with120 negligible oscillation. The tracking error for the dynamic response of the speed loop is presented in Figure 22. Among the designed controllers, the PI control strategy showed sensitivity, 100 large slow convergence, and oscillation to load variation. However, the FLC based on LM algorithm, FLC based on SD technique, SMC, and HC (Adaptive Sliding Mode Controller (ASMC)) provided 80 less oscillation, and fast convergence when compared to PI. allowable sensitivity, 94 60 40
92
0
0.5
1
1.2
1.4
1
1.5 t (sec)
2
2.5
Figure 21. Load disturbances in presence of speed variation.
3
Moreover, HC strategies provided an overshoot of 0.04 rad/s that settled at 𝑡𝑡 = 1.4 s and 0.08 rad/s undershoot with negligible oscillation. The tracking error for the dynamic response of the speed loop is presented in Figure 22. Among the designed controllers, the PI control strategy showed sensitivity, large slow convergence, and oscillation to load variation. However, the FLC based on LM algorithm, FLC based on SD technique, SMC, and HC (Adaptive Sliding Mode Controller (ASMC)) provided Energies 2018, 11, 2339 21 of 25 allowable sensitivity, less oscillation, and fast convergence when compared to PI. 180
ω
160 140 ω (rad/sec)
ref
PI Adaptive PI FLC based on LM FLC based on SD SMC HC
120 100 80
94
60 40
92
0
0.5
1
1.2
1.4
1
1.5 t (sec)
2
2.5
3
Figure 21. Load disturbances in presence of speed variation.
Figure 21. Load disturbances in presence of speed variation. Energies 2018, 11, x FOR PEER REVIEW
21 of 25
2 PI Adaptive PI FLC based on LM FLC based on SD SMC HC
Speed Loop Error (rad/sec)
1 0 -1 -2 0.05
0
-3
-0.5 -4
0
-1 -5
1
1.05
1.1
-6
-0.05
-7 1.8 -8
0
0.5
1
1.5
2
2.2
2
2.4 2.5
3
t (sec) Figure 22. The speed loop error response for designed control schemes. Figure 22. The speed loop error response for designed control schemes.
5.4. Parameter and Load Variations 5.4. Parameter and Load Variations The step-change in parameters generated the worst-case scenario for the motor under IFOC. The The step-change in parameters generated the worst-case scenario for the motor under IFOC. variation in the parameters was as follows: rotor resistance 𝑅𝑅𝑟𝑟 was 200% of rated value, mutual The variation in the parameters was as follows: rotor resistance Rr was 200% of rated value, inductance 𝐿𝐿𝑚𝑚 was 80% of rated value, inertia 𝐽𝐽𝑒𝑒𝑒𝑒 to was 130% of rated value, 𝑇𝑇𝐿𝐿 was 50% of rated mutual inductance Lm was 80% of rated value, inertia Jeq to was 130% of rated value, TL was 50% value, 𝑅𝑅𝑠𝑠 was 120% of rated value, stator inductance 𝐿𝐿𝑠𝑠 was 80% of rated value, and rotor of rated value, Rs was 120% of rated value, stator inductance Ls was 80% of rated value, and rotor inductance 𝐿𝐿 was 90% of rated value, at time 𝑡𝑡 = 1 s as presented in Figure 23. In comparison to inductance Lr 𝑟𝑟was 90% of rated value, at time t = 1 s as presented in Figure 23. In comparison to traditionally tuned PI controllers, the FLC based on LM, FLC based on SD, SMC, and HC provide traditionally tuned PI controllers, the FLC based on LM, FLC based on SD, SMC, and HC provide smooth, robust, permissible overshoot with faster response, rise time, and settling time for the smooth, robust, permissible overshoot with faster response, rise time, and settling time for the parameter uncertainties and load disturbances as presented in Figure 23. parameter uncertainties and load disturbances as presented in Figure 23. 210 ω
200
c)
190
ref
PI Adaptive PI FLC based on LM FLC based on SD SMC
inductance 𝐿𝐿𝑚𝑚 was 80% of rated value, inertia 𝐽𝐽𝑒𝑒𝑒𝑒 to was 130% of rated value, 𝑇𝑇𝐿𝐿 was 50% of rated value, 𝑅𝑅𝑠𝑠 was 120% of rated value, stator inductance 𝐿𝐿𝑠𝑠 was 80% of rated value, and rotor inductance 𝐿𝐿𝑟𝑟 was 90% of rated value, at time 𝑡𝑡 = 1 s as presented in Figure 23. In comparison to traditionally tuned PI controllers, the FLC based on LM, FLC based on SD, SMC, and HC provide smooth, robust, Energies 2018, 11, 2339 permissible overshoot with faster response, rise time, and settling time for 22 ofthe 25 parameter uncertainties and load disturbances as presented in Figure 23. 210 ω
200
190 ω (rad/sec)
ref
PI Adaptive PI FLC based on LM FLC based on SD SMC HC
180 185.8 185.6
170
185.4 185.2
160
185 150
1 0
0.5
1.5 1
2
2.5
1.5
2
2.5
3
t (sec) Figure23. 23.The Thespeed speedloop loopparameter parametervariations. variations. Figure
The designed control scheme’s performance is critically and analytically compared with recent The designed control scheme’s performance is critically and analytically compared with recent literature in Table 2. Additionally, the dominant features are guaranteed in regards to load literature in Table 2. Additionally, the dominant features are guaranteed in regards to load disturbances, disturbances, parameter variation, chattering phenomena, electrical faults perturbations, and parameter variation, chattering phenomena, electrical faults perturbations, and robustness for the √ proposed control scheme. Moreover defines presence and × defines deficiency of the stated property. Table 2. Comparison of the designed control strategies with the literature. Ref.
CS
[1] [4] [19] [20] [30] [31]
ADSMCS NF and PI FSM and PI ASMC FL and PI AFSMC and PI
Our Work
PI Adaptive PI FLC based on LM FLC based on SD SMC HC
IFOC √ √ √ √ √ √
PV √ √
LV √
× √ √ √
× √ √ √ √
√ √ √ √ √ √
√ √ √ √ √ √
√ √ √ √ √ √
EFP
× × × × × × √ √ √ √ √ √
CP √ √
× × × × √ × × × × ×
R √ √ √ √
× √ × √ √ √ √ √
CS: Control strategies, PV: Parameter Variation, IFOC: Indirect Field Oriented Controller, EFP: Electrical Faults Perturbations, LV: Load Variation, CP: Chattering Phenomena, R: Robustness, ADSMCS: Adaptive Dynamic Sliding Mode Control System, NFC: Neuro-Fuzzy Controller, PI: Proportional and Integral, FSM: Fuzzy Sliding Mode, ASMC: Adaptive Sliding Mode Controller, AFSMC: Adaptive Fuzzy Sliding Mode Controller, FLC: Fuzzy Logic Controller, LM: Levenberg-Marquardt, SD: Steepest Descent, SMC: Sliding Mode Controller, HC: Hybrid Controller.
6. Conclusions and Future Work Optimal control strategies such as Adaptive PI, FLC based on LM and SD, SMC, and HC are designed for the speed loop of the IFOC IM drive that ensures the robustness in the presence of load disturbance, speed variation, electrical fault perturbations, and parameter uncertainties. The designed adaptive PI control strategy results illustrates better dynamic response with a rapid settling time and low overshoot for sudden changes of the IFOC IM drive. Since the exact system model is not
Energies 2018, 11, 2339
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required, the FLC based on LM and SD techniques is stable, robust, and insensitive to operating conditions and parameter variations. The SMC is a powerful non-linear control scheme that assures a remarkably fast dynamic response, robustness, external disturbance rejection, and immunity to parameter variation. The proposed HC incorporates the favorable characteristics of the SMC and PI-fuzzy by assuring fast dynamic response, global stability, and reduced chattering in the torque and speed loop. The performance of the designed control schemes is superior to that of fine-tuned PI control schemes for parameter and load variation. Furthermore, the order at which the best performance for the proposed controllers is noted for HC, FLC based on LM, SMC, FLC based on SD, and adaptive PI. The dominant features of the designed optimal control approach in comparison with the conventionally tuned PI control strategies are: low overshoot, rise time, and settling time, fast dynamic response, less sensitivity to load disturbances and electrical fault perturbations, robust to speed variations and parameter uncertainties, reduced chattering, low voltage dips, and less power consumption. The fault tolerant capability of the design control strategy is also guaranteed in the simulation result. In the near future, the proposed control schemes will be extended to position, flux (isd ), and torque (isq ) control loops for IFOC IM. Moreover, the simulated results will be validated using the hardware TMS320F28335 DSP board or DSPACE (Delfino Texas Instrument, Mansfield, TX, USA). Furthermore, Adaptive L1, H-infinity, High Order Adaptive SMC, and Feedback Linearization control strategies will be implemented. Additionally, the proposed optimal control schemes will be effectively adopted for a Doubly Fed Induction Motor (DFIM) and a Doubly Fed Induction Generator (DFIG), and the outcomes will be experimentally validated for robustness. Instead of MOSFET and IGBT switching devices (Delfino Texas Instrument, Mansfield, TX, USA), Silicon Carbide (SiC) and Gallium Arsenide (GaN) will be used to enhance the efficiency of the inverter. Author Contributions: K.Z., W.U.D., and M.A.K. proposed the main idea of the paper. K.Z. implemented the mathematical derivations, simulation verifications, and analyses. The paper was written by K.Z. and was revised by A.K., U.Y., T.D.C.B. and H.J.K. All the authors were involved in preparing the final version of this manuscript. All work was supervised by H.J.K. Funding: This Research was supported by BK21PLUS, Creative Human Resource Development Program for IT Convergence. Conflicts of Interest: The authors declare no conflict of interest.
Appendix Table A1. Machine nominal parameters. IM Parameters
Values
Rated Power Phases Line Voltage System Frequency Full Load Slip Number of Poles Switching Frequency Stator Resistance Stator Leakage Resistance Rotor Resistance Rotor Leakage Resistance Moment of Inertia Mutual Inductance Full Load Current Full Load Speed
3 HP/2.4 kW 3 460 V (L-L, rms) 60 Hz 1.72% 4 20 kHz 1.7 Ω 5.25 Ω 1.34 Ω 4.57 Ω 70 kg·m2 139 Ω 4A 1750 rpm
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Table A2. Control schemes’ constants. Control Strategies
Parameter
Values
PI
kp ki
0.2446 3.5298
Adaptive PI
k1 k2
0.5 0.75
FLC based on LM
c1 c2 σ1 σ2 b1 b2 λ µ
0.2 0.3 0.11 0.33 0.7 0.01 0.57 0.66
FLC based on Steepest Descent
α1 α2
0.0001 0.00199
Sliding Mode
k β
0.2–1.5 0.5–4
Hybrid
k1 k2
0.2 0.88
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