Optimized Pole and Zero Placement with State Observer for LCL-type Grid-connected Inverter Mingyu Xue, Yu Zhang, Fangrui Liu, Yong Kang, Yongxian Yi State Key Laboratory of Advanced Electromagnetic Engineering and Technology (AEET) Huazhong University of Science and Technology, Wuhan, China Email:
[email protected] Abstract—Due to the attractive harmonic elimination capability, LCL filter is widely adopted in the grid-connected converters. Combined with complete state feedback control, PI current controller of the converters offers almost full controllability, resulting in super grid current waveform. By proposing an optimal pole and zero placement strategy for the PI state-space controller in this paper, the dominant poles and non-dominant poles are carefully specified to ensure system stability even in worst cases of low LCL resonant frequency and weak grid. With the proposed method, the high-order system behaves like a typical second-order system. The obtained maximum bandwidth varies with the filter size and the control burden is moderate. The computer aid design (CAD) based selection of control parameters is also provided with simplicity and adaptability. Moreover, only grid current is required as the state variable with the developed reduced-order state observer. Experimental results are provided to verify the feasibility and validity of the proposed method.
I. I NTRODUCTION The voltage source PWM grid-connected converter [1], [2] is widely used in modern industry, like distributed generation whose sources are mostly renewable energy (solar, wind etc.), rectifier, active power filter (APF) and so on. To pursue green power as well as to meet the stringent harmonic standard, the conventional L filter is replaced by the LCL filter, for its attractive harmonic elimination capability. However, this brings challenges to suppress the resonance. Diverse strategies, especially resonance damping, have been proposed for the LCL filter based grid-connected converters. The active damping methods are preferred to the passive ones [3] for their higher energy and filtering efficiency. Among the active ones, there are capacitor voltage lead-lag compensation [4], [5], capacitor voltage high-pass feedback [6], capacitor current feedback [7]–[10] and weighted-average-current feedback [11]. The capacitor current feedback is nowadays of main stream, for it is simple and effective. The LCL-type inverter is able to work well in most conditions with active damping, however, in case of low resonance frequency or weak grid with relatively large impedance, the stability will fade [12], [13]. Furthermore, condition like low voltage ride through (LVRT) put more stress on the dynamics. Thus, it is insufficient to stabilize the system only. From the pole point of view, the potential instability results from the imperfect pole placement, for lack of freedom degrees. Hence, the bandwidth and stability margin are in contradiction. State feedback methods [14]–[18] provide a promising solution. While some of them rely on linear quadratic regulate
978-1-4577-0541-0/11/$26.00 ©2011 IEEE
L1
Vdc Vi
Fig. 1.
L2
R1 I i Vcf
Cf
R2 I g Vg
Single-phase Grid-connected Inverter with LCL Filter
(LQR) method, which is somewhat empirical and obscure, the PIstatespace method [16] is more straightforward and practicable for it directly inherits the discrete zero and poles of L-type grid-connected converter as dominant ones. However, it has not taken the achievable bandwidth as well as the control burden into consideration. With fixed poles and zero, the performance, described by bandwidth and overshoot, is only related to the sampling time, whatever the filter is. High bandwidth is sometimes unsuitable, e.g., a larger size filter should have slower dynamics. The control burden should also be medium, or system will be susceptible to perturbation. Besides, the tedious symbolic expression for parameter calculation makes it difficult to be popularized. For different control strategies, i.e., the one beat delay (OBD) is compensated by state observer or feedback as an extra state variable, and for different control objects, i.e. grid current feedback or inverter current feedback, the symbolic expressions should be derived accordingly, requiring much effort. In order to solve above problems, a more general direct pole placement method of varying the natural resonance frequency (ωn ) is proposed, by which the design procedure turns more flexible and adaptive. The obtained maximum bandwidth varies with filter size and the stability margin sustains suitable. The control burden is reduced. The provided computer aid design (CAD) based parameter calculation makes design convenient. Moreover, a reduced-order state observer is developed to reduce the sensors which are used for the measurement of state variables, while the common noise and filter delay problems are also avoided. Experimental results are provided to validate the analysis and the controller design procedure. II. S YSTEM M ODELING The topology is shown in Fig. 1 where the dc link can be simplified as a battery with constant voltage.
377
A. Modeling in s-domain
Vg (k )
T
, the input Take the state vector as x = Ii Vcf Ig T vector as u = Vi Vg (Vi is the average output voltage of the inverter). The output vector is C = [ 0 0 1 ] when the grid current is controlled, or C = [ 1 0 0 ] when the inverter current is controlled. The continuous state space equation can be derived by Kirchhoff’s current law as x˙ = Ax + Bu
(1)
y = Cx where
H2 *
Vi (k )
z 1
Vi (k ) H1
⎢ A=⎣
− L11
1 −R L1
1 Cf
0
1 L2
0
0
⎤
⎡
1 L1
⎢ − C1f ⎥ ⎦,B = ⎣ 0 2 0 −R L2
0 0 − L12
y (k )
C
G Fig. 2. Presentation of single-phase grid-connected inverter with LCL filter in discrete state space
Vi * (k )
I g * (k 1) I g (k 1)
⎤ ⎥ ⎦.
GPI
State Observer
-
K
Discrete Generalized Control Plant
I g (k )
a f
xˆ(k 1)
C
Fig. 3. Control block diagram for discrete state feedback with OBD compensated
The inverter output voltage can be expressed as Vi = Vi∗ e−sTd ,
z 1I3
-
⎡
x(k )
(2)
Vi∗
and Td represent the command of normalized where inverter output voltage and the system delay. Such delay can be modeled as the sum of OBD (T , sample interval) and PWM transport delay (0.5T ).
III. D ISCRETE S TATE F EEDBACK C ONTROL The OBD in the main loop can either be compensated or be feedback as an extra state. The control strategies will be briefed accordingly. Without loss of generality, the grid current is taken as control object.
B. Modeling in z-plane In digital system, the controller only acts at the sample instant, and holds constant during the sample interval. On the other hand, though Vg varies continuously all the time, it can be assumed to be constant during the sample interval if only T is small enough. Thus the discrete state space equation can be yielded from (1) by the mean of zero-order-hold (zoh), as x(k + 1) = Gx(k) + Hu(k) y(k) = Cx(k), where
(3)
−1 G = eAT = L−1 (sI3 − A) , H=
| H2
H1
=
0
T
L−1 is the inverse Laplace transform. Eq. (2) can be discretized as ∗
The OBD in the main loop is compensated by predictive state feedback with the help of observer, as shown in Fig. 3, where are feedback by the vector the predicted states a K K K Ii Vcf Ig . GP I = KP + KI (1 − z −1 ) Kf = is the discrete PI controller. With the integer of GP I taken into account, it is clear that the system shown as Fig. 3 is of fourth-order. B. OBD feedback
G(τ )dτ B.
Vi (k) = Vi (k + 1).
A. OBD compensated
(4)
The discrete generalized control plant can be expressed as follows with the OBD being an extra state. x(k + 1) G H1 x(k) = 0 0 Vi (k + 1) Vi (k) (5) 0|3×1 H2 ∗ + Vi (k) + Vg (k) 1 0 The corresponding presentation in discrete state space is shown as Fig. 2.
The extended state vector is defined as xA = [ x Vi ]T , As shown feedback by the vector in Fig. 4, all the states are Kfb = KIi KVcf KIg KVi . Here the reduced order state observer is employed to reduce the sensors, thus Ii and Vcf are replaced by the observer output Iˆi and Vˆcf respectively. Since the OBD increases the order, the system shown as Fig. 4 is of fifth-order.
I g * (k )
GPI
-
I g (k )
State Observer
Iˆi (k ) ˆ V (k ) cf
Vi * (k )
-
Discrete Generalized Control Plant
I g (k )
K bf I (k ) g
Vi (k ) Fig. 4. Control block diagram for discrete state feedback with OBD feedback
378
1
C. Discussion
0.8π/T
0.4 0.9π/T
0.2
π/T π/T
0 −0.2
0.9π/T
0.1π/T
−0.4 0.8π/T
−0.6
−1 −3.5
0.2π/T
0.7π/T
−0.8
−3
−2.5
−2
−1.5
0.3π/T
0.6π/T 0.4π/T 0.5π/T −0.5 0 0.5
−1
1
Real Axis
(a) pole and zero map 1.4
IV. P OLE P LACEMENT
1.2
1
Amplitude
The pole placement can be accomplished either by LQR [14] or by direct pole placement. The former one sets the weighing matrix at first, usually empirically, and then obtains the control parameters with the closed-loop poles arriving at last. The stability is guaranteed, however, the main drawback is that the performance is not clearly related to the well known control issues, i.e. bandwidth and overshoot. In contrast, the latter one overturns the design procedure. It specifies the closed-loop poles at first, and then calculates the control parameters. Since the performance is highly dependent on the poles, this method is directly oriented to the final behavior. This paper is focused on the direct one.
0.6
0.2
0
This method subtly inherits the discrete poles and the PI zero of L-type converter regulated by symmetrical optimum (SO) method. Compared with other pole placement methods with integer only, the additional PI zero helps improve the performance. The desired poles and PI zero can be re-expressed as [16], (6)
where z1 and p1 are the PI zero and its damping pole respectively, and p2,3 are the dominant conjugate poles. It is noticeable that all the poles and the PI zero are constant, which makes specification straightforward. However, this risks some potential penalties as follows: 1) The ratio between bandwidth and LCL resonance frequency (ωr ): Fig. 5 shows the perfomance of the controller, indicating the performance is only adaptive to the sampling frequency, i.e., the smaller T, the faster response, or the higher bandwidth. The filter, based on which the controller are regulated, has no impact on the performance at all. However, according to the rule of thumb that the bandwidth should be less than half of ωr , this method misses this restriction. Thus in case of low resonance frequency, one may find that the bandwidth approximate the turn point of the filter unsuitably,
T
0.8
0.4
A. PIstatespace
z1 = 0.8889, p1 = 0.7822, p2,3 = 0.6080 ± j0.1412
0.1 0.3π/T 0.2 0.3 0.4 0.2π/T 0.5 0.6 0.7 0.1π/T 0.8 0.9
0.7π/T
0.6
Imaginary Axis
Case OBD compensated can achieve higher bandwidth. However, the output current quality highly depends on the accurateness of the predictor, for the feedback in the outerloop is predicted rather than sampled. That is, even zero track error is obtained in the outer-loop, the output current may still be distorted. In contrast, when OBD is feedback, the error indicates the tracking performance definitely. The observer can either be the Luenberger’s or be Kalman filter. Although the latter one serves to filter white noise, it will affect the main loop design while the former one does not, which is determined by the character of separate pole design. Since the hardware uncertainty, data processing error and the deviation between observer output and the real one caused by noise filtering (especially for Kalman filter) are almost inevitable, the OBD feedback is recommended here.
0.5π/T 0.6π/T 0.4π/T
0.8
0
1
2
3
4
5
6
7 −3
Time (sec)
x 10
(b) step response versus T Fig. 5. Performance of the PIstatespace method controlled system: OBD feedback and grid current the control object
which should be avoided. Otherwise, if the bandwidth is not sufficiently lower than ωr , the average model will be invalid [17], resulting the performance shown in Fig. 5 not reliable. 2) The burden on the controller: The control effort, which is representd by Vi∗ , is usually incorporated in the optimal function when LQR is employed [14]. However, it is not taken into consideration in the PIstatespace method, which means the controller may be underberden or overburden. To help understand this phenomena, a function evaluating the control effort is defined as T
1
Q = (KP 2 + KI 2 + Kf Kf ) 2
(7)
Thus the control effort versus AC capacitor with fixed inductors is shown as Fig. 7. One can see that the controller burden increases sharply when Cf surpasses point A, leading to overburden finally.
379
1.4
PIstatespace
1.2
Amplitude
&RQWUROHIIRUW 4
1
Cf
0.8
0.6
B
0.4
The proposed method
0.2
A
0
0
0.2
0.4
0.8
1
1.2
1.4
1.6
1.8 −3
Time (sec)
x 10
(a) step response versus Cf
&P )
Fig. 6.
0.6
1.4
Control effort versus AC capacitor with fixed inductors
1.2
B. The proposed method of varying ωn
(−ζ±j
p2,3 = e
√
T
0.8
0.6
0.4
1−ζ 2 )ωn T
z1 = 1 − 0.15
Amplitude
1
To solve above problems, a more general direct pole placement method of varying ωn is proposed here. The PI zero and its damping pole are still used to keep the merits brought by the proportional part of PI. All the poles and PI zero are specified by
0.2
2π [1 − real(p2 )] ωn T
(8) 0
p1 = 0.9z1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (sec)
where ζ is the damping ratio, usually set as 0.707. ωn is set as min(0.5ωr , 0.1 · 2πfs ) , considering the frequency region of the valid average model. However, one should keep p2,3 in the right half panel of z-plane. Eq. (8) is empirically derived from the observed law that z1 should lie between p2,3 and point z = 1, and move in the same way as p2,3 do toward the point z = 1 or z = 0, but at smaller speed, in order to sustain suitable overshoot. The non-dominant poles should be away from the dominant ones as far as possible. Unlike the continuous pole placement whose control burden increases sharply with the absolute value of the non-dominant poles, the discrete pole placement does not encounter this problem. Thus one can put the nondominant poles at the origin of the z-plane. It can be seen that with the proposed method, the bandwidth varies with the AC capacitor, obeying the principle that larger size filter has slower dynamics. Control burden is also significantly reduced as shown in Fig. 7, promising the system more robust to perturbation. PIstatespace method, however, has fixed product of ωn and T with regard to (8), indicating it well adaptive to T rather than the filter.
5 −3
x 10
(b) step response versus T Fig. 7. Performance of the proposed method controlled system: OBD feedback and grid current the control object
C. The Proposed CAD based Parameter Calculation Since the parameter calculation in [16] is tedious, this part is devoted to simplifying and improving the procedure by using CAD. The key is to reform the model including the controller to be standard controllable. The case OBD feedback is taken for example, so (5) is regarded to. The further extended matrixes incorporating the PI controller are given as ⎡ ⎤ G A | HA 0|4×1 ⎢ − − − ⎥ ˆ ˆ G(k) = ⎣ (9) ⎦,H = 1 0 | 0 and the pole vector is given as P = [ p1 p2 p3 p4 p5 ]. ˆ as the feedback gain vector for pole placement Note K for standard controllable expressed system, one can find ˆ = acker(G, ˆ H, ˆ P ), where acker() is the Ackermann K
380
TABLE I S ETUP PARAMETERS Symbol
Quantity
Po
Rated Output Power
Value 1 kW
VDC
DC Bus
360 V
Vg fg
Grid voltage Grid frequency
220 Vrms 50 Hz
Li Lg Cf
Inverter-side Inductor Grid-side Inductor AC Filter Capacitor
1 mH 1 mH 4.4 μF
fc fs
Switching Frequency Sampling Frequency
10 kHz 20 kHz
(a) Ii
(b) Vcf
Fig. 8. Experimental results of the reduced-order state observer: the upper is sampled and the lower is observed
function embedded in MATLAB. The control parameters are subsequently obtained by ˆ + 0|1×4 1 )· Kfb KI = (K −1 HA GA − I4 (1 + α)CA GA − CA α (1 + α)CA HA (10) and KP = αKI , where α = z1 /(1 + z1 ), CA = [ C 0 ]. The above parameter selection rule is adaptive, i.e., if the inverter current is controlled, only C,P and α should be revised; if OBD is compensated, GA , HA should be replaced by G, H1 respectively. For more detail of the deduction, one is referred to [19]. V. E XPERIMENT A prototype with parameters listed in Table I is built to verify the proposed method. Unipolar sinusoidal pulse width modulation (USPWM) is employed, with sampling action taken at the bottom and top of the carrier signal. The control chip is 32-bit fixed-point 100MHz TMS320F2808. Case of OBD feedback with grid current controlled is focused on, as Fig. 4 shows. With the help of reduced order state observer, only the grid current as state is sensed. So are the grid voltage and the DC bus. Grid synchronization is realized by filtering the grid voltage with a second order generalized integer. The reduced order state observer is realized with small size code and it takes only 0.78μs to run. This one has been found easier to tune than the full order one, resulting from the fewer couplings between the variables. Another merit is the system gets more robust to noise as well as signal filtering delay which is usually a problem when real-time sampling is employed. Fig. 8 shows the performance of the observer. To suppress the disturbance from the grid voltage to the grid current, PR+HC is employed instead of the PI controller, with parameters inherited from the above one. Gcon (s) = KP + Ki +
Ki 3
s2
2ζ ωg s + 2ζ ωg s + ωg 2 2ζ nωg s
(11) 2
n=3,5,7,11
s2 + 2ζ nωg s + (nωg )
Fig. 9. Experimental results of the LCL-type grid-connected inverter controlled by the proposed method: from up to low are Vg , Ii and Ig respectively. Ig =1.4Arms.
where Ki = KI /T , ωg = 2πfg , ζ is the damping ratio of the quadratic generalized integer. The result is shown in Fig. 9. The THD of grid voltage and grid current are 3.82% and 1.49% respectively. VI. C ONCLUSION State feedback control offers good steady and dynamic performance for the LCL-type grid-connected inverter. The typical state space control strategies are briefed, according to whether the OBD is compensated or feedback. OBD feedback is recommended, with regard to accurateness and ease. The publicized PIstatespace method is not adaptive to the filter, indicating possible unsuitable bandwidth as well as control burden. Besides, the procedure of control parameter calculation is tedious. A method of varying ωn is provided with empirical guideline to solve above problems. Not only the adaptability is well improved, but also the control burden is significantly reduced, promising robust behavior for the
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inverter. Besides, the parameter calculation is simplified with CAD. Experimental results are provided to valid the proposed method, taking example of case OBD feedback, where reduced order state observer is employed, and grid current is taken as control object. ACKNOWLEDGMENT This work was supported by the Natural Science Foundation of China under Award 51007026. R EFERENCES [1] R. Rodrłguez, W. Dixon, R. Espinoza, J. Pontt, and P. Lezana, “PWM regenerative rectifiers: State of the art,” IEEE Trans. Ind. Electron., vol. 52, no. 1, pp. 5–22, Feb. 2005. [2] F. Blaabjerg, R. Teodorescu, and M. Liserre, “Overview of control and grid synchronization for distributed power generation systems,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1398–1409, Oct. 2006. [3] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCLfilter-based three-phase active rectifier,” IEEE Trans. Ind. Appl., vol. 41, no. 5, pp. 1281–1291, Sep./Oct. 2005. [4] M. Liserre, R. Teodorescu, and F. Blaabjerg, “Stability of photovoltaic and wind turbine grid-gonnected inverters for a large set of grid impedance values,” IEEE Trans. Power Electron., vol. 21, no. 1, pp. 263–272, Jan. 2006. [5] V. Blasko and V. Kaura, “A novel control to actively damp resonance in input LC filter of a three-phase voltage source converter,” IEEE Trans. Ind. Appl., vol. 33, no. 2, pp. 542–550, Mar./Apr. 1997. [6] M. Malinowski and S. Bernet, “A simple sensorless active damping solution for three phase PWM rectifier with LCL filter,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1876–1880, Apr. 2008. [7] F. Liu, Y. Zhou, S. Duan, J. Yin, B. Liu, and F. Liu, “Parameter design of a two-current-loop controller used in a grid-connected inverter system with LCL filter,” IEEE Trans. Ind. Electron., vol. 56, no. 11, pp. 4483– 4491, Nov. 2009.
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