Control of Three-Phase Voltage Source Converters with the Derivative-Free Nonlinear Kalman Filter

Transcription

1 Intell In Syst 26 2:2 DOI.7/s y ORIGINA PAPER Control of Three-Phase Voltage Source Converters with the Derivative-Free Nonlinear Kalman Filter G. Rigatos P. Siano 2 N. Zervos C. Cecati Receive: 25 October 25 / Revise: 25 December 25 / Accepte: 7 January 26 / Publishe online: 5 February 26 Springer ScienceBusiness Meia Singapore 26 Abstract The paper is concerne with proving ifferential flatness of the three-phase voltage source converter VSC moel an its resulting escription in the Brunovsky canonical form. For the linearize canonical moel of the converter a feeback controller is esigne. At a secon stage, a novel Kalman Filtering metho Derivative-free nonlinear Kalman Filtering is introuce. The propose Kalman Filter is reesigne as isturbance observer for estimating aitive input isturbances to the VSC moel. These estimate isturbance terms are finally use by a feeback controller that enables the DC output voltage track esirable setpoints. The efficiency of the propose state estimation-base control scheme is teste through simulation experiments. Keywors Differential flatness theory Observer-base control Nonlinear ynamical systems Disturbance estimator Derivative-free nonlinear Kalman Filter B G. Rigatos grigat@ieee.org P. Siano psiano@unisa.it N. Zervos nzervos@isi.gr C. Cecati c.cecati@ieee.org Unit of Inustrial Automation, Inustrial Systems Institute, Rion, 2654 Patras, Greece 2 Dept. of Inustrial Eng., University of Salerno, 8484 Fisciano, Italy Department of Information Engineering, Computer Science an Mathematics, University of Aquila, Via Vetoio, 67 Aquila, Italy Introuction The paper proposes a nonlinear control scheme for threephase voltage source converters VSCs where estimation of isturbances an variations of the loa current is performe with the use of a new nonlinear filtering approach, the socalle Derivative-free nonlinear Kalman Filter. VSCs, are three-phase filtere rectifiers, that are wiely use in the electric power gri mainly for power flow control. VSCs are the main builing blocks of power flow controllers in transmission lines. For example, VSCs are containe in unifie power flow controllers, or istribution-static synchronous compensators. VSCs enable control of the amplitue an phase angle of the AC terminal voltages. Moreover, their biirectional power flow capabilities allow VSCs to perform real an/or reactive power flow control in AC transmission lines []. The ynamic moel of VSCs is a nonlinear one an several nonlinear control methos have been propose for it. inearization roun certain operating points with the computation of Jacobian matrices an control with the use of linear feeback control methos have to cope with the approximate linearization errors [2,]. Operation range is restricte an a relatively big capacitor is neee for keeping a constant DC-voltage in presence of a varying loa. Initial nonlinear control approaches consier the representation of the VSC ynamics in the q reference frame an use PI compensators. Passivity-base control methos have been propose in [4]. Neural/fuzzy control methos for VSCs have been analyze in [5 8]. Back-stepping control approaches have been propose in [9]. State estimation-base control for VSCs has been stuie in [ 2]. The power matching moulation approach has been propose in [,4], while the virtual flux control metho has been analyze in [5,6]. Other control approaches consier input output linearization, as well as input-state linearization [7]. The latter

2 22 Intell In Syst 26 2:2 methos transform the nonlinear system into a ecouple linear one. It is also known that one can attempt transformation of the nonlinear VSC moel into the linear canonical Brunovsky form through the application of ie-algebra. With the application of ie-algebra methos it is possible to arrive at a escription of the system in the linear canonical form if the relative egree of the system is equal to the orer of the system. However, this linearization proceure requires the computation of ie erivatives partial erivatives on the vector fiels escribing the system ynamics an this can be a cumbersome computation proceure. Moreover, ifferential flatness theory has been propose for VSC control [8 2]. Differential flatness theory is currently a main irection in nonlinear ynamical systems an enables linearization for a wier class of systems than the one succeee with ie-algebra methos [2 2]. To fin out if a ynamical system is ifferentially flat, the following shoul be examine: i the existence of the so-calle flat output, i.e., a new variable which is expresse as a function of the system s state variables. The flat output an its erivatives shoul not be couple in the form of an orinary ifferential equation ODE, ii the components of the system i.e., state variables an control input shoul be expresse as functions of the flat output an its erivatives [24 27]. In certain cases the ifferential flatness theory enables transformation to a linearize form canonical Brunovsky form for which the esign of the controller becomes easier. In other cases by showing that a system is ifferentially flat, one can easily esign a reference trajectory as a function of the so-calle flat output an can fin a control law that assures tracking of this esirable trajectory [25,26]. This paper is concerne with proving ifferential flatness of the three-phase VSC moel an its resulting escription in the Brunovsky canonical form [28]. It is shown that for the linearize converter s moel it is possible to esign a feeback controller. At a secon stage, a novel Kalman Filtering metho, the Derivative-free nonlinear Kalman Filter, is propose for estimating the non-measurable elements of the state vector of the linearize system. With the reesign of the propose Kalman Filter as a isturbance observer, it becomes possible to estimate isturbance terms which are ue to variations of the loa current an to use these terms in the feeback controller. By avoiing linearization approximations, the propose filtering metho, improves the accuracy of estimation, an results in smooth control signal variations an in minimization of the tracking error of the associate control loop [29 2]. The structure of the paper is as follows: in inearization of the Converter s Moel Using ie Algebra section the ynamic moel of the VSC is analyze. Moreover, it is shown how linearization of the moel of the VSC can be performe by applying ie-algebra theory. In Differential Flatness of the Voltage Source Converter section it is proven that the moel of the VSC is a ifferentially flat one an that it can be transforme to the linear canonical form. In Kalman Filter- Base Disturbance Observer for the VSC Moel section it is shown that the Derivative-free nonlinear Kalman Filter can be reesigne in the form of a isturbance observer so as to enable estimation of isturbance terms which are ue to loa current variations. In Simulation Tests section simulation tests are provie to evaluate the performance of the propose nonlinear control scheme. Finally, in Conclusions section concluing remarks are state. inearization of the Converter s Moel Using ie Algebra Dynamic Moel of the Voltage Source Converter The VSC moel in the rotating q reference frame is given by [,2]: i Ri ωi q v V c 2 u, i q ωi Ri q v q V c 2 u 2, C DC V DC V DC R c 4 i u 4 i qu 2, where i, i q are the line currents i a, i b, i c after transformation in the q reference frame, an equivalently v,v q are the phase voltages v a,v b,v c after transformation in the q reference frame. Variable V c enotes the DC voltage output of the converter, u η an u 2 η q stan for control inputs. The line losses an the transformer conuction losses are moelle by R an the inverter switching losses are moele by R c. Moreover, v q is taken to be. In Fig. the electric circuit of the VSC is epicte. Denoting x [i, i q, V c ] T as the state vector, y [e c, i q ] T as the output an u [η,η q ] T as the input vector, the MIMO nonlinear one of the VSC is written in the state-space form ẋ f x Gxu, y hx, 2 where R x ωx 2 v f ωx R x 2, G [g g 2 ] x C D CR c x x x x 4C c 4C c,

3 Intell In Syst 26 2:2 2 Differential Flatness of the Voltage Source Converter Definition of Differentially Flat Systems Differential flatness is a structural property of a class of nonlinear systems, enoting that all system variables such as state vector elements an control inputs can be written in terms of a set of specific variables the so-calle flat outputs an their erivatives. The following nonlinear system is consiere: ẋt f xt, ut. 8 Fig. Electrical circuit of the voltage source converter h h h 2 ec i q 4 x 2 x2 2 C c 2 x2. 4 x 2 inearization of the Converter s Moel Using ie-algebra inearization of the converter s moel will be performe using ie-algebra an with the computation of the associate ie erivatives [,4]. The following state variables are efine: z h x, z 2 f h x, an z h 2 x. After intermeiate computations see Appenix: inearization of the Converter s Moel Using ie-algebra section one arrives at the following equations ż z 2, ż 2 v, ż v 2, 5 or in state-space escription ż z v ż 2 z 2, v 2 ż z z y z 2, 6 y 2 where the new control inputs are efine as z v 2 f h x g f h xu g2 f h 2 xu 2, v 2 f 2 g 22 u 2. 7 Therefore, the initial nonlinear system of the VSC is transforme into the linear canonical form. The linearize system is controllable an observable. The time is t R, the state vector is xt R n with initial conitions x x, an the input is ut R m. Next, the properties of ifferentially flat systems are given [2 27,5]. The finite imensional system of Eq. 8 can be written in the general form of an ODE, i.e., S i w, ẇ, ẅ,..., w i, i, 2,...,q. The term w is a generic notation for the system variables [these variables are for instance the elements of the system s state vector xt an the elements of the control input ut] while w i, i, 2,...,q are the associate erivatives. Such a system is ifferentially flat if there are m functions y y,...,y m of the system variables an of their time-erivatives, i.e., y i φw, ẇ, ẅ,...,w α i, i,...,m satisfying the following two conitions [2 28,5]: There oes not exist any ifferential relation of the form Ry, ẏ,...,y β which implies that the erivatives of the flat output are not couple in the sense of an ODE, or equivalently it can be sai that the flat output is ifferentially inepenent. 2 All system variables i.e., the elements of the system s state vector w an the control input can be expresse using only the flat output y an its time erivatives w i ψ i y, ẏ,...,y γ i, i,...,s. An equivalent efinition of ifferentially flat systems is as follows: Definition: The system ẋ f x, u, x R n, u R m is ifferentially flat if there exist relations h: R n R m r R m, φ: R m r R n an ψ: R m r R m, 9 such that y h x, u, u,...,u r, x φ y, ẏ,...,y r, an u ψ y, ẏ,...,y r, y r.

4 24 Intell In Syst 26 2:2 This means that all system ynamics can be expresse as a function of the flat output an its erivatives, therefore the state vector an the control input can be written as xt φ ut ψ yt, ẏt,...,y r t yt, ẏt,...,y r t, an. It is note that for linear systems the property of ifferential flatness is equivalent to that of controllability. ẏ f x 2 R 2 x2 2 2 v x x 2 R, c 7 y f2 x 2. 8 Solving Eq. 7 with respect to x 2 one obtains x 2 R c [ 2 Rx2 2 Ry2f2 2 ] v x ẏ f. 9 By substituting Eq. 9 into Eq. 2 one gets Differential Flatness of the Voltage Source Converter s Moel It will be shown that the ynamic moel of the VSC is a ifferentially flat one, i.e., it hols that all state variables an its control inputs can be written as functions of the flat outputs an their erivatives. Moreover, it will be shown that by expressing the elements of the state vector as functions of the flat outputs an their erivatives one obtains a transformation of the VSC moel into the linear canonical Brunovsky form. The following flat outputs are efine y f h x C c 2 2 x2, 2 y f2 h 2 i q. It hols that ẏ f 2 R x 2 x2 2 2 v x R c x 2, 4 while eriving once more with respect to time one gets ÿ f R2 x 2 2 x2 2 R v x 2x2 C c Rc 2 R x x 4 v x x x 2R c C c Rx2 x x 2x 2R c C c Moreover, ẏ f2 u u 2. 5 ωx R x 2 x u 2. 6 It can be confirme that all state variables an the control inputs of the VSC moel can be written as functions of the flat outputs y f, y f2 an of their erivatives. It hols that y f2 x 2 i q. Moreover, using the efinition of the flat outputs it hols y f 4 x2 4 y2 f 2 C c 2 R c [ 2 Rx2 2 Ry22 2 ] v x ẏ f Cc R c R x 2 4 C cr c v x 4 4 y2 f 2 C cr c R y 2 f 4 2 C cr c ẏ f y f. 2 By computing the roots of the binomial given in the above equation it becomes possible to express state variable y as a function of the flat outputs an their erivatives. Thus, by keeping the maximum of the binomial s root s one obtains x f a y f, ẏ f, y f2. 2 Using the relation for x escribe in Eq. 2 into Eq. 9 one has [ x 2 R c 2 Rf a y f, ẏ f, y f2 2 Ry2 f 2 2 v f a y f, ẏ f, y f2 ẏ ] x R c [ 2 Rf a y f, ẏ f, y f2 2 Ry2 f 2 2 v f a y f, ẏ f, y f2 ẏ ] 2 x f b y f, ẏ f, y f2. From the first line of the state-space escription of the system given in Eq. 2 an Eqs. 4 it hols that ẋ u R x ωx 2 v x u R x ωx 2 v ẋ x u f c y f, ẏ f, y f2. 2 Similarly, from the secon line of the state-space escription of the system given in Eq. 2 an Eqs. 4 it hols that

5 Intell In Syst 26 2:2 25 ẋ 2 ωx R x 2 x u 2 u 2 ωx R x 2 ẋ 2 x u 2 f y f, ẏ f, y f2. 22 Consequently, all state variables an the control inputs in the moel of the VSC can be written as functions of the flat outputs an their erivatives. Thus, the VSC moel is a ifferentially flat one. inearization of the Converter s Moel Using Differential Flatness Theory Using the efinitions of the flat outputs y f an y f2 for the VSC moel, an consiering the new state variables z y f, z 2 ẏ f an z y f2 one obtains ż z 2, ż 2 R2 x 2 2 x2 2 R v x 2x2 C c Rc 2 R x x 4 v x x x 2R c C c Rx2 x x 2x u 2, 2R c C c ż ωx R x 2 or equivalently ż z 2, ż 2 v, u x u 2, 2 ż v 2, 24 z v, ż v The control inputs which enable convergence to the esirable setpoints are v z k ż ż k2 v 2 ż k2 z z z z,. 27 The control law of Eq. 27 succees the following tracking error ynamics ë k ė k 2 e, ė k 2 e, 28 which results into lim t e t an lim t e t. It can be notice that the linearize moel of the converter obtaine after the application of ifferential flatness theory is the same with the one obtaine with the use of the ie erivative approach. It hols that v 2 f h x g f h xu g2 f h xu 2, v 2 f h 2 x g h 2 xu g2 h 2 xu 2, 29 or, in matrix form v 2 f h x v 2 f h 2 x g f h x g2 f h x g h 2 x g2 h 2 x v It hols that u u 2. which can be also written in state-space form ż z v ż 2 z 2, v 2 ż z z ẏ z ẏ 2 z As alreay note, the linearize system is controllable an observable. The new control inputs v i, i, 2arethe same as the ones efine in the ie algebra-base approach [see Eq. 5 in Appenix: inearization of the Converter s Moel Using ie-algebra section]. The linearize moel of the VSC can be escribe by the following two equations v 2 R2 2 x2 x2 2 R v x 2x2 C c Rc 2 ωx R x 2 R x x 4 v x x x Rx 2 x 2R c C c x 2x 2R c C c x u u 2. Accoring to the above, the VSC ynamic moel can be also written in a more compact form as ṽ f Mũ ũ M ṽ f. 2 Outlining the previous linearization approach one starts from Eqs. 2 an which provie the flat outputs of

6 26 Intell In Syst 26 2:2 the system. By successive erivations of these flat outputs with respect to time one arrives at an input output linearize equivalent escription which can be also written in the linear canonical Brunovsky form. Thus linearization of the system s ynamics is achieve an from that point on the solution of the feeback control an of the state estimation problems becomes possible. The flat outputs for a ynamical system are chosen to be functions of its state vector elements. In a large part of the relevant bibliography there is no systematic approach for selecting flat outputs, as there is no systematic way to choose a yapunov function for a control loop. However, as note in [5], recently there have been evelope methos which enable to compute analytically the flat outputs of ynamical systems is base on expressing the system in the so-calle implicit form an in computing the associate ifferential matrix [24]. To solve state estimation problems in the VSC moel the Derivative free nonlinear Kalman Filter will be introuce next. The Derivative-free nonlinear Kalman Filter stans for a new contribution to the fiel of nonlinear estimation an uner certain conitions it is proven to maintain the optimality features of the stanar Kalman Filter. By proving that the monitore system is ifferentially flat, its transformation to the linear canonical Brunovsky form becomes possible. This enables to solve both the control an the state estimation problem. The novelty of the article is primarily in the use nonlinear estimation metho, which is the Derivative-free nonlinear Kalman Filter. The filter consists of i a nonlinear transformation iffeomorphism which enables to write the state-space moel of the system into the linear canonical form, ii application of the Kalman Filter recursion to the equivalent linearize moel, an iii an inverse transformation, base again on ifferential flatness theory, which enables to obtain state estimates for the initial nonlinear moel. This filtering metho is a new an genuine result in nonlinear estimation theory which uner specific conitions retains the optimality features of the stanar Kalman Filter. Moreover, the use of the aforementione filter as a isturbances estimator is also a novel element in this research work. The filter estimates simultaneously the non-measurable state vector elements of the converters an isturbances or moelling error terms that affect its functioning. Comparing to other nonlinear estimators the new filter provies state estimates of improve accuracy minimum variance. Moreover, in [5] it has been emonstrate that this filter is computationally faster than other nonlinear filters extene an unscente Kalman Filter or particle filter. Kalman Filter-Base Disturbance Observer for the VSC Moel The simultaneous estimation of the non-measurable elements of the VSC state vector e.g., ẏ f as well as the estimation of aitive isturbance terms e.g., associate with variations of the loa current i is possible with the use of a isturbance estimator [6 4]. Next, it will be consiere that aitive input isturbances e.g., ue to loa variations affect the VSC moel. Thus, it is assume that the thir row of the state-space equations of the VSC of Eq. 2 inclues a isturbance term ẋ x C c R c x 4C D C u x 2 4C D C u 2. By escribing the system s ynamics using the ifferential flatness theory approach an using the efinition of the input v given in Eq., the isturbances effects appear as follows ÿ f v 2 x, 4 R c which means that the aitive isturbance term is now escribe by T 2 R c x. 5 The isturbance T may also comprise aitional uncertainty terms associate with the numerical values of the parameters of the VSC moel. It is assume that the aggregate ynamics of term T is escribe by its thir orer erivative T f y f, ẏ f, y f2. 6 Thus, it hols that z y f, z 2 ẏ f, z y f2, z 4 T, z 5 T an z 6 T. The ynamics of the extene state-space moel is written as ż z 2 ż 2 v T ż v 2, ż 4 z 5 ż 5 z 6 ż 6 f y f, ẏ f, y f2, 7 or in matrix form one has z à z Bṽ, ỹ C z, 8 ż z ż 2 z 2 ż z ż 4 z 4 ż 5 z 5 ż 6 z 6

7 Intell In Syst 26 2:2 27 y y 2 The associate state estimator is v v 2 f y f, ẏ f, y f2 z z 2 z z 4 z 5 z 6,. 9 ˆ z à o ˆ z B o ṽ K o ỹ C ˆ z, 4 where à o, B o, C o, 4 while the estimator s gain K o R 6 2 is obtaine from the stanar Kalman Filter recursion [4 44]. Defining as Ã, B, an C, the iscrete-time equivalents of matrices à o, B o an C o, respectively, the Derivative-free nonlinear Kalman Filter can be esigne for the aforementione representation of the system ynamics [2,]. The associate Kalman Filter-base isturbance estimator is given by Measurement upate It is of worth to give a brief comparison between global linearization an control for the VSC moel using ifferential flatness theory an ie algebra. One shoul point out that whereas necessary an sufficient conitions for the use of ie algebra-base control exist only if the system is input-to-state linearizable, for the use of flatness-base control necessary an sufficient conitions are the system to be ifferentially flat which also means input output linearizable. Obviously, input output linearizable systems are a wier class than input-to-state linearizable systems an consequently flatness-base control can be applie to a wier class of nonlinear ynamical systems. Simulation Tests To evaluate the performance of the propose nonlinear control scheme, that uses Kalman Filtering to estimate the nonmeasurable isturbances of the VSC moel, simulation experiments have been carrie out. Different DC voltage setpoints V DC have been assume. Moreover, ifferent external isturbance terms e.g., ue to loa perturbations have been consiere to affect the VSC ynamic moel. The control loop use in the VSC control is shown in Fig. 2. It can be notice that the aggregate control signal comprises the flatness-base control input an the term for the annihilation of the isturbances which is base on the estimation provie by the Kalman Filter. Several cases of VSC operation uner ifferent perturbation terms have been presente. The isturbance ynamics was completely unknown to the controller an its ientification was performe in real time by the isturbance estimator. It is shown that the Derivative-free nonlinear Kalman Filter reesigne as a isturbance observer is capable of estimating with accuracy the unknown isturbance input. [ ] K k P k C T C P k C T R, [ ] ˆ zk ẑ k K k C zk C ˆ z k, 42 Pk P k K k C P k, Time upate P k à kpkã T k Qk, ˆ z k à kˆ zk B kṽk. 4 Fig. 2 Control loop for the VSC comprising a flatness-base nonlinear controller an a Kalman Filter-base isturbances estimator. The aggregate control input comprises

8 28 Intell In Syst 26 2:2 i time sec Tm Tm est time i q.5 u time sec V c 2 u time sec a b Fig. a Control of the state variables blue line of the VSC in case of reference setpoint re line, an b estimation of isturbance input first iagram green line an variation of the control inputs secon an thir blue lines i time sec Tm Tm est time i q.5 u time sec V c 2 u time sec a b Fig. 4 a Control of the state variables blue line of the VSC in case of reference setpoint 2 re line, an b estimation of isturbance input first iagram green line an variation of the control inputs secon an thir blue lines The associate results are presente in Figs., 4, 5 an 6. In these iagrams the setpoints are rawn with the re line, the real values of the state vector elements of the converter are epicte with the blue line, while the state estimates which are provie by the Derivative-free nonlinear Kalman Filter are printe with a green line. Several reference setpoints have been efine for the VSC state variables, i.e., currents i, i q an the output voltage V c an as it can be observe from the associate iagrams, the propose control scheme resulte in fast an accurate convergence to these setpoints. The isturbance observer that was base on the Derivativefree nonlinear Kalman Filter was capable of estimating the

9 Intell In Syst 26 2:2 29 i time sec Tm Tm est time i q.5 u time sec V c 2 u time sec a b Fig. 5 a Control of the state variables blue line of the VSC in case of reference setpoint re line, an b estimation of isturbance input first iagram green line an variation of the control inputs secon an thir iagrams blue line i time sec Tm Tm est time 2 i q.5 u time sec V c 2 u time sec a b Fig. 6 a Control of the state variables of the VSC blue line in case of reference setpoint 4 re line, an b estimation of isturbance input first iagram green line an variation of the control inputs secon an thir iagrams blue line unknown an time-varying input isturbances affecting the VSC moel. The simulation experiments have confirme that the propose state estimation-base control scheme not only enables implementation of VSC control through the measurement of a small number of variables e.g., the ones appearing in the flat outputs y f an y f2 but also improves the robustness of the VSC control loop in case of isturbances. The improvement in the performance of the control loop that is ue to the use of a isturbance observer base on the Derivative-free nonlinear Kalman Filter is explaine as follows: i compensation of the isturbance terms which

10 Intell In Syst 26 2: x x t a t b Fig. 7 Convergence of the output voltage V c blue line to the reference setpoint re line a without using the isturbance observer an b when using the isturbance observer are generate by parametric uncertainty or unknown external inputs an ii more accurate estimation of the isturbance terms because the filtering proceure is base on an exact linearization of the system s ynamics an oes not introuce numerical errors as for example in the case of the extene Kalman Filter. This is shown in Fig. 7. The isturbances epicte in Figs., 4, 5 an 6 of the article represent the cumulative effects of moelling uncertainty an of external perturbations. Moelling uncertainties can be ue to parametric variations in the electric circuit of the VSC, while the external perturbations can be inuce by voltage variations in the gri e.g., voltage sags, harmonics istortions, etc.. It is shown that by using the Derivativefree nonlinear Kalman Filter as a isturbance estimator it is possible to estimate simultaneously the non-measurable state vector elements of the converter an the aggregate perturbation inputs. By knowing such isturbance inputs it is possible to compensate for them by introucing an aitional term to the feeback control input. As an outline of the previous analytical an experimental results it can be state that by proving that the VSC moel is ifferentially flat, its transformation to the linear canonical Brunovsky form becomes possible. This enables to solve both the control an the state estimation problem. The novelty of the article is primarily in the use nonlinear estimation metho, which is the Derivative-free nonlinear Kalman Filter. This filtering metho is a new an genuine result in nonlinear estimation theory which uner specific conitions retains the optimality features of the stanar Kalman Filter. Moreover, the use of the aforementione filter as a isturbances estimator is also a novel element in this research work. The filter estimates simultaneously the non-measurable state vector elements of the converter an isturbances or moelling error terms that affect its functioning. Aitionally, one can compare control base on global linearization methos an the ifferential flatness theory approach, against control base on approximate linearization. It has been confirme that flatness-base control is of improve robustness. Finally, comparing to aaptive control methos the propose flatness-base control approach is computationally simpler. Conclusions The paper has propose a nonlinear control scheme for three-phase VSCs. Estimation of isturbances terms ue to variations of the loa current has been performe with the use of a new nonlinear filtering approach, the so-calle Derivative-free nonlinear Kalman Filter. First, it was proven that the ynamic moel of the three-phase VSC is a ifferentially flat one, an this enable its escription in the Brunovsky linear canonical form. It has been shown that for the linearize converter s moel it is possible to esign a state feeback controller. At a secon stage, a novel Kalman Filtering metho, the Derivative-free nonlinear Kalman Filter, has been propose for estimating the non-measurable elements of the ynamic moel of the VSC.

11 Intell In Syst 26 2:2 It has been shown that by avoiing linearization approximations, the propose filtering metho, improves the accuracy of estimation, an results in smooth control signal variations an in minimization of the tracking error of the VSC control loop. Moreover, with the reesign of the propose Kalman Filter as a isturbance observer, it became possible to obtain estimates of isturbance terms associate with variations of the loa current. The VSC s control input was generate by incluing in the state-feeback control law an input that is base on the estimate of the isturbance terms. Simulation tests have been provie to evaluate the performance of the nonlinear control scheme. A comparison between the approach that was base on ifferential flatness theory an ie algebra-base linearization of the converter s moel has been performe. 2Rc x x R2 while it also hols 2 f h x C c R c x 2 x2 2 R v x 2x2 C c Rc 2, g f h x z 2 g z 2 g 2 z 2 g g f h x x x 2 x Rx 2 v x Rx 2 2 x x g f h x R c 4C c R x x 4 v x x x, 2R c C c Appenix: inearization of the Converter s Moel Using ie-algebra inearization of the converter s moel can be also performe using ie-algebra an with the computation of the associate ie erivatives [,4]. The following state variables are efine: z h x, z 2 f h x, an z h 2 x. Thus one gets f h x h f h f 2 h f f h x x x 2 x 2 x R x ωx 2 v 2 x 2 ωx R x 2 C c x x 2 R x 2 x2 2 C c R c z 2 f h x 2 v x R c x 2. an also g2 f h x z 2 g 2 z 2 g 22 z 2 g 2 x x 2 x g2 f h x Rx 2 v Rx 2 2 x2 x R c 4C c Rx 2x Therefore, one has ż z 2, x 2x 2R c C c. x g2 f h x ż 2 2 f h x g f h xu g2 f h xu It can be confirme that it hols ż z 2. Inee one has z h x therefore Similarly, one has ż 2 2 f h x g f h xu g2 f h 2 xu 2, 44 where 2 f h x z 2 f z 2 f 2 z 2 f 2 f x x 2 x h x Rx 2 v R x ωx 2 v Rx 2 ωx R x 2 ż z ẋ z ẋ 2 z ẋ ż x x 2 x 2 x R x ωx 2 v C c x x C c R c 2 x 2 ż 2 Rx2 ωx R x 2 2 ωx x 2 2 v x 2 Rx2 2 2 ωx x 2 R C x 2. Thus, one finally obtains ż 2 R x 2 x2 2 2 v x R c x 2. 46

12 2 Intell In Syst 26 2:2 Equation 44 can be also confirme. Inee, using that z 2 f h x 2 Rx2 x2 2 2 v x R c x 2 one obtains ż 2 z 2 ẋ z 2 ẋ 2 z 2 ẋ ż 2 x x 2 x Rx 2 v R x ωx 2 v x u Rx 2 ωx R x 2 x u 2 2Rc x x x u x 2 u 2 C c R c 4C c 4C c ż 2 R2 x 2 x2 2 R v x 2x2 C c Rc 2 R u x x 4 v x x x 2R c C c Rx2 x x 2x 2R c C c u 2, 47 which through the previous relations about 2 f h x, g f h x an g2 f h x confirms finally Eq. 44. The thir state variable is also examine, that is z h 2 x i q. It hols that f h 2 x h 2 f h 2 f 2 h 2 f x x 2 x f h 2 x f f 2 f f h 2 x ωx R x Equivalently, one gets g h 2 x h 2 g h 2 g 2 h 2 g x x 2 x g h 2 x g g 2 g g h 2 x, 49 an similarly one has g2 h 2 x h 2 g 2 h 2 g 22 h 2 g 2 x x 2 x g2 h 2 x g 2 g 22 g 2 g2 h 2 x x. 5 Therefore, it hols ż f h 2 x g h 2 xu g2 h 2 xu 2 ż ωx R x 2 u x u 2, ż ωx R x 2 x u 2. 5 Equation 5 correspons to the secon line of the statespace equations of V c an therefore confirms the relation ż f h 2 x g h 2 xu g h 2 xu Therefore one arrives at the following equations ż z 2, ż 2 2 f h x g f h xu g2 f h 2 xu 2, ż f 2 g 22 u 2, 5 where the new control inputs are efine as v 2 f h x g f h xu g2 f h 2 xu 2, v 2 f 2 g 22 u Consequently, one gets ż z 2, ż 2 v, ż v 2, 55 or in state-space escription ż z v ż 2 z 2, v 2 ż z z y z y 2 z Therefore, the initial nonlinear system of the VSC is transforme into the linear canonical form. References. Zhou, Z., Wang, C., iu, Y., Hollan, P.M., Igic, P.: oa current observer base fee-forwar DC bus voltage control for active rectifiers. Electr. Power Syst. Res. 84, Blasco, V., Kaura, V.: A new mathematical moel an control of a three-phase AC DC voltage source converter. IEEE Trans. Power Electron. 2, Tsai, M.T., Tsai, W.I.: Analysis an esign of three-phase AC-to- DC converters with high power factor an near-optimum feeforwar control. IEEE Trans. In. Electron. 46, Escobar, G., Chevreau, D., Ortega, R., Menes, E.: An aaptive passivity-base controller for a unity power factor rectifier. IEEE Trans. Control Syst. Technol. 94, Cecati, C., Dell Aquila, A., ecci, A., iserre, M.: Implementation issues of a fuzzy-logic-base three-phase active rectifier employing only voltage sensors. IEEE Trans. In. Electron. 522,

13 Intell In Syst 26 2:2 6. Cecati, C., Ciancetta, F., Siano, P.: A multilevel inverter for photovoltaic systems with fuzzy logic control. IEEE Trans. In. Electron. 572, Cecati, C., Ciancetta, F., Siano, P.: A FPGA/fuzzy logic-base multilevel inverter. In: Proceeings of ISIE 29, 8th IEEE International Symposium on Inustrial Electronics, pp Cecati, C., Dell Aquila, A., iserre, M., Ometto, A.: A fuzzy-logicbase controller for active rectifier. IEEE Trans. In. Appl. 9, Allag, A., Hammoui, M., Mimoune, S.M., Aya, M.Y., Becherif, M., Miraoui, A.: Tracking control via aaptive backstepping approach for a three-phase PWM AC DC converter. In: IEEE ISIE 27 International Symposium on Inustrial Electronics. ee, K., Jahns, T.J., ipo, T.A., Blasko, V.: New control metho incluing state observer of voltage unbalance for gri voltagesource converters. IEEE Trans. In. Electron. 576, eon, A.E., Solsona, J.S., Busaa, C., Chiacchiarini, H., Valla, M.I.: High-performance control of a three-phase voltage-source converter incluing feeforwar compensation of the estimate loa current. Energy Convers. Manag. Elsevier 5, Huerta, F., Pizzaro, D., Cobreces, S., Roriguez, F., Giron, C., Roriguez, A.: QG Servo controller for the current control of C gri-connecte voltage-source converters. IEEE Trans. In. Electron. 59, Capece, S.., Cecati, C., Rotonale, N.: A sensorless control technique for low cost AC/DC converters. In: IEEE Inustry Applications Conference 2, 8th IAS Annual Meeting, vol., pp Capece, S.., Cecati, C., Rotonale, N.: A new three-phase active rectifier base on power matching moulation. In: IEEE IECON 2, 29th Annual Conference of the Inustrial Electronics Society, vol., pp Malinkowski, M., Kazmierkowski, M.P., Hansen, S., Blaabjerg, F., Marques, G.D.: Virtual-flux-base irect power control of threephase PWM rectifiers. IEEE Trans. In. Appl. 74, Malinkowski, M., Jasinksi, M., Kazmierkowski, M.P.: Simple irect power control of three-phase PWM rectifier using spacevector moulation DPC-SVM. IEEE Trans. In. Electron. 52, Menalek, N., Al-Haa, K., Fnaiech, F., Dessaint,.A.: Nonlinear control technique to enhance ynamic performance of a shunt active power filter.iee Proc.Electr.Power Appl.54, Gensior, A., Ruolph, J., Gülner, H.: Flatness-base control of three-phase boost rectifiers. In: th European Conference on Power Electronics an Applications, EPE 25, Dresen, Germany, September Gensior, A., Sira-Ramirez, H., Ruolph, J., Gulner, H.: On some nonlinear current controllers for three-phase boost rectifiers. IEEE Trans. In. Electron. 562, Song, E., ynch, A.F., Dinavahi, V.: Experimental valiation of a flatness-base control for a voltage source converter. In: Proceeings of the 27 American Control Conference, New York, USA Ruolph, J.: Flatness Base Control of Distribute Parameter Systems. Steuerungs- un Regelungstechnik, Shaker Verlag, Aachen Sira-Ramirez, H., Agrawal, S.: Differentially Flat Systems. Marcel Dekker, New York Rigatos, G.G.: Moelling an Control for Intelligent Inustrial Systems: Aaptive Algorithms in Robotics an Inustrial Engineering. Springer, New York évine, J.: On necessary an sufficient conitions for ifferential flatness. Appl. Algebra Eng. Commun. Comput. 22, Fliess, M., Mounier, H.: Tracking control an π-freeness of infinite imensional linear systems. In: Picci, G., Gilliam, D.S. es. Dynamical Systems, Control, Coing an Computer Vision, vol. 258, pp Birkhaüser, Basel Villagra, J., Anrea-Novel, B., Mounier, H., Pengov, M.: Flatness-base vehicle steering control strategy with SDRE feeback gains tune via a sensitivity approach. IEEE Trans. Control Syst. Technol. 5, aroche, B., Martin, P., Petit, N.: Commane par platitue: Equations ifférentielles orinaires et aux erivées partielles. Ecole Nationale Supérieure es Techniques Avancées, Paris Martin, Ph., Rouchon, P.: Systèmes plats: planification et suivi es trajectoires. Journées X-UPS, École es Mines e Paris, Centre Automatique et Systèmes, Mai Bououen, S., Boutat, D., Zheng, G., Barbot, J.P., Kratz, F.: A triangular canonical form for a class of -flat nonlinear systems. Int. J. Control 842, Rigatos, G., Siano, P., Zervos, N.: PMSG sensorless control with the use of the erivative-free nonlinear Kalman filter. In: IEEE ICCEP 2, IEEE International Conference on Clean Electrical Power, Alghero, Sarinia, Italy, June 2. Rigatos, G.G.: A erivative-free Kalman filtering approach to state estimation-base control of nonlinear ynamical systems. IEEE Trans. In. Electron. 59, Rigatos, G.G.: Nonlinear Kalman filters an particle filters for integrate navigation of unmanne aerial vehicles. Robot. Auton. Syst. 6, Khalil, H.K.: Nonlinear Systems, 2n en. Prentice Hall, Upper Sale River Mahmou, M.A., Rota, H.K., Hussain, M.J.: Full-orer nonlinear observer-base excitation controller esign for interconnecte power systems via exact linearization approach. Electr. Power Energy Syst. 4, Rigatos, G.: Nonlinear Control an Filtering Approaches Using Differential Flatness Theory: Applications to Electromechanical Systems. Springer, Switzerlan Chen, W.H., Ballance, D.J., Gawthrop, P.J., Reilly, J.O.: A nonlinear isturbance observer for robotic manipulators. IEEE Trans. In. Electron. 474, Cortesao, R., Park, J., Khatib, O.: Real-time aaptive control for haptic telemanipulation with Kalman active observers. IEEE Trans. Robot. 225, Cortesao, R.: On Kalman active observers. J. Intell. Robot. Syst. 482, Gupta, A., Malley, M.K.O.: Disturbance-observer-base force estimation for haptic feeback. ASME J. Dyn. Syst. Meas. Control, Article number Miklosovic, R., Rake, A., Gao, Z.: Discrete implementation an generalization of the extene state observer. In: Proceeings of the 26 American Control Conference, Minneapolis, Minnesota, USA Harris, C., Hong, X., Gan, Q.: Aaptive Moelling, Estimation an Fusion from Data. Springer, Berlin Bassevile, M., Nikiforov, I.: Detection of Abrupt Changes: Theory an Applications. Prentice-Hall, Englewoo Cliffs Rigatos, G., Zhang, Q.: Fuzzy Moel Valiation Using the ocal Statistical Approach. Publication Interne IRISA No. 47, Rennes Kamen, E.W., Su, J.K.: Introuction to Optimal Estimation. Springer, New York 999