Maximum Torque-per-Amp Control for Traction IM Drives: Theory and Experimental Results

Transcription

1 Maimum Torue-per-Amp Control for Traction IM Drives: Theory an Eperimental Results Serhiy Bozhko, Serhii Dymko, Serhii Kovbasa, Sergei Peresaa, Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 RD, Unite Kingom Department Electrical Engineering, National Technical University of Ukraine KPI, Peremogy av. 37, 356 Kyiv, Ukraine Corresponing author: Abstract A novel maimum torue per Ampere (MTPA) controller for inuction motor (IM) rives is presente. It is shown to be highly suite to applications that o not eman an etremely fast ynamic response, for eample electric vehicle rives. The propose MTPA fiel oriente controller guarantees asymptotic torue (spee) tracking of smooth reference trajectories an maimises the torue per Ampere ratio when the evelope torue is constant or slow varying. An outputfeeback linearizing concept is employe for the esign of torue an flu subsystems to compensate for the torue-epenent flu variations reuire to satisfy the MTPA conition. As a first step, a linear approimation of the IM magnetic system is consiere. Then, base on a stanar saturate IM moel, the nonlinear static MTPA relationships for the rotor flu are erive as a function of the esire torue, an a moifie torue-flu controller for the saturate machine is evelope. The flu reference calculation metho to achieve simultaneously an asymptotic fiel orientation, torue-flu ecoupling an MTPA optimization in steay state is propose. The metho guarantees singularity-free operation an can be use as means to improve stator current transients. Eperimental tests prove the accuracy of the control over a full torue range an show successful compensation of the magnetizing inuctance variations cause by saturation. The propose MTPA control algorithm also emonstrates a ecoupling of the torue (spee) an flu ynamics to ensure asymptotic torue tracking. In aition, a higher torue per Ampere ratio is achieve together with an improve efficiency of electromechanical energy conversion. Keywors inuction motor, torue, control, tracking, saturation. I. INTRODUCTION During recent ecaes there has been a growing tren within many applications to replace Inuction Machine (IM) with Permanent Magnet Synchronous Machine (PMSM) ue to their higher efficiency, torue an power ensity []. However the cost of PMSM is significantly higher than IM ue to the use of rare-earth magnetic materials which have a very limite origin an their cost is continuously increasing. The tenency to reuce the use of epensive rare-earth magnets in inustrial an electrical traction rives has riven a renewe interest for research into avance esign an control concepts for IM [],[3]. Fiel-oriente vector control (FOC) [], avance FOC [5] an Direct Torue Control (DTC) [6] of IMs have been establishe a e-facto inustrial stanar for high an meium ynamic performance applications. Vector controlle an DTC IM rives typically operate with constant flu magnitue even at low values of prouce torue which results in goo ynamic performance. However conversely, the machine efficiency an power factor can be low, especially for small torue values. The IM torue is a prouct of the flu amplitue an the torue component of the stator current, proviing a egree of freeom for reuction of the power conversion losses or for attaining other performance criteria. The optimization techniues typically reporte in publications ajust the flu level as a function of the electromagnetic torue using various optimization proceures. The flu regulation restricts the rive ynamic performance hence this approach can be employe in applications not reuiring an etremely fast response, for eample, in electric vehicle rives where the rive only operates at rate torue for a limite proportion of time. A number of control strategies to optimize ifferent performance objectives are known [], [3], [7]-[], incluing minimization of active an total losses, power factor maimization, MTPA control, maimum torue per voltage control an maimum power transfer. The establishe optimization methos are esigne for steay-state operation (i.e. the rive is operating in constant torue). Dynamic behaviour optimization uring torue transient is only consiere in very few papers [], [3]. MTPA control [] minimizes the stator current for a given machine torue. Maimizing the machine torue by having limite source voltage an inverter current capability improves the electromechanical system performance. This is particularly beneficial for traction systems. Uner the MTPA control strategy, the torue controller ajusts the flu reference to increase the efficiency at low loas. As a result of this optimization, the torue per Ampere ratio is maimize an aitionally the achievable values of motor efficiency are close to those obtaine using the minimum active losses optimization criterion [], []. The basic MTPA control objective is achieve by controlling stator current torue an flu components, epresse in terms of rotor flu reference frame, to be eual. This leas to IM operation with constant slip freuency which is eual to the reciprocal of the rotor time constant. The MTPA relations are erive from the conition of the IM when proucing constant electromagnetic torue. A few theoretical results base on vector an scalar control concepts are known: moifie fiel-orientate control [], non-holonomy approach [5] an voltage freuency control [6]. However, simultaneous control of machine torue an flu results in poor torue ynamics; moreover, these ynamics cannot be specifie ue to the compleity an nonlinearity of the controlle plant (IM). For all the optimization techniues above an important issue for variable flu operation is the machine saturation effect. This effect results in varying machine inuctances hence the assumption of linear magnetic circuits, common for stanar optimization routines, is no longer vali. In aition, algorithms for flu estimation will no longer provie accurate information reuire for torue an flu controls. For MTPA control these issues have been stuie in [7]-[9]. In [7] a moification of [] is presente using an IM moel which accounts for the effects of magnetizing an leakage saturation. The esire stator current amplitue an slip freuency are

2 approimate as nonlinear functions of the torue reference. Fiel oriente control with a stanar MTPA approach for spee regulation of electric vehicles is propose in [8]. Torue an flu components of the stator current references are compute at the base of the MTPA curve as functions of spee controller output, proportional to the motor torue reference. MTPA algorithm [9] base on irect-flu vector control (DFVC) [] provies fast stator flu regulation using irect ais voltage control within a stator flu oriente reference frame. Fast torue regulation is achieve by controlling the torue component of the stator current vector (uarature), while the flu current component (irect) is not controlle. An aitional control action is neee to limit the stator current amplitue. The flu reference reuire in orer to achieve the MTPA conition in [9] is given by a nonlinear static function of the esire torue. The nonlinear saturation effect is taken into account in [7]-[] using store compute or measure ata. At present, publishe stuies aress the asymptotic torue regulation problem for constant torue references. However, a comple nonlinear torue-flu ynamic is generate by MTPA optimization [], [5]-[7] making this approach unsuitable for technological applications where accurate torue tracking control is reuire, for eample, in orer to enhance passenger comfort uring vehicle motion. Tracking of the smooth references is a more general solution of the torue control problem an can be consiere as etension of the fast torue regulation typically achieve with fast flu an torue current subsystems having high gain flu an current controllers [8], [9]. Torue tracking is a necessary reuirement in orer to successfully track the esire spee trajectories in spee control moe. This paper aresses the problem of asymptotic torue tracking control with MTPA optimisation for saturate IMs. In [] this problem was investigate assuming linear magnetic circuits for MTPA optimisation an controller esign. This stuy, in orer to improve the torue-flu tracking performance, takes into account the effect of saturation within the controller esign. The key contribution of this paper is a novel torue-flu tracking controller esign that simultaneously provies asymptotic torue tracking of the smooth reference trajectories in the whole range of machine torues an tracking of the torue-epenent flu references in orer to achieve MTPA optimisation in steay state. Torue-flu ecoupling allows the flu reference trajectories to be formulate as a static or ynamic functions of the torue reference hence avoiing a singularity at torue zero-crossing an improving stator current transients. Flu tracking allows to set the initial machine ecitation level close to zero hence preserving singularity-free operation. The propose approach is base on output-feeback linearising control an applie to both inirect an irect (observer base) fiel orientation. The theoretical finings of this stuy an the effectiveness of the propose approach are confirme by thorough eperimental valiation. This paper is an epane an further evelope version of the earlier conference paper []. The paper is organize as follows. The IM moel an control problem formulation are given in Sections II an III. The torue tracking MTPA controllers for linear approimation of the magnetising curve are esigne in Section IV. An etension of the MTPA torue-flu controller for saturate IM is given in Section V. The spee controller with MTPA is presente in Section VI. In Section VII the eperimental test results are reporte followe by the Conclusions of the stuy. II. INDUCTION MACHINE MODEL For the purpose of this stuy the /λ-saturate IM moel reporte in [3], [] has been employe. The moel assumes that only the magnetizing inuctance L m is saturate hence the leakage inuctances are constant, an neglects the crosssaturation inuctance so static an ynamic magnetizing inuctances are eual. The following efinitions are use: - static inuctance of the magnetizing circuit: L i i i () m m m m m where ψ m(i m) magnetizing curve, an i m magnetizing current; - stator an rotor inuctances, respectively: L i L i L m m m L i L i L m m m where L σ=const an L σ=const are stator an rotor leakage inuctances. Uner these assumptions, the two-phase moel of saturate IM in an arbitrary rotating reference frame,, is given as follows: T i i L T T / J m i i i u / m m m m m i i i u / m m m m m m m m m m m m m L i i L i i where u, u are stator voltage components (here an throughout the paper subscripts an enote vector variable components in the reference frame),, i are stator currents, ψ, ψ efine the rotor flu, ω is the rotor spee, T is the electromagnetic torue, T L is the loa torue an ε is the angular position of the reference frame with respect to a fie stator reference frame (ab) in which physical variables are efine. Slip freuency is efine as ω =ω - ω, an J is the total rotor inertia. One pole pair is assume without loss of generality. In the moel (3) constants (all positive) relate to IM electrical parameters are given by: i R / L m m L i L i m m m m m R L i m m m m m m L i L i L i m m m m m m 3Lm im L im Here R, R are stator an rotor resistances respectively. The ine m in () is use to enote the parameter s epenency on magnetizing current i m. It is important to note that the traitional moel of the nonsaturate IM can be erive from (3), assuming linear () (3) ()

3 magnetic circuits, i.e. not epening on i m, hence the following applies: L m(i m)=l m where L m=const the machine magnetizing inuctance at the rate flu, an α m=α, β m=β, γ m=γ, σ m=σ, μ m=μ. III. CONTROL PROBLEM STATEMENT In this stuy the torue tracking control problem is formulate as follows: Consier the IM moel (3) an assume that: A. The stator currents an rotor spee are available for measurement. All motor parameters are known an constant. All saturation-epenent parameters are known function of a magnetizing current. A. The torue reference trajectory is a smooth an boune function together with its first an secon time erivatives. Uner these assumptions, the control problem is to esign a torue controller which guarantees the following control objectives: CO. Asymptotic torue tracking with all internal signals boune, i.e. limt t where T T T is torue tracking error; CO. Maimization of Torue per Ampere ratio in steay state: T (5) ma T I, I i i (6) where I is a stator current magnitue. CO3. Asymptotic fiel orientation, i.e. lim (7) t The following sections report the propose solution to the formulate control problem. IV. TORQUE CONTROLLER DESIGN FOR LINEAR MAGNETIZING CURVE (MTPAL) This Section eals with the esign of torue-flu controllers that simultaneously guarantee an asymptotic tracking of the permissible torue references, the rotor flu orientation an flu-torue ecoupling. Flu tracking allows: a) to esign the flu reference trajectories as a static or ynamic function of esire torue in orer to achieve the MTPA conition in steay state an improve stator current transients; b) to avoi singularity (when flu is zero) selecting the flu reference an controller initialization. At the initial stage two controllers assuming linear magnetic circuits were esigne; an MTPA controller with inirect fiel orientation an an MTPA controller with irect fiel orientation employing a rotor flu observer. For both these cases current-fe control is assume. Following these, a fullorer irect fiel-oriente MTPA control is propose, incluing proof of its asymptotic stability. A. Inirect fiel orientation for current-fe IM Inirect fiel orientation allows for vector control esign to achieve high IM rive performance. In stanar configuration with inepenent torue an flu references it is simpler in comparison to irect orientation methos from the point of view of practical implementation. The propose solution eploits the concept of inirect fieloriente IM control [] with flu control given by: m L i L m i where ψ> is a smooth flu reference trajectory. The controller (8) (9), for the current-fe conition, guarantees that both flu magnitue tracking () an asymptotic fiel orientation () are globally achieve: t (8) (9) lime () lime t () where e an e are flu tracking error epresse in terms of the reference frame: e, e () The torue tracking error euation can be erive using (3) an efinitions (5), () as follows: T i ei ei T (3) from which a feeback-linearising torue controller can be constructe as: i T / () Euations (8), (9) an () efine the inirect fiel-orientation base flu-torue controller. Uner the action of the propose controller the error ynamics can be erive as: T e i e i e e e e e e (5) From bounness of flu an torue references an their erivatives it follows that currents in (9) an () are also boune. The euilibrium point (e, e )= is globally eponentially stable, an conseuently the torue error accoring to the first euation in (5) goes eponentially to zero, i.e. limt. From the analysis above it follows that t torue tracking (CO), the fiel orientation (CO3) an flu reference (ψ >) tracking are achieve. It shoul be note that asymptotic inirect fiel orientation an conseuently torue an flu ecoupling for time-varying flu references are possible only if flu tracking is achieve. Conition ψ > guarantees that controller is free of singularities. B. Direct fiel orientation for current-fe IM In this subsection irect fiel-oriente torue-flu tracking control employing an asymptotic rotor flu observer is presente. The reuce-orer flu observer for a linear case of (3) is efine as:

4 ˆ ˆ L i, m L i ˆ ˆ m (6) where is an observe flu value. This observer guarantees [] that the flu estimation errors ˆ, (7) ˆ ecay eponentially to zero provie. Using (7), the torue error euation can be foun as follows: T i ˆ i i T, (8) from which a feeback-linearising torue controller can be constructe as: i T / ˆ. (9) Using (3), (7), (8) an (9), the torue-flu error ynamic is erive as T T i ˆ () () () ˆ (t) For boune T an, the current i is boune as well. In aition, the subsystem ()-() is globally eponentially stable: t lim, (3) Hence, the torue error T in () eponentially ecays to zero if is boune. ˆ (t) In orer to provie the specifie estimate flu ynamics the following PI controller is employe to control estimate flu magnitue in (6): i kp Lm k i () where: ˆ is flu (estimate) tracking error; k ψp an k ψi are the controller proportional an integral gains respectively. From (6) an (), the error ynamic can be erive as k k i p (5) If () then all solutions of (5) (t) t. From this conition follows that ˆ (t) (t), an conseuently t is boune, lim T,. The analysis above proofs that control objectives (CO), (CO3), flu reference (ψ >) tracking an flu vector estimation are globally achieve. From conition ˆ (t) (t) it also follows that controller (6), (9) is free of singularities. Note that asymptotic irect fiel orientation oes not reuire flu an torue tracking properties. Nevertheless, for small ψ > (neee for MTPA optimization) an fast references the errors in flu regulation become critical. The flu tracking guarantees that (t) t, hence singularities are avoie. C. Flu reference selection to achieve MTPA conition The torue-flu tracking capabilities of the controllers (8),(9), () an (6), (9), () allow the selection of flu reference trajectories such that the maimization of Torue per Ampere ratio in steay state (CO) is achieve. MTPA conitions for all flu references ψ > in (9) an () are: a) erivative is boune b) flu reference in steay state satisfies the following: where m L i (6) i is the steay state value of i an ψ > is a small flu to avoi singularities in (8), (), (6) an (9) at zero torue reference T =. From (6), (9), an () it follows that the MTPA ratio in steay state is achieve uner the conition: i i / L (7) m where is the steay state value of an ψ /L m is a small constant. Note that the known result [] is a subset of the erive conitions (6), (7). Accoring to assumption A, the torue reference T an its erivative are boune, hence the flu reference trajectory ψ > may be compute irectly from (6) an () with, i i as a solution of the uaratic euation:: T, T L T. (8) 3 The time erivative of LT sign(t ) 3 T can be foun as:. (9) From (9), () an stability analysis above it follows that the selecte flu reference trajectory (8) satisfies the steay state conition (7) an has a boune erivative. Hence, all three control objectives are successfully achieve: torue tracking (CO), fiel orientation (CO3) an maimization of the torue per Ampere ratio in steay state (CO). Base on the iscussion above, a more general solution to the flu reference trajectory selection can be propose. The flu control plant is st orer (current fee conition) while the torue subsystem has no ynamics. Hence, some filtering of the flu reference is reuire in orer to reuce spikes in stator current flu component uring transients. One possible solution for flu reference filtering irectly follows from (9) an () (for irect fiel orientation from () an (9) if (t) ) as an output of the nonlinear ynamic system: L T 3. (3) In a more general case the flu reference is efine as an output r of the smoothing filter which processes the reference efine by (8). Such a filter allows the reconstruction of the

5 flu reference erivatives hence reucing the computational buren. For eample, a n -orer filter can be employe:,, k k k r (3) where k an k are filter tuning gains. Selecting unity amping k k / the conition ψ > is satisfie. The filter reconstructs the st an n erivatives of the flu reference (the n erivative is reuire for the full-orer control to be consiere in the net subsection). If reuire, the filter ynamic can be esigne as fast as possible by selection of high gains. Large enough values of k an k will provie, accoring to singular perturbation theory [6], the flu reference (8). The resulting control system, with the MTPA controller, has a single input, namely the torue reference T, an three outputs: the torue T, the rotor flu magnitue an its angular position. It shoul be note that controllers base on irect an inirect fiel orientation will provie the same ynamic performance for the same reference trajectories. In the net subsection the reuce orer solution is etene to a full-orer algorithm in preparation for the subseuent esign of an MTPA controller for a saturate machine within Section V. D. Full orer controller esign In the Sections above it was assume that the machine is current-fe, i.e. an i currents are the control signals. Consiering practical IM rive implementations, the currents in (3) represent only their esire ynamics: the reference trajectories an i are given by (9), () for inirect fiel orientation an by (), (9) for irect fiel orientation. These trajectories are the references for the inner current control loops. As the PI controllers typically employ high gains in orer to achieve a very fast response this justifies the assumption that the current ynamic in (3) can be neglecte. As a result, appropriate current tracking controllers must be esigne. In this Section, the full-orer torue controller is erive from the propose reuce-orer controller by the aition of current control loops which use the back-stepping proceure [7]. For both inirect an irect fiel oriente controllers the esign proceure is similar. The proceure is emonstrate below for the irect fiel oriente strategy. The current loops controls are esigne as follows: an u i i ˆ i k i (3) ip u i i ˆ i k i ip k i ii (33) where i i i an i i i are current tracking errors, k ip, k ip are the current controller s proportional gains, k ii is the integral gain an is the integral component of the -ais current controller. It shoul be note that the current reference erivatives in (3), (33) are known functions of T an T accoring to assumption A above. Consier the moifie flu observer given by ˆ ˆ L i m L i ˆ i ˆ m (3) where γ > is the observer correction gain. It is important to note that the correction term provies the close loop properties an therefore the observer is robust with respect to the variation in rotor resistance variations at nonzero machine spee [8]. i ˆ From (3), (3) (3) the system error ynamic is efine by three subsystems: - flu estimation subsystem: i i k i i - estimate flu regulation subsystem: i p k L i k m - torue current regulation subsystem i k i i k i ii where ki k, ip ki k. ip (35) (36) (37) In orer to investigate the stability of the system (35) (37) the following uaratic form of the flu subsystem (35) must first be consiere V i (38) The time erivative of (38) along the trajectories (35) can be erive as follows: V k i i (39) Uner the conition ki i () the form of (38) becomes a Lyapunov function satisfying conitions V, V () Hence, accoring to Lyapunov stability criteria it can be conclue that the euilibrium point T,,i () is globally eponentially stable. Since the subsystems (35), (36), an (37) are connecte in series (illustrate by Fig. ), the conclusion is that for a boune spee signal the euilibrium point T,, i, i T,, T, (3) is globally eponentially stable. This implies that the torue tracking error T(T,i,i,, ) ecays eponentially to zero, while also achieving asymptotic fiel orientation an

6 MTPA in steay state. Hence the control objectives (CO)- (CO3) are met. Flu estimation subsystem (35) Estimate Flu Regulation Subsystem (36) Torue Current Regulation Subsystem (37) Fig.. Error ynamics structure Tuning parameters of the controller (), (3)-(35) are: flu controller proportional k ψp an integral k ψi gains, proportional an integral gains k ip, k ip,k ii of current controllers an the observer gain γ. Stanar tuning for the linear n -orer systems (36) an (37) are use, an relation between k ip an γ is given by (). It shoul be note that the full-orer current controllers (3) an (33) reuire reference current erivatives. These are comple nonlinear epressions which epen on the torue reference trajectory an its st an n -orer time erivatives. When implementing in practical systems, if the reuire torue trajectories are smooth, it is possible to simplify the current controllers by neglecting these erivatives provie that the current controllers have high gains. In orer to avoi operating the machine in saturation moe the range of shoul be limite such that the flu oes not ecee its rate value. As iscusse in Subsection C, tracking performance of the propose controller provies some freeom in flu reference selection as a ynamic function of esire torue. This feature can be regare as a mean for current transients optimisation consiering only the reference signals T,, i, an not a full comple system ynamics. V. NONLINEAR MTPA TORQUE CONTROL (MTPAS) The MTPAL control algorithms esigne in Section IV assume linear magnetic circuits. In practice, these are nonlinear (ue to the saturation effect) an this can lea to errors in torue control an to eviation from the MTPA conition. This Section proposes irect MTPA vector control consiering the non-linearity of the machine magnetising curve in orer to avoi the abovementione issues. Assuming an ieal rotor flu orientation in steay state, the - ais component of the magnetizing flu is negligible an the -ais rotor current is zero [9]. Hence, i m i, m () an the IM torue euation becomes:, T i i m i L i i (5) m where is a rotor flu magnitue. The torue per Ampere ratio can be written as T TI. (6) From (5), the torue component of the stator current is T i (7) L i i hence m m i / T L i i i L i T (8) T m m m m Solving the euation T i the following relationship can be erive: i T (9) (5) where η is a small flu current which prouces the initial ecitation in orer to avoi a singularity in the IM control, an η ()=. Euation (5) will maimize the Torue per Ampere ratio in steay state (when T=const an assuming μ m=const). As an illustrative eample, Fig. shows the magnetizing curve ψ m(i m) as well as the MTPA relations (5) for an optimal flu-torue relation ψ o(t) (5) calculate for the 5kW inuction machine in Appeni II. Fig.. Magnetizing curve, magnetizing inuctance an MTPA relationship for the IM in Appeni II It is important to note that for saturate IM the stanar MTPA relationship = i [] is no longer vali. In orer to maimize the torue per Ampere ratio the flu reference shoul be ajuste accoring to the reuire torue, this is epicte in Fig. for the eample machine. Summarizing this Section, the propose moifie full-orer torue control algorithm inclues flu controller (), flu observer (3), torue controller (9), an current controllers (3) (33). The algorithm takes into account the machine magnetizing curve consiering the moel (3) constants as given by (). From the practical assumption that there is a constant relationship between any pair of L (i m), L (i m) an L m(i m) it can be conclue that: L i L const m m m m const m (5) Asymptotic stability for the propose torue control system with the moifie controller (9), (), (3) (3) can be easily shown using the same Lyapunov analysis as presente in previous Section. The block iagram of the torue control system with the feeback linearising controller (9), (), (3) (3) for saturate IM is shown in Fig.3.

7 PWM T ψo Filter (3) T ˆ k s α m k p i Torue Controller L Flu Controller i m - i - i i f,,, i -ais Current Controller (3) -ais Current Controller (33) i f T, ˆ, ˆ u u ab u a u b S abc 3 Energy Source POWER CONVERTER 3 Magnetizing Inuctance Calculator αl m - s α m ˆ Flu Observer L (i ) m αl m -γ β m m s i ab i a i b 3 i abc 3 Spee Sensor IM Fig. 3. Block iagram of the torue control system with feeback linearizing controller VI. SPEED CONTROLLER There are little applications in which machine torue control in eplicit form is reuire for eample, in spooling an tensioning rives, electric traction, etc. However in most applications spee control is reuire, hence this Section consiers the esign of an eternal spee controller for the torue control systems esigne previously To begin, assume a smooth an boune (together with the first two time erivatives) spee reference trajectory ω an unknown constant loa torue. Uner these conitions a spee controller [3], in combination with the above propose eponentially asymptotically stable torue-flu subsystem, (9), (), (3) (3) provies asymptotic spee tracking an constant loa torue estimation. The spee controller [3] is given by where T J TL ˆT L k i k p ˆ is the spee tracking error, (5) ˆT L is the loa torue estimation component of the constant T L/J, k ωp an k ωi are spee controller proportional an integral gains respectively, an τ is the small time constant of the spee filter. In (5) a st -orer linear filter is introuce in orer to erive the torue reference erivative that is reuire for implementation of the torue controller. If the current controller reuires a n -orer torue reference erivative then a n -orer linear filter shoul be employe. The ynamics of spee control loop with spee controller (5) is given by the linear time-invariant thir orer system [3] which has three tuning parameters k ωp, k ωi, τ to be selecte to guarantee asymptotic stability an transient performance. VII. EXPERIMENTAL VERIFICATION This Section reports the results of eperiments conucte to support the analytical finings of the previous Sections. The following eperiments have been conucte: - torue tracking performance comparison for ifferent control strategies using the small-power (5.5kW) IM rive system etaile in Appeni I - torue an spee control performance for ifferent control strategies using the meium-power (5kW) IM traction rive etaile in Appeni II - comparison stuy or the ynamic performances of the MTPA controllers uring fast torue trajectory tracking. The eperimental results are reporte below. A. Comparison of torue tracking performances The task of this eperiment was to assess the effect of magnetic saturation on torue tracking performance for three ifferent control strategies, namely: - Inirect Fiel-Oriente Control with constant flu (IFOC) [5]; - Full-orer МТPА control with linear magnetic curve representation (MTPAL) accoring to(9), (3)-(3); - Full-orer MTPA control (9), (3)-(3) accounting for IM saturation (MTPAS). The core iea of this eperiment was to emonstrate both the error in torue regulation uner MTPA control, if the saturation is not accounte for (MTPAL), an the capability of the MTPAS algorithm to achieve an error margin which is compatible with IFOC (however IFOC oes not provie MTPA criteria).

8 During these initial tests, the flu controller () was remove an the flu current component reference was set as follows: - for MTPAL control: i L i (53) m - for MTPAS control: i T (5) This methoology also enables utilisation of the close-loop flu control esigne in Section IV. This will be emonstrate within separate tests reporte later in this paper. During the initial tests the following test scenario was applie: - initial interval (t<.5s) for setting flu reference for MTPA controllers ( =.5Wb); an for setting a rate ecitation ( =.Wb) for IFOC strategy - starting at t=.5s, an incremental series of torue reference steps was applie as shown in Fig.(a); the increments are 7Nm each with the uration.5s such that at t=6.55s the torue reference reaches the machine rate value of 35Nm - at t=7.75s the torue reference is reuce to zero. It shoul be note that in this eperiment the machine rotor spee, when processing of the torue tracking reference, has been stabilize at ra/s with the spee-controlle loa machine. For each strategy uner comparison, the torue tracking error was measure using ata from the torue sensor. The results are presente in Fig.. It can be conclue from Fig. (b), that the IFOC algorithm provies asymptotic torue tracking. Uner MTPAL control (Fig. (c)) asymptotic torue tracking is only achieve when the rotor flu is close to its rate value. As the flu reuces, the flu estimation error increases (ue to saturation) therefore an error in calculate i current appears leaing to a torue tracking error. For torue reference 7 Nm the error is approimately 5% t,s.5.5 a) Torue Reference, Nm c) Torue Tracking Error, Nm MTAL t,s t,s Fig.. Torue tracking: (a) torue reference trajectory; (b) () torue errors provie by IFOC, MTPAL an MTPAS controls The MTPAS control takes into account the effect of magnetic saturation; therefore, the torue tracking error is nearly zero in the full range of torue references, as the results in Fig.() confirm. Hence, the propose MTPAS controller provies the same accuracy as the IFOC in the whole torue regulation range. From the eperimental results in Fig. it is clear that the propose MTPAS strategy successfully compensates for the IM saturation effect. B. Torue an spee controller performances b) Torue Tracking Error, Nm IFOC ) Torue Tracking Error, Nm MTAS t,s The eperiments etaile in this Section investigate the performance of the propose MTPA strategy for meiumpower motors using a 5kW IM-base test rig. The parameters of the test rig are given in Appeni II. Torue tracking The torue tracking performance was teste for IFOC an MTPAS controls. The following scenario was applie: - initial interval (t<.5s) for establishing the minimum flu =.Wb - at t=.5s a Nm torue reference step (3% of the rate torue) is applie followe by multiple Nm steps, each 3s apart, until Nm is reache (67% of rate value) at t=3.5s, as illustrate by Fig. 5 - at t=6.5s the torue reference is reuce to zero - uring the interval.5s to 3s the torue reference is a sinusoial function with magnitue 8Nm an angular freuency.8 ra/s. This trajectory can be regare as slow in comparison to IM magnetic system ynamics. For this eperiment a DC machine was use as a system loa. The machine was set into ynamic braking moe with the ecitation wining supplie by the eternal DC voltage source an the armature connecte to the eternal resistance. This configuration provies the loa torue proportional to the rotor spee hence allows to avoi overrun uring torue trajectory processing an also to create a small loa torue for the spee tracking tests reporte in the net subsection. Torue Reference, Nm t, s Fig. 5. Torue reference trajectory The responses of the flu an torue stator current components are epicte in Fig. 6 along with the machine spee. Since the spee trajectories of the IFOC (Fig. 6(a)) an MTPAS controlle (Fig. 6(b)) systems are ientical, it can be conclue that the evelope torue is eual for both strategies while stator current behaviour is ifferent. The stator current magnitue, torue per Ampere ratio, machine active power an cumulative consume energy for both strategies are compare in Fig. 7. a) Stator - an -ais currents, A I t, s c) Stator - an -ais currents, A 8 I 6 I t, s I t, s Fig. 6. Stator currents an spee behaviour uring torue trajectory tracking: IFOC (top), an MTPAS (bottom) The results in Figs 6 an 7 clearly emonstrate the avantages of MTPAS control with respect to stanar IFOC with constant ecitation. The MTPAS controller not only provies a higher torue per Amp ratio but also reuces active power consumption hence improving the rive energy efficiency. During the test scenario etaile above, the IFOC-base b) Rotor spee, ra/s ) Rotor spee, ra/s t, s

9 system consume 7kJ of energy while the MTPAS-base one consume only.kj, i.e. a 6.5% improvement is achieve. Spee tracking This eperiment investigate the performance of the propose torue controller within an outer spee control loop (5). The spee controller parameters are set as follows: represents one of the most avance solutions to the MPTA control problem. Both eperiments were performe with.kw inuction motor with parameters given in Appeni III. Parameters of this motor are similar to one use in [9]. The following test scenario was applie: - the machine is preliminary eite with minimum flu =.5 Wb; Stator - an -ais currents, A i i IFOC Estimate Rotor Flu, Wb IFOC t, s 6 8 6t, s Fig. 7. Transients uring torue tracking in the 5 kw IM traction rive kω=6, kωi=9, τ=.s. The eperiment was conucte using the following scenario: - uring the initial phase (t<.5s) the motor is operate at very low spee (.5ra/s) in orer to avoi the effect of Coulomb friction - at t=.5s the spee reference trajectory shown in Fig. 8(a) was applie. The maimum IM torue uring this test was limite to 3% of its rate value. In this region the propose MTPA optimization was epecte to emonstrate an appreciable effect. The measure spee tracking error is nearly zero for both controllers, as reporte in Fig. 8(b) hence the torue trajectories are regare ientical. Corresponing transients of stator current an rotor flu (estimate) are epicte in Fig.9. It can clearly be seen from Fig.9 that the flu current component in the MTPA system is significantly reuce in comparison to the IFOC system. As a result the stator current magnitue shown in Fig.8(c) is much smaller for the same torue profile t, s - 6 a)spee Reference, ra/s b)spee Tracking error, ra/s t, s c)stator Current, A IFOC MTPAS t, s Fig. 8. Spee trajectory tracking by 5kW IM traction rive C. MTPA controllers with flu regulation This test was conucte in orer to emonstrate the performance of MTPA control with close-loop flu regulation (MTPASF) esigne in Section IV, with the flu controller (). This strategy has also been compare against the performance elivere by a DFVC system [9] which t, s Fig. 9. Stator currents an estimate flu behaviour uring spee trajectory tracking: IFOC (top) an MTPAS (bottom) - then the torue reference trajectory shown in Fig. was applie; torue reference starts from zero an reaches Nm (6% of the rate value) in. s; at time t=. s there is a reference reversal to - Nm uring. s; an at t=.5 s the torue reference is reuce to zero. During the eperiment the rotor spee was stabilize with the loa machine at ra/s. The applie test conitions are similar to those in [9], but with smaller critical value of =.5Wb. The selecte torue reference trajectory has been chosen in orer to compare the system ynamics provie by the algorithms, incluing torue zero-crossing, as well as operation in steay-state. Both algorithms have been teste using the same tunings for i current controllers (k ip=7, k ii=5) an for flu controllers (k ψp=, k ψi=). The proportional gain of current controller in MTPASF was set to k ip=7. Uner such tuning the compare algorithms have similar ynamic of flu an i -current control loops Stator - an -ais currents, A i i MTPAS Torue reference, Nm 6 8 6t, s t, s Fig.. Torue reference for DFVC an MTPASF Fig. compares the transient behaviour of the DFVC in Fig.a, as well as MTPASF with filtere static flu reference computation (8), (3) in Fig.b, an with ynamic flu reference as (3) in Fig.c. It can be conclue from Fig., that the stator current magnitue is the same for all compare algorithms in steay state. During ynamics, the DFVC algorithm (Fig. a) provies faster flu changes resulting in significant stator current spikes, in particular at zero torue crossings, when the flu is reuce to its minimum value. In MTPASF with static flu reference formulation the esirable flu ynamic is efine by filter (3) coefficients hence can be ajuste as reuire. For eample, the results Estimate Rotor Flu, Wb MTPAS

10 Stator Flu Reference, Wb t, s Stator - an ais current, A i 5 i Rotor Flu Reference, Wb t, s Stator - an ais current, A i 5 i Rotor Flu Reference, Wb t, s Stator - an ais current, A i 5 i t, s t, s t, s Stator Current Magnitue, A Stator Current Magnitue, A Stator Current Magnitue, A Steay state level 6 Steay state level 6 Steay state level t, s t, s t, s a) b) c) Fig.. Transients uring tracking of smooth torue reference: a) DFVC; b) MTPASF with filtere static flu reference; c) MTPASF with ynamic flu reference shown in Fig.b are taken with k =3, an k =5 proviing natural freuency eual to 65 ra/s. If the filter natural freuency is increase to 5 ra/s (k =5 an k =565), MTPASF provies the control ynamics similar to DFVC shown in Fig.a. For MPTASF controller with ynamic flu reference (3) the transients are shown in Fig.c from which it can be seen that smaller current magnitue is reuire to track the same torue trajectory. It shoul be note that there are no current surges ue the reuce rate of flu changes. In aition, the effect of stator current increase uring tracking of sinusoial torue trajectories T =T msin(πft) is illustrate by eperimentally-taken freuency responses shown in Fig.. Maimum (peak) stator current as a function of the reference freuency is shown in Fig.a, an the stator current meansuare value over the reference perio - in Fig.b. The characteristics were measure for two reference magnitues: T m=nm (.7pu) и T m=8nm (.53pu), an the current limitation has been set to pu. As it is clearly seen, torue tracking of sinusoial references using MTPASF with flu reference (3) reuires smaller stator current at higher reference freuencies. Note that further increase of torue reference freuency for MPTASF results in flu reference oscillating aroun the constant value with the small magnitue. Hence, this mean approaching the constant-flu operation. The freuency at which such uasi-constant-flu operation can be assume is efine by the freuency response of the non-linear filter (3) hence epens on the machine parameters an on the torue reference magnitue ( that epens on T in the right-han sie enominator). As it was foun for the employe machine, at Hz torue reference the flu reference was oscillating within.5% of its steay-state value this is small enough to consier this operational moe as a constant-flu moe. 5 Tm=8 Nm 5 Peak current, A Tm= Nm a) 8 6 Tm=8 Nm Tm= Nm DFVC MTPASF Stator current, RMS, A DFVC MTPASF Current limitation level Rate current RMS f, Hz f, Hz b) Fig.. Freuency responses of DFVC an MPTASF. VIII. CONCLUSIONS In this paper a novel MTPA fiel-oriente control algorithm for IM rives base on the output-feeback linearising techniue has been esigne an eperimentally verifie. The nonlinear controller guarantees asymptotic torue (spee) tracking of smooth reference trajectories an maimises the torue per Ampere ratio when the machine is operating with constant or slow varying torue. The two torue-flu controllers base on inirect an irect fiel orientation employing a reuce-orer flu observer are esigne. The torue-flu controllers an MTPA criterion take into consieration the effect of magnetic saturation in orer to provie an improvement of the torue-flu tracking accuracy uring the whole range of machine torues. Since the maimisation of torue per Ampere ratio is similar to the criterion of active losses minimisation, the machine efficiency at light loas is improve. The eternal spee tracking controller for the torue control system is esigne an presente as well. The methoology of the flu reference calculation as static or ynamic function of the reuire torue is given, which allows to achieve MTPA optimisation in

11 steay state, guarantees singularity free operation with small initial ecitation for zero torue an can be use as mean to improve current transients. An intensive eperimental stuy of the propose solution, an comparison against stanar irect fiel-oriente control with constant flu operation, has prove that for the same torue an spee transient performances a significant stator current reuction is achieve in uasi steay-state operation. The effectiveness of the main flu saturation compensation is emonstrate eperimentally. For faster torue references, eperiments have prove that by appropriate flu reference selection one can achieve satisfactory stator current transients even for small initial flu ecitation. The propose IM rive control system is an attractive solution for technological applications where fast ynamic response is not reuire, for eample in electric traction rives. APPENDIX I The first test rig was a Rapi Prototyping Station (RPS) as shown in Fig. A. The RPS was base on a 5.5kW inuction motor, epicte as IM# in Fig. A, controlle by a 38V/5A PWM-inverter. The IM was mechanically couple to a vector-controlle loa machine. The Inverter switching freuency was.5 khz. The motor spee an torue were measure by a 5ppr optical encoer an a torue sensor (Lorenz Messtechnik DR-). The sampling time was s. IM # technical characteristics: Pn=5.5kW, In=A, Vn=38V, fn=5hz, ωn=5 ra/s, R=.9Ω, R=.65Ω, Lσ=Lσ=.6H, Lm=.7H. Controller parameters: kip=kip=7, kii=5, =8-3, k=, k=.5 5, kψp=3, kψi=5. ~38 V PC R C + Voltage sensor PWM CS CS IM # is is ω Torue sensor Loaing IM Encoer TMS3F8335 base DSP-controller (3 bit, floating point) IM vector control system Fig. A. Eperimental setup of electromechanical system with testing an loaing inuction motors APPENDIX II The secon test rig was base on a 5kW inuction motor powere by an inustrial traction inverter an mechanically couple to a DC machine loa. The Inverter switching freuency was.5khz. The motor spee was measure by a 5p/rev optical encoer. The sampling time was s. 5kW IM technical characteristics: Pn=5kW, In=98A, Vn=38V, fn=5hz, ωn=5 ra/s, R=.5Ω, R=.Ω, Lσ=Lσ=.5H, Lm is given by Fig.. Controllers parameters: kip=kip=5, kii=65, = -3, k=, k=.5 5. APPENDIX III.kW IM technical characteristics: Pn=.kW, In=5. A, Vn=38V, fn=5hz, ωn=5 ra/s, R=3.5Ω, R=.5Ω, Lσ=Lσ=.9H, Lm=.79 H. Loa control ~38 V REFERENCES [] M.J.Melfi, S.Evon an R.McElveen, Inuction versus permanent magnets motors, IEEE In. Applicat. Magazine, vol. 5, pp. 8-35, Nov/Dec 9. [] D.Biswas, K.Mukherjee an N.C.Kar A Novel approach towars Electrical Loss Minimization in Vector Controlle Inuction Machine Drive for EV/HEV in Proc. Transportation Electrification Conference an Epo, 8- Jun., Dearborn,, pp.-5. [3] W.Sung, J.Shin, an Y.Jeong, Energy-Efficient an Robust Control for High-Performance Inuction Motor Drive With an Application in Electric Vehicles IEEE Trans. on Vehicular Technology, vol. 6, no. 8, pp ,. [] B.K.Bose Moern power electronics an AC Drives, Upper Sale River, NJ: Prentice Hall,. [5] J. Chiason Moelling an high performance control of Electrical Motors. IEEE Press, 5. [6] G. S. Buja an M. P. Kazmierkowski, Direct torue control of PWM inverter-fe AC motors - a survey IEEE Trans. on Inustrial Electronics, vol. 5, no., pp ,. [7] H.-T. Lee, Li-Ch. Fu an H.-S.Huang Sensorless Spee Tracking of Inuction Motor With Unknown Torue Base on Maimum Power Transfer IEEE Trans. on Inustrial Electronics, vol. 9, no., pp. 9-9,. [8] A. M. Bazzi an P. T. Krein, Review of methos for real-time loss minimization in inuction machines, IEEE Trans. on Inustry Applications, vol. 6, no. 6, pp ,. [9] Y. Wang, T. Ito, R. D. Lorenz, Loss manipulation capabilities of eabeat irect torue an flu control inuction machine rives IEEE Trans. on Inustry Applications, vol. 5, no. 6, pp , 5. [] S. A. Ohano, R. Bojoi, A. Boglietti, Ş. G. Roşu, G. Griva Maimum efficiency per torue irect flu vector control of inuction motor rives IEEE Trans. Inustry Applications, vol. 5, no. 6, pp. 5, 5. [] O.Wasynchuk, S.D.Suhoff, K.A.Corsine, J.Tichenor, P.Krause, I.Hansen, L.Taylor A maimum torue per Ampere control strategy for inuction motor rives IEEE Trans. on Energy Conversion, vol.3, no., pp.63-69, 998. [] S. N. Vukosavic, an E. Levi, A Metho for Transient Torue Response Improvement in Optimum Efficiency Inuction Motor Drives IEEE Trans. on Energy Conversion, vol. 8, no., pp. 8 93, 3. [3] J-F. Stumper, A. D otlinger, an R. Kennel Loss Minimization of Inuction Machines in Dynamic Operation IEEE Trans. on Energy Conversion, vol. 8, no. 3, pp , 3. [] A.Consoli, G.Scarcella, G.Scelba Inuction Motor Sensorless Control Base on a Maimum Torue per Ampere Approach Recor of the Inustry Applications Conference, 3. 38th IAS Annual Meeting. Conference, vol. 3, pp. 5-. [5] B.Grcar, P.Cafuta, G.Stumberger, A.M.Stankovic, A.Hofer Non- Holonomy in Inuction Machine Torue Control IEEE Trans. on Control Systems Technology, vol. 9, no., pp ,. [6] M.Cacciato, A.Consoli, G.Scarcella, G.Scelba Inirect Maimum Torue per Ampere Control of Inuction Motor Drives proc. of Power Electronics an Applications 7, Aalborg, Denmark, pp. -. [7] C.Kwon an S.D.Suhoff, An improve maimum torue per amp control strategy for inuction machine rives in th Annual IEEE Applie Power Electronics Conference an Eposition, March, 5, pp [8] B.Gu, J.-Sh. Lai Control of Inuction Machine with Etene Range of Maimum Torue Capbility for Traction Drives in Proc. Energy Conversion Congress an Eposition, 7- Sep., Phoeni,, pp [9] R.Bojoi, Z.Li, S.A.Ohano, G.Griva, A.Tenconi Unifie irect-flu vector control of inuction motor rives with maimum torue per ampere operation in Proc. 3 Energy Conversion Congress an Eposition, 5-9 Sep., Denver, 3, pp [] G. Pellegrino, R. I. Bojoi, an P. Guglielmi, Unifie Direct-Flu Vector Control for AC Motor Drives, IEEE Trans. on Inustry Applications, vol. 7, no. 5, pp. 93,. [] S.Dymko, S.Peresaa, R.Leihol Torue Control of Saturate Inuction Motor with Torue per Ampere Ratio Maimization in Proc. IEEE International Conference on Intelligent Energy an Power Systems, -6 Jun., Kyiv,, pp.5-56.

12 [] Bozhko S, Dymko S, Kovbasa S, Peresaa S, eitors. MTA control for traction IM rives: Theory an eperimental results. 5 International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion an Roa Vehicles (ESARS); March 5. [3] E.Levi A Unifie Approach to Main Flu Saturation Moelling in D- Q Ais Moels of Inuction Machines IEEE Trans. on Energy Conversion, vol., no. 3, pp. 55-6, 995. [] E.Levi Impact of Cross-Saturation on Accuracy of Saturate Inuction Machine Moels IEEE Trans. on Energy Conversion, vol., no. 3, pp. -6, 997. [5] S.Peresaa an A.Tonielli, High-performance robust spee flu tracking controller for inuction motor, Int. J. Aapt. Control Signal Process., vol., pp. 77,. [6] P.V.Kokotovic, H. K. Khalil an J.O Reilly Singular Perturbation Methos in Control: Analysis an Design, Acaemic Press, 987. [7] P. V. Kokotovic, The joy of feeback: nonlinear an aaptive, IEEE Control Systems Magazine, vol.. pp. 7 7, June 99. [8] Peresaa S., Tilli A. an Tonielli A. Theoretical an eperimental comparison of inirect fiel-oriente controllers for inuction motors // IEEE Trans. on Power Electronics, vol. 8, no.. pp. 5 63, 3. [9] E. Levi, S. Vukosavic, V. Vuckovic Saturation Compensation Schemes for Vector Controlle Inuction Motor Drives in Proc. st Annual IEEE Power Electronics Specialists Conference, Jun., San Antonio 99, pp [3] S.Peresaa, A.Tilli an A.Tonielli New passivity base spee-flu tracking controllers for inuction motor in Proc. IEEE IECON Annual Conf. of the IEEE Inustrial Electronics Society, Nagoya,, pp. 99-.