ThC /09/$ AACC A. El magri, F. Giri, A. Abouloifa, I. Lachkar, F.Z. Chaoui

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1 9 American ontrol onference Hyatt Regency Riverfront, t. ouis, O, UA June -, 9 Th7. NONINEAR ONTRO OF AOIATION INUDING YNHRONOU OTOR AND A/D/A ONVERTER A formal analysis of speed regulation and power factor correction A. El magri, F. Giri, A. Abouloifa, I. achkar, F.Z. haoui Abstract We are considering the problem of controlling synchronous motors driven through A/D rectifiers and D/A inverters. The control objectives are threefold: (i) forcing the motor speed to track a erence signal, (ii) regulating the D ink voltage, (iii) enforcing power factor correction (PF) with respect to the power supply net. First, a nonlinear model of the whole controlled system is developed in the Park-coordinates. Then, a nonlinear multiloop controller is synthesized using the backstepping design technique. A formal analysis based on yapunov stability and average theory is developed to describe the control system performances. In addition to closed-loop global asymptotic stability, it is proved that all control objectives (motor speed tracking, D link voltage regulation, and unitary power factor) are asymptotically achieved up to small harmonic errors (ripples).. The above results are confirmed by simulations which, besides, show that the proposed regulator is quite robust with respect to uncertain changes of load torque. P I. INTRODUTION ERAANENT magnet synchronous (P) motors are more suitable for electric traction compared with induction motors. Indeed, they possess a better mass/power ratio, develop a much higher power level and present a more satisfactory efficiency. In effect, the Joule losses in P motors are much less important as these involve no field and rotor currents. The spectacular development of power electronics technology, over the last recent years, has resulted in reliable power electronic converters which make it possible to drive synchronous machines in varying speed mode. Indeed, speed variation can only be achieved for these machines by acting on the supply net frequency. Until the recent development of modern power electronics, there was no effective solution to A machine speed control because there was no simple way to vary the net frequency. On the other hand, in the electric traction domain, the used power nets are either D or A but mono-phase. Theore, three-phase D/A inverters turn out to be the only possible interface (between railway nets and -phase A motors) due to their important capability to ensure a flexible voltage and frequency variation. The above considerations illustrate the major role of modern power electronics in the recent development of electrical traction applications. As mentioned above, a three-phase D/A inverter used in traction is supplied by a power net that can be anuscript received eptember, 8. All authors are with the GREY ab, University of aen, aen, France. orresponding author: F. Giri, fouad.giri@ unicaen.fr either D or mono-phase A. In the case of A supply, the (mono-phase) net is connected to the three-phase D/A inverter through a transformer and A/D rectifier (Fig ). The connection line between the rectifier and the inverter is called D link. The system consisting of the A/D converter, the D/AD inverter and the P motor has to be controlled to achieve varying speed erence tracking. The point is that such system behaves as a nonlinear load vis-à-vis to the A supply line. Then, undesirable current harmonics are likely to be generated in the A line. These harmonics reduce the rectifier efficiency, induce voltage distortion in the A supply line and cause electromagnetic compatibility problems. The pollution caused by the converter may be reduced resorting to additional protection equipments (transformers, condensers ) and/or over-dimensioning the converter and net elements. However, this solution is costly and may not be sufficient. To overcome this drawback, the control problem must have as objective not only motor speed control but also rejection of current harmonics. The last objective is erred to power factor correction (PF), []. Previous works on synchronous machine speed control simplified the control problem neglecting the dynamics of the A/D rectifier and so making the focus only on the set D/A inverter - otor. A wide range of control solutions have thus been proposed. These involved as well simple techniques such as field-oriented control (FO) [7] and N techniques such as feedback linearization (F) [], direct torque control (DT) [8] or sliding mode () [9]. Ignoring the A/D rectifier in the development of a control strategy, is criticized at least from two viewpoints. First, such development relies on the assumption that the D voltage provided by the A/D rectifier is perfectly regulated. The problem is that a perfect regulation of the rectifier output voltage can not be met ignoring the rectifier load which is nothing other than the set D/A inverter - otor. The second drawback of the previous control strategy lies in the entire negligence of the PF requirement. It is not judicious, from a control viewpoint, to consider separately the association inverter - otor, on one hand, and the rectifier, on the other hand. In the present work, we are developing a new multiloop control strategy that deals simultaneously with both controlled subsystems: the A/D converter and the combination D/A inverter - otor. The main feature of our control design is threefold: i. A input current loop is first designed so that the /9/$. 9 AA 7

2 coupling between the power supply net and the A/D rectifier operates with a unitary power factor; ii. A second loop is designed to regulate the output voltage of A/D rectifier so that the D link between the rectifier and inverter operates with a constant voltage; iii. A bi-variable loop is designed to enforce the motor velocity to track its varying erence value and to regulate the d-component of stator current to zero in order to optimize the absorbed stator current. All loops are designed using the backstepping technique and yapunov design, []. A theoretical analysis will prove that the four-loop controller thus described actually stabilizes (globally and asymptotically) the controlled system and does achieve its tracking objectives with a good accuracy. ore precisely, it is shown that the steadystate tracking errors corresponding to rectifier input current and rectifier output voltage, motor speed and stator current d-component are harmonic signals and their amplitudes depend on the supply net frequency: the larger the net frequency the smaller the error amplitudes. It follows in particular that the motor regulation objective and the PF requirement are actually ensured, up to harmonic errors of insignificant amplitude, provided the net frequency is large enough. This formally establishes the existence of the so-called ripples, which are usually observed in similar practical applications, and proves why this phenomenon is generally insignificant. These theoretical results are obtained making a suitable use of different automatic control tools e.g. averaging theory and yapunov stability []. The paper also includes a simulation study confirming the above theoretical results and, besides, shows that the controller compensates well to disturbing effects due load changes. The paper is organized as follows: the controlled system (including the A/D/A converter and the synchronous motor) is modeled and given a state space representation; the control objectives in ection ; the controller design and the closed-loop system analysis are presented in ection ; the controller performances and robustness are illustrated ection through numerical simulations; a conclusion and a erence list end the paper. To alleviate the paper presentation, a list of notations is given hereafter. Notation list stator winding inductance R resistance of the stator windings i d, i q d- and q- axis currents v d, v q d- and q- axis voltages ω angular velocity of the rotor p number of pole pairs T load torque J combined inertia of rotor and load f combined viscous friction of rotor and load flux motor constant K Fig.. chematic representation of single phase A supply powering -phase A motor II. ODEING THE AOIATION A/D/A ONVERTER- YNHRONOU OTOR The controlled system is illustrated by Fig. It includes an A/D boost rectifier, on one hand, and a combination D/A converter-synchronous motor on the other hand. The circuit operates according to the well known Pulse Width odulation (PW) principle. ~ ve( A. odeling the PW A/D Rectifier The transformer secondary is connected to a H-bridge converter which consists of four IGBT s with anti-parallel diodes for bidirectional power flow arrangement. This subsystem is described by the following set of differential equations: die ve = s vdc (.a) dt dv dc = sie is (.b) dt where i e is the current in inductor, v dc denotes the voltage in capacitor, i s designates the input current inverter, v e is the supply net sinusoidal voltage ( ve =. E.cos( ωe ) and s is the switch position function taking values in the discrete set {, }. pecifically: if is ON and is OFF s = if is OFF and is ON It is not suitable for control design due to the switched nature of the control input s. As a matter of fact, existing nonlinear control approaches apply to systems with continuous control inputs. Theore, control design for the above inverter will be based upon the following average version of (.a-b): dx v = e u x (.a) dt ie A-D s.ie Vdc is U U U U U U D-A P dx = u x i s (.b) dt where x = ie, x = vdc and u = s denote, respectively, the average values of i e, v dc and s over cutting periods. id, iq Fig.. A/D/A drive circuit with three-level inverter ia ib ic ω 7

3 B. odeling the combination PW D/A convertersynchronous motor uch modeling is generally performed in the d-q rotating erence frame because the components i d and i q then turn out to be D currents. According to [], the model of the synchronous motor, expressed in the d-q coordinates, is given by: dω F K T = ω iq (.a) dt J J J diq R K = iq pω id ω vq (.b) dt di d R = id pω iq vd (.c) dt The inverter d- and q-voltage can be controlled independently. To this end, these voltages are expressed in function of the corresponding control action (see e.g. []): vq = vdc u (.a) vd vdc u (.b) i s = ( u iq u id ) / (.c) where u = uq, u = ud are the average modulation indexes in the d- and q-axis, respectively. imilarly, let us introduce the state variables x = ω, x = iq, x = id. Then, substituting (a-b) in (a-c) yields the following state space representation of the combination invertersynchronous motor : dx F K T =. x x dt J J J (.a) dx R K =. x. p. x. x. x u x dt (.b) dx R =. x p. x. x u. x (.c) dt The state space equations obtained up to now constitute a state-space model of the whole system including the A/D/A converters combined with the synchronous motor: dx v = e u x (6.a) dt dx = u x ( ux u ) dt (6.b) dx F K T =. x x dt J J J (6.c) dx R K = x p xx x ux dt (6.d) dx R = x p xx ux dt (6.e) III. ONTROER DEIGN A. ontrol objectives The first control objective is to force the speed ω to track a erence signal ω. The second objective is to constrain the input current rectifier to be sinusoidal and in phase with the A supply voltage (PF). But, there are three control inputs at hand, namely u, u and u. Then, we will further seek two additional control objectives. pecifically: - controlling the continuous voltage v dc so as it tracks a given erence signal v dc (generally constant, equal to the nominal voltage entering the inverter) - regulating the current i d to a erence value i d, equal to zero in order to guarantee the absence of d- axis stator current The last requirement is explained by the fact that the developed torque is given by the relation T =. p( K iq ( d q ) idiq )/ (see e.g. [6]). Accordingly, torque control should be performed acting on both i d and i q. But, for the surface-magnet synchronous motor, the large effective airgap means that d q = i.e. i d does not really influence T and so it is sufficient to regulate it to zero. B. ontrol loop design for current i e The PF objective means that the input current rectifier should be sinusoidal and in phase with the A supply voltage. We theore seek a regulator that enforces the current x to tack a erence signal x of the form: x = k v e (7) At This point k is any positive (time-varying) parameter. Introduce the current tracking error: z = x x (8) In view of (6.a), the above error undergoes the following equation: ve =. u. x x& (9) To get a stabilizing control law for this first-order system, consider the quadratic yapunov function V =. z. It can be easily checked that the time-derivative V & is a negative definite function of z if control input is chosen to be: u = c. z ( ve / ) x& x with c () ( ) / Proposition. onsider the control subsystem (6.a) and the control law (). The erence x is assumed available and derivative. The inner closed-loop system undergoes the following equation: = c.z with c >. It is clearly seen that the error z converges exponentially fast to zero, whatever the initial conditions.. ontrol loop design for the voltage v dc The aim of the outer loop is to generate a tuning law for the ratio k so that the output voltage v dc be regulated to a given erence value v dc. ) Relation between k and x The first step in designing such a loop is to establish the relation between the ratio k (control inpu and the output > 7

4 voltage x. This is the object of the following proposition. Proposition. onsider the power converter described by (6.a-b) and the i e control loop defined by (). One has the following properties: ) The output voltage x varies, in response to the tuning ratio k, according to the equation: dx = ( k ve zve ) ( ux ux) () dt x ) The squared voltage ( y = x ) varies, in response to the tuning ratio k, according to the equation: dy k v f ( x, dt = e () where f ( x, = ( zve x( ux ux)) () Proof. ) The input power in the A/D side is expressed by Pe = x ve. The delivered power at the load (capacity and inverter) is given by P trans = u. x. x. Using the power conservation argument ( P = P ), one has: trans x. ve = u. x. x () Using () and the fact that the input current expression is x = k. ve z, yields : u x = ( k ve z ve) x, which together with (6.b) establishes () ) ets introduce the variable change y = x in (). Deriving y with respect to time and using () yields the model () and completes the proof of proposition. ) quared D-link voltage regulation The ratio k stands up as a virtual control input in the system (). The erence signal y = ˆ v of the e dc squared output capacitor voltage ( x = vdc ) is chosen to be constant, equal to the nominal input voltage of the inverter. Then, it follows from () that the tracking error z = y undergoes the following equation: z & y = E k E k cos( e f ( x, y& ω () To get a stabilizing control law for this system, consider the following quadratic yapunov function: V = (6). z It is easily checked that the time-derivative V & can be made a negative definite function of the state z by letting: ( c z f ( x, t & ) k E k E cos( ω e = ) y (7) where c > is a design parameter. An approximate simple solution is: k = ( c z f ( x, y& )/ E (8) In view of such choice, it follows from (8), (7) and () that z undergoes the differential equation: = cz k E cos(ωe (9) Remark. The signal k is treated by a pilter to obtain its derivative signal (then the time-derivative of x is available). D. ontrol loop design for motor speed ω A control law for the remaining (actual) control input, namely u, will now be determined based on equations (6c-d) in order to guarantee speed erence tracking. To this end, let z denote the speed tracking error: z = x ω () In view of (6.c), the above error undergoes the following equation: F K T =. x. x ω& () J J J In (), the quantity α = ( K / J ). x stands up as a (virtual) control input for the z -dynamics. et α denote the desired trajectory (yet to be determined) of α. It is easily seen from () that if α = α with: F T α = c z x &. ω () J J Then one would get = cz with c > is a design parameter. This would clearly ensures asymptotic stability of () with respect the yapunov function: V =. z () In effect, the time derivative of V would then be: V & = z = cz < () As α = ( K / J ). x, is a virtual control input, one can not set α = α. Nevertheless, the above expression of the desired trajectory is retained and a new error is introduced: z = α α () Using ()-(), it follows from () that the z - dynamics undergoes the following equation: z & = cz z (6) The next step consists in determining the control input u so that the errors ( z, z ) vanish asymptotically. The trajectory of the error z is obtained by operating a timederivation on (), that is: = ( K / J ). x& α& (7) Using () and (6c-d) in (7) yields: K = β ( x) γ c ux (8) J where K R K F FK β ( x) =. x p. x. x. x. x. x (9) J J J FT T& γ ( = & ω () J J Then, the error equation (6) and (8) can be rewritten in a more compact form: 7

5 z & = cz z K = β ( x) γ c ux () J To determine a stabilizing control law for (), let us consider the quadratic yapunov function candidate: V = V. z () Using (6), the time derivative of V can be rewritten as: V & = cz zz z () This shows that, for the z, z -system to be globally asymptotically stable, it is sufficient to choose the control u so that V & = cz cz (with c > ). In view of (), this amounts to let: = cz z () omparing () and () yields the following backstepping control law: u = ( J/K ) ( c c ) z ( c ) z β ( x) γ x () ( ) / E. d-axis current loop design The d-axis current x undergoes equation (6.e) in which the following quantity: v = p. x. x u. x / (6) acts as a virtual input. As the erence signal i d is zero, it follows that the tracking error z = x undergoes the equation: z & = ( R / ) z v (7) To get a stabilizing control signal for this first-order system, consider the following quadratic yapunov function: V =. z (8) It can be easily checked that the time-derivative V& = cz is a negative definite function of z if the (virtual) control input is let to be: v = ( ( R / ) c ) z with c > (9) Now, it is readily observed that the actual control input u is obtained substituting (9) in (6) and solving the resulting equation. Doing so, one gets: R u = cz z p xx () x Proposition. onsider the control system consisting of the subsystem (6c-e) and the control laws () and (). The resulting closed-loop system undergoes, in the z, z z )-coordinates, the following equation: (, c z = c z () c z It is readily seen that () is globally asymptotically stable with respect to the yapunov function V =.( z z z ). As () is linear, then the error vector ( z, z, z ) converges exponentially fast to zero, whatever the initial conditions. Theorem. onsider the system including the A/D/A power converters and the synchronous motor connected in tandem, as shown in Fig.. For control design purpose, the system is represented by its average model (6a-e). et the erence signals v dc, ω and i d be selected such that v dc >, ω and i d =. onsider the controller defined by equations (), () and () where all design parameters, namely c, c, c, c and c are positive. Introduce the notations: = [ z z z z z ] T ; ε = / ω e z Then, one has the following properties: ) The resulting closed-loop system undergoes the following equation: z &( = A z( g( z, () with c A = c c c ;. k. E g( z, = c, z, z cos(ω e ) The tracking errors z and z vanish asymptotically ) et the erence signals v dc and ω be nonnegative and periodic with period Nπ / ωe, where N is any positive integer. Then exists a positive real ε and ξ such that for all < ε < ε : the tracking error z is a harmonic signal that continuously depend on ε and: z (t, ε ) < ξε onsequently: lim z ( t, ε ) = () ε Proof. In order to prove the theorem, consider the equation () and introduce the time-scale change τ = ω e t and the state variable w ( τ ) = z( which implies w& ( τ ) = ( / ωe. Then, equation () gives: w & ( τ ) = ε A w( τ ) ε g( τ, w, ε ) (a) with: g( τ, w, ε ) = ( ( / ) k E cos( τ ) ) T (b) The stability of system () will be now be analyzed with using tools from the averaging theory []. As v dc and ω are periodic with period N π / ωe and N π / ωe respectively, with N and N are any positive integer numbers. The average system is essentially obtained averaging the function g ( τ,, ) with respect to its first argument, over the interval [, π]. From (b) it is readily seen that the average value of g ( τ,, ) is precisely equal to zero. Hence, the average version of (a) is: w & ( τ ) = ε A w( τ ) (6) In order to get stability results regarding the system of interest, i.e. (a), it is sufficient to analyze the linear average system (). It is clear that matrix A is Hurwitz (all its eigenvalues have negative real parts) because the 7

6 coefficients c to c are positive. Then, the origin w = is an equilibrium point of the average system (). Now, invoking averaging theory (see e.g. Theorem. in []), we conclude that there exists a positive real constants ε and ξ such that, for all < ε < ε, () has a unique, exponentially stable, π-periodic solution w ( z, ε ) with the property w ( z, ε ) = z( t, ε ) ξε. This proves the theorem. multiloop nonlinear controller. Presented approach guarantees (Theorem) well line side power quality, controllable and stable (average) D-link voltage (v dc ) as well as, the power factor at A input mains is close to unity (PF) in the entire operating range of the drive. The convergence of the rotor speed and d-axis current, towards their erences values, is guaranteed. These results have been confirmed by a simulation study which, further, showed the robustness of controller performances with respect to load changes. REFERENE Fig.. ontrol system including A/D/A converter and a P IV. IUATION The experimental setup, described by Fig., has been simulated in atlab/imulink environment. The involved elements have the following characteristics:. upply network: ve ( =. E cos( ωe ; E=v/Hz. A/D/A converters: =mh; =.mf;. ynchronous motor: =9.mH; R=.6Ω; K =.9; J=.76Nm/rd/s², F=.89Nm/rd/s; p=. The erence values of the state variables are chosen as: v dc = V; ω steps from to rad/s at t=.s; a constant load torque of Nm is applied to the drive at t=.s and then back to Nm at t=.7s. i d =. The following values of the controller parameters turned out to be suitable: c =, c =, c =8, c =9, c =8. The controller performances are illustrated by Figs to 6. Fig shows that a unitary power factor is achieved after a transient period following each change in erence values or load torque. Figs and 6 show that the tracking quality is quite satisfactory for all controlled variables ( v dc, ω, id ). The response time is less than. s. The disturbing effect, due to load torque change, is also well compensated by the regulator. V. ONUION In this paper we have considered the problem of controlling the power electronic A/D/A converters with synchronous motor load. The system dynamics have been described by the averaged th order nonlinear statespace model (6a-e). Based on such a model, the yaponuv stability and averaging theory are used to establish the [] ingh B., G. Bhuvaneswari, and V. Garg, Improved Power Quality A-D onverter for Electric ultiple Units in Electric Traction, Power India onference, 6 IEEE, 6 pp [] Khalil H, NONINEAR YTE, Prentice Hall,. [] enthil Kumar N., aravanan K., peed control of P drive using VI, Industrial Electronic onference (IEON), Busan, outh Korea, pp Vol., [] ichael J, Ryan D and Rik W, odeling of inewave Inverters: A Geometric Approach, IEON, Proceedings of the th Annual onference of the IEEE, pp.96 - vol [] Wallmark O., On control of permanent-agnet synchronous otors in Hybrid-Electric Vehicle Application, Technical Report, chool of Electrical Engineering, halmers University of technology Goteborg, weden,. [6] uhammad H, Rashid, Power electronics handbook, Academic press,. [7]. Jasinski,. ichowlas,. P. Kazmierkowski Direct control for A/D/A converter-fed induction motor with active filtering function, ompel, The International Journal for omputation and athematics in Electrical and Electronic Engineering, Vol., pp. -, 6. [8] aleh, K.I.; ohammed, O.A.; Badr,.A. Field oriented vector control of synchronous motors with additional field winding, Energy onversion, IEEE Transaction on Vol 9, pp. 9,. [9] Yang Z.P.,.H. Wang,.. iu, Hou, Y.B. ai, Variable structure control with sliding mode for self-controlled synchronous motor drive speed regulation, IEEE International ymposium on Industrial Electronics, vol., pp 6 6, 99. [] Kuroe Y., K. Okamura, H. Nishidai, T. aruhashi, Optimal speed control of synchronous motors based on feedback linearization, International onference on Power Electronics and Variable-peed Drives, pp 8, Ie (A).Ve (V) Time (s) Fig.. Input current and voltage waveform.6.. Reference speed Rotor speed (rad/s) d-axis urrent (A) Time (s) Fig. 6. peed ω(rad/s) and d-axis current id(a) 6 Time (s) Fig.. Dynamic response of the D-link voltage:v dc 7