DESIGN OF ROBUST AND FREQUENCY ADAPTIVE CONTROLLERS FOR HARMONIC DISTURBANCE REJECTION IN A SINGLE-PHASE POWER NETWORK T. Gouraud , M. Guglielmi, F. Augery Institut de Recherche en Cybernetique de Nantes, U.M.R. No 6597, Ecole Centrale de Nantes, BP 92101, 44321 Nantes Cedex 03, France. Fax : (33) 02 40 37 25 22, e-mail :
[email protected] y GE44, Bd de l'Universite, B.P.406, 44602 Saint Nazaire cedex, France. Keywords : Vibration control, Adaptive.
Abstract This paper deals with the control of a voltage source inverter which rejects the harmonic disturbances in a power network. Two controllers are studied : the rst one uses the method of the internal model of Morari whereas the second one is based on the pole placement principle. But the closed-loop system is very sensitive to slight changes of the network frequency. As this latter one can vary in a small interval, the controllers are adapted to the fundamental frequency tracked by an extended Kalman ltering. We prove that the regulator made of the frequency estimator and the adapted controller based on the implicit pole assignment is stable. Finally, the robustness to uncertainties of parameter system and the adaptability to the frequency variations have been shown by simulations.
1 Introduction The problem of harmonic disturbance rejection arises in many technological elds and takes particular features depending on the speci c context [9, 1, 2]. Herein, we focus on the problem of the active lter control in the power networks, where the aim is to cancel the harmonic current as it is recommended in the IEEE 519 standard [11]. The main sources of harmonics are the power electronic circuits presenting a non-linear impedance to the power supply such as switch mode power converters, triac-based controllers, recti ers, dimmers. These equipments generate non-sinusoidal currents which may produce overvoltages and overheatings, and disturbances of the communication equipments. So it is necessary to reduce this phenomenon which can be solved by passive lter made of a bank of LC lters. But, these ones are not suitable to time-varying loads and they can lead to resonance phenomena at unexpected frequencies. So, since a few years,
a more ecient alternative is studied : active lter based on either voltage or current source inverter providing an injected current which reduces the harmonic requirements of the load [17]. Previous studies and solutions for active lter control do not appear to have achieved the optimum harmonic reduction [4, 17]. The sensitivity of the closed-loop behavior is one of the most important problem, as mentioned in [18, 8]. In practice, the frequency is not perfectly wellknown and may vary. This paper deals with the design of controllers which take into account variations of the network frequency. Both proposed regulators are adapted to frequency variations estimated by a non-linear observer. For the control system based on pole assignment, the stability of the closed-loop is proved. For the other one, the stability proof is not so easy and we have only run simulations which show the stability versus the frequency. Furthermore, the robustness to system uncertainties is shown for both controllers. In the present paper, section 2 is devoted to the problem statement and its modelization. Section 3 presents a controller based on the theory of the internal model of Morari whereas section 4 develops a design method deduced of the pole assignment approach. In section 5, these controllers are adapted to the fundamental frequency variations, which are determined by a real-time estimator. Finally, simulation results showing their robustness to system uncertainties and insensitivity to frequency variations are proposed in section 6.
2 Modelization Figure 1 shows the equivalent electrical model of a power system with an active lter and a non-linear load. The voltage source vs and the impedance represent the network whereas the non-linear load can be approximated by a current sink iL . The active lter uses a H-bridge inverter and a reactor impedance RF , LF . The inverters are controlled in a Pulse Width Modulation (PWM) mode
RS
DDDDDDDD
mvS 6
L S 6
iS -iL iF
vC
RF
DDDDDDDD 6
vF
d
u- G(s) = B(s)
-
A(s)
6
L F m ? m
? -
b G(s)
Kk (s)
m ? +
-
-IS
m ?
db
?
Figure 1: Con guration of the active power lter. RS = 0:250 , LS = 0:46 mH, RF = 0:123 m , LF = 9:60 mH, Figure 2: Structure of the controller based on the IMM RP = 23 m , LP = 1:2 mH, CP = 14:4 F. for a single harmonic disturbance with the storage capacitor as a voltage source. The rst order lter (RF , LF ) provides a trade-o between system bandwidth (maximum eliminated harmonic order) and reduction of switching ripple. Therefore, when the switching frequency is much higher than the harmonic frequencies being compensated, the active lter can be approximated by a voltage source vF and the reactor impedance RF , LF . So, the transfer function between the network current iS and the lter voltage vF to be controlled has the following form [8]: IS = (R + R ) +1 s (L + L ) (VF VS ) F S F S R + s L F F + (R + R ) + s (L + L ) IL F S F S B(s) = A(s) (VF VS ) + d(s) (1) As the non-linear load current is modelized by a sum of a fundamental component and harmonics, the output d of the linear lter (RF +RRSF)++ss L(LFF +LS ) is described by the following modelization : d(t) =
X
k
ak sin(2kf0 t + 'k )
6
u- G(s) = B(s)
m
6
m
6+
-
A(s)
? +
d
m
b G(s)
ppp pp K (s) K2 (s) M
IS
? +
-b ih2
b ihM
Estimation of the fundamental and harmonic disturbance
m ?
db
?
Figure 3: Structure of the controller based on the IMM for M harmonic disturbances
a control system based on the Internal Model of Morari b (IMM) [16]. Including a model transfer G(s) of the process B(s)=A(s), one can subtract the e ect of the input on the measurement Is yielding an estimation d^ of the disturbance d. First of all, we have studied the closedloop when the disturbance is just a single sinusoidal signal (d(t) = ak sin(2kf0 t)). So as to reject d, the sensitivity Hd (s) = Is =d has to be forced to zero and the (2) function controller K(s) satis es the following condition :
where the amplitudes ak and the initial phases 'k are un- Hd (j 2kf0) = 0 with Hd (s) = 1 Kk (s)Gb(s) b(s)) 1+Kk (s) (G(s) G known and considered as deterministic, whereas the funb | 2kf0 ) = 1 ) K ( j 2kf ) G( k 0 damental frequency f0 slowly varies around its nominal value (fn0 = 50Hz). Here, the aim is to design a controller that generates an Thus, we have chosen for Kk (s) a lter such as : appropriate input vF of the active lter to force iS to be Kk (s) = k 11 + ss TTk a sinusoidal current. From a didactic point of view, the k h continuous framework of the controllers will be reported i b 2kf0 )) tan Arg( G(| in the paper. The control design methodology is similar 1 with Tk = 2kf (3) for the discrete-time version. 2 0
3 Control design based on internal model of Morari
b 2kf0 ) k = G(|
1
Since the controller resulting from the IMM only depends b on the plant model G(s), the control system must be roFirst of all,the frequency f0 is considered to be con- bust to uncertainties of parameter system. The robustness stant and known. Figure 2 shows the block-diagram of to uncertainties is studied by simulations in section 6.
Without any diculty, this approach can be extended to a disturbance composed of several harmonic frequencies. The input u of the plant transfer is the sum of the harmonic components ltered by a transfer function Kk (s) ( g. 3). Thus, the estimated disturbance db is split up into its harmonic components and a transfer Kk (s) is linked to each frequency kf0 . Therefore, a harmonic estimator has to be included into the closed-loop to separate db into its harmonic component parts (bih2 ; ; bihM ). Figure 3 shows the block-diagram of this control system. To estimate the harmonic components of d,b the amplitudes, the initial phases and the fundamental frequency have to be estimated and tracked. Although it is a usual problem, its inherent non-linearity, due to frequency variations, makes it tricky. There exists several techniques to solve the estimation problem [10], but they are not suitable to real-time applications. Only handful techniques tackle to the tracking problem. Some of them are extensions of estimation techniques [15], whereas others are devoted the tracking problem such as the recursive maximum likelihood estimator or the Extended Kalman Filter (EKF) [7, 9]. Recently, sucient conditions for the stability of the EKF estimator have been developed in [14]. These conditions ensure that the estimation error can be maintained within guaranted bounds. These design considerations are required to prevent lter divergence. Therefore, the EKF observer has been chosen to track the amplitudes, the phases and the fundamental frequency of d.b
4 Implicit pole placement control design Iref -
T (s)
-
i-
6
d
S
1
(s)
R(s)
u B(s) A(s)
-
i-S + ?
I
?
Figure 4: Block diagram of pole assignment controller An other approach than the internal model of Morari is to estimate the unknown disturbance d and to include that estimate in the control law. Therefore, the design of this controller requires a model of the disturbance, whereas the principle of the internal model of Morari needs a model of the plant. The general linear structure for disturbance rejection may be represented by : T(s) I U = R(s) I + s S(s) S(s) reg with deg(S) deg(R); deg(S) deg(T ) and the coecient of the highest power in S is unity. The block diagram
of the closed-loop is shown on gure 4 which is described by the following relation: Is = B(s)T(s) (4) D(s) Ireg + Hd (s)d D(s) = A(s)S(s) + B(s)R(s) Hd(s) = A(s)S(s) D(s) (5)
Disturbances are handled by introducing additional speci cations on the admissible control law RST. Therefore, to reject harmonic disturbances without changing the fundamental frequency, R,S and T may be chosen so that Hd (s) is equal to 0 for the harmonic and fundamental frequencies when the reference input is set to the fundamental component of the load current. Moreover, the error between the reference input and the measurement must be equal to 0. But these conditions leads jHd (|2f)j to be greater than 1 for the other frequencies (Bode-Freudenberg theorem [5]). Therefore, in order not to increase the frequencies which are not rejected, the control targets have to be reduced : the sensitivity function is approximately set to 1 for the fundamental frequency and 0 for harmonic frequencies. This may be achieved by requiring that R, S and T have the following form : Hd (|2k f0 ) 0; =) S(s) = S (s) S 0 (s) 2 2 2 2 Y 2 k f0 (6) with S (s) = s + 2 rrk2 s++4rk2k+2 f4 2 k1 e = Is Ireg = 0
k
0
(7) =) D(s) B(s)T(s) = s + (2f0 ) E(s) (8) where the rk parameters have been chosen to manage the trade-o between the attenuation of harmonics and the increase of the other frequencies. As the amplitude of harmonics decreases with the order, high order harmonic could be less attenuate than the lower ones. Therefore, we advise to choose rk as a decreasing function of harmonic order. The remaining polynomials R, S 0 and T may be determined by several methods such as H1 or H2 [4, 18]. Herein, we propose a methodology based on the pole assignment method. When comparing the polynomial approach and the solution obtained by a feedback and a state observer, it is possible to split up the closed-loop characteristic polynomial D into two polynomials P C (D = P C). P is the feedback polynomial deduced from poles of the system in open-loop, whereas C is the polynomial of the state observer. The observed states are ones of the system model and ones of the harmonic disturbances and the fundamental frequency. Therefore, C is obtained from the poles of the plant transfer B=A augmented by zeros of the characteristic polynomial of harmonics (S ) and the fundamental frequency(Rf0 ). As advised in [3], R and S 0 can be computed from poles of the observer and the feedback and from two parameters Tc and Tf giving respectively the dynamic of the feedback and the observer. This methodology is known to lead 2
2
to a controller being robust to uncertainties of parameter system. Like R and S 0 , the polynomial T is the solution of a Diophantine equation (8). This one takes into account that the reference input is a sinusoidal signal with the frequency f0 . In order to reject the harmonic disturbance, we have not augmented the model of the plant (A=B) with the model of the oscillating disturbances. But, the model of the exogenous signal is included in the observer and the constraint of the harmonic disturbance rejection is set by the factorization of S. Therefore, we explicitly apply the internal model principle of Wonham, which leads to a stable regulator.
5 Adaptability to frequency variations Both controllers are very sensitive to fundamental frequency variations, which have to be taken into account for the design. Since the polynomials R(s), S(s) and T(s) (6) (8) are functions of the fundamental frequency so as the transfer function Kk (s) (3), both control laws can be adapted to frequency variations. For the controller based on the internal model of Morari, the analytic relation linking the parameters k , Tk of the lter Kk (s) and the fundamental frequency f0 have been shown by equation (3). Thus, thanks to this non-linear function, it is easy to adapt the system control to the fundamental frequency variations, estimated by an harmonic observer such as the Extended Kalman lter described in section 3 [7]. For the controller based on the implicit pole assignment principle, no obvious analytic relation linking the linear control law R, S, T and f0 has been produced : the polynomials R, S and T are the solution of two Diophantine equations A(s)S(s; f0 )+B(s)R(s; f0 ) = D(s; f0 ) and equation (8). Due to the large required memory to solve this problem becomes too important for more than two harmonics, we did not succeed to determine an analytic relation for a large number of harmonics by means of a computer algebra system (Mapple software). Otherwise, it is possible to solve the Diophantine equation using either polynomial [12] or matrix computation [13]. But, these methods lead to algorithms needing a large amount of computational burden and they can not be used for our real-time application. Therefore, as the frequency varies around a nominal value, we can approximate the coecients of the polynomials R(s; f0 ), S(s; f0 ) and T(s; f0 ) by a second order expansion around the nominal value of the fundamental frequency fn0 . For instance, S(s; f0 ) is approximated by : S(s; f0 ) = S(s; fn0 ) + (f0 fn0)S(s; fn0 ) + 21 (f0 fn0 )2 2S(s; fn0 )
and the fundamental frequency is estimated by the EKF estimator ( gure 5). fb0 Iref-
-
?
u -
RST
EKF
B(s) A(s)
IL d
i
+ IS - ? -
?
Figure 5: Block digram of pole assignment controller adapted to frequency variations Since the frequency estimate is independent of the controller, the stability of the estimator and the control system can be separately studied. As mentioned above, the EKF estimator is stable. Furthermore, thanks to a theorem of Gentina and Borne [6], the adapted controller is stable if there exists a polynomial M(s) with constant coecients computed from the closed-loop characteristic polynomial D(s; f0 ) such as : 8i < n mi jdi (:)j mn 1 > dn 1 with n = Deg(D(s; :)) and D(s; f0 ) = A(s)S(s; f0 ) + B(s)R(s; f0 ) X = sn + dk (f0 )sk M(s) = sn + Y
=
k
