Linear Quadratic Optimal Control of a Single-Phase Grid-Connected Inverter With an LCL Filter Hao Tang, Rongxiang Zhao*, Shengqing Tang, Zheng Zeng College of Electrical Engineering Zhejiang University Hangzhou 310027, Zhejiang Province, P. R. China E-mail:
[email protected] Abstract—A linear quadratic optimal control strategy for a single-phase grid-connected inverter with LCL filter is presented in this paper. Firstly, the mathematical model of the gridconnected inverter is investigated. Then, its linearized small signal state-space model is obtained, and a linear quadratic regulator (LQR) is achieved by the solution of algebraic Riccati equation (ARE). Finally, some simulations have been done on the MATLAB/Simulink platform in detail, which verifies the excellent performance of the proposed approach in steady-state and transient operation conditions.
A two-current-loop control strategy, which consists of an outer grid current controller and an inner capacitor current regulator, is proposed in [5]. The problem of this strategy is the need for complex tuning procedure, which can be found in [6]. An interesting method documented in [7] can degrade the inverter control system from third-order to first-order by splitting the capacitor of the LCL-filter into two parts proportionally. Literature [8] proposed a weighted average current (WAC) feedback control strategy, which is an extension to the LCL control strategy presented in [7].
Keywords-grid-connected inverter; linear quadratic regulator (LQR); single-phase system; LCL filter.
In this paper, an LQR controller is chosen for a singlephase grid-connected inverter with an LCL-filter. The LQR control law essentially provides a multivariable proportional regulator. Based on the small signal model of the gridconnected inverter, the optimal gain matrix of the LQR controller can be obtained by solving a related algebraic Riccati equation [9]. To reduce the system cost, a capacitor voltage observer is proposed. The detailed design procedure of the LQR controller is also included in this paper. The performance of the inverter system is verified by the simulation results on the Matlab/Simulink platform.
I.
INTRODUCTION
In recent years, research interest in renewable sources has grown rapidly due to some of their advantages, such as the elimination of harmful emission and inexhaustible resources of the primary energy. The pulse width modulated (PWM) gridconnected inverter has played a very important role in the integration of renewable energy sources into the utility grid [1][3]. Traditionally, L-filters are used as the interfaces between the utility grid and the grid-connected inverters to attenuate the current harmonics caused by the pulse width modulated (PWM) method. However, the alternative LCL filters exhibit better performance in reducing switching current harmonics and has become more and more popular. To achieve the same performance of damping switching current harmonics, the inductance required in a LCL-filter based grid-connected inverter is reduced compared with L-filter. In another word, the switching frequency can be lower with the same inductance. In spite of such advantages, the grid-connected inverter with an LCL filter is a third-order system which may cause instability at the resonant frequency, so it requires more complex current control strategies [4].
II. THE STATE-SPACE MODEL Fig. 1 shows the system topology of a photovoltaic gridconnected inverter. The topology is mainly made up of five parts: a photovoltaic module, a boost converter used for the maximum power point tracking (MPPT), a single-phase fullbridge PWM inverter, a low-pass LCL-filter to connect the inverter output to the utility grid through a triac and an LQR controller to regulate the inverter. Boost Converter PV Module
Many attempts have been taken to design a better controller for grid-connected inverters with LCL filters. In [4], a passive resistor is added to the LCL filter to damp the resonance. This method is very easy to implement, but the system efficiency is reduced due to the extra power loss on the damping resistor.
978-1-4673-0158-9/12/$31.00 ©2012 IEEE
+ VPV −
Full-bridge Inverter + Vdc −
i1 +
uo −
L1 iC + uc C −
L2
i2 Triac
System measurements PWM
Vm
LQR
Fig. 1. System topology of photovoltaic grid-connected inverter.
372
us
A mathematical model describing the grid-connected PWM inverter can be writing in the following form: di L1 1 + R1i1 =uo − uc dt
(1)
di2 + R2 i2 = uc − u s dt
(2)
L2 C
duc = i1 − i2 dt
(3)
where
III.
SMALL-SIGNAL MODEL
The small-signal model can be deduced from the linearization of the large-signal model (4) around an operating point. The operating point can be regarded as the references of the state variables. In order to obtain the small-signal model, (4) can be written as: (6)
x = f ( x , uo , us )
where f ( x , uo , us ) = Ax + B1uo + B2 us
(7)
L1
inverter side inductance;
Assuming that the operation point and deviation are ( xref , uoref , usref ) and (∆x, ∆uo , ∆us ) respectively, the small-
L2
grid side inductance;
signal model is given by
R1
equivalent series resistor of L1;
R2
equivalent series resistor of L2;
i1
inverter output current;
i2
grid side current;
uc
capacitor voltage;
uo
inverter output voltage;
us
grid side voltage.
∆x =
Substituting (7) into (8) yields ∆x = A∆x + B1∆uo + B2 ∆us
(4)
x =Ax + B1uo + B2 us
where
1 L1 B1 = 0 , B2 = 0
0 − R2 L2 −1 C
−1 L1 1 L2 0
(10)
∆x = A∆x + B1∆uo
which is used to design the LQR controller in the section IV. IV.
LQR CONTROLLER DESIGN
A. Controllability and observability Before designing the LQR controller, we must make sure that the state-space model is controllable and observable. Using the parameters list in the Table I, the state matrix A and control matrix B1 are expressed as
−10 = A 0 5 × 104
0 −1 L 2 0
−1000 1000 0
0 −10 5 × 10
4
T
B1 = [100 0 0]
The output equation can be expressed as
y = Cx
(9)
The utility grid is assumed to be an ideal power source and the deviation ∆us is zero, thus (9) can be reduced to
There are three energy-storage elements in the LCL-filter, thus the three state variables are i1, i2, and uc. Choose i2 as the system output, the state equation can be written as
− R1 L1 0 1 C
(8)
where ∆x = x − xref , ∆uo = uo − uoref , and ∆us = us − usref .
Note that the grid inductance is neglected, as it is much smaller than the filter inductance.
i1 i , A x == 2 uc
∂f ∂f ∂f ∆x + ∆uo + ∆us ∂x ∂uo ∂us
(11)
(12)
Thus, the controllability matrix is given by (5)
103 2 B1= AB A B 0 1 1 0
where y = i2 and C = [ 0 1 0] . The above state-space model is valid for large-signal operation. In the steady-state condition, the state variables i1, i2, and uc are all alternate current variables, thus the state-space model cannot be used for the LQR controller design directly.
−104 0 5 × 10
7
−5 × 1010 5 × 1010 −5 × 108
(13)
It is obvious that the controllability matrix has full rank, so the state-space model is controllable, and the observability does not need to be checked because complete state feedback is available.
373
∆uo
∆x
B1
+
+
∆x
∫
= uCref L2
i1ref = iCref + i2 ref = C -K
Fig. 2. The control block diagram of the grid-connected inverter.
B. The Optimal Gain Matrix The quadratic cost function of the linear system (10) is given by
1 2
∫ ( ∆x Q∆x + ∆u +∞
T
0
T o R∆uo
) dt
(14)
where Q is the error weighted matrix: a 3 × 3 constant, symmetric, positive semi-definite matrix, and R is the control weighted matrix: a 1×1 symmetric, positive definite matrix. The first item in (14) stands for the control accuracy, while the second one for cost of the control effort.
+ us
(21)
K = R −1 B1T P
(15)
where P, the 3×3 constant, positive definite, symmetric matrix, is the solution of the matrix algebraic Riccati equation (ARE)
AT P + PA − PB1T R −1 B1T P + Q = 0
(16)
From (15) and (16), K can be expressed as
k3 ]
(17)
The optimal control law is given by
∆uo* =− K ∆x
(18)
Substituting (17) back into (18) yields ∆uo =−k1∆i1 − k2 ∆i2 − k3 ∆uC
= i1ref C
(22)
dus + i2 ref dt
(23)
C. Capacitor Voltage Observer In order to perform the complete state feedback, two currents sensors are needed to sense the inductor currents i1 and i2, and a voltage sensor to sense the capacitor voltage uC. To reduce the system cost and simplify the hardware design, an observer for the capacitor voltage is introduced. From (3), the capacitor voltage can be observed by the following equation
= uC V.
1 (i1 − i2 )dt C
∫
(24)
SIMULATION RESULTS AND ANALYSIS
The Matlab/Simulink model based on the Fig.2 has been developed to evaluate the proposed control method. The system parameters are listed in TABLE I. Different values of the matrixes Q and R have been tested through simulations using Matlab/Simulink platform. The set of values found to guarantee a fast response and zero steadystate error are 0 0 10 Q = 0 10000 0 , R = 0.01 0 0 10 Using the Matlab built-in function “lqr(A,B,Q,R)”, the optimal gain can be obtained as
(19)
K = [913.56 86.27 64.45]
which can be written as expanded form
(25)
Thus the optimal control is expressed as ∆uo = −913.56 × ∆i1 − 86.27 × ∆i2 − 64.45 × ∆uC
∆uo = −k1 (i1 − i1ref ) − k2 (i2 − i2 ref ) − k3 (uC − uCref ) (20) Equation (20) shows that the LQR control law essentially provides a multivariable proportional regulator. Based on (19) and (20), a control block diagram is presented in Fig. 2, where the state vector Δx is controlled to be zero by the completer state feedback, which means that the state variables in (4) track their references without steady-state error.
d 2i2 ref dus + L2C + i2 ref dt dt 2
Considering the parameter of the filter inductor L2 and capacitor C listed in the TABLE I, the second item in (22) can be omitted, which results in a simple form of i1ref
The optimal gain matrix K is given by
K = [ k1 k2
dt
and A
J=
di2 ref
TABLE I.
PARAMETERS OF THE GRID-CONNECTED INVERTER SYSTEM
Filter inductance ESR of filter inductance Filter capacitance DC bus voltage Switching frequency Grid voltage (line to line, RMS) Grid frequency
i2ref and i2 are the reference grid current and output grid current in (20), respectively. From Fig. 1, UCref and i1ref can be expressed as
374
(26)
L1=L2=1mH R1=R2=0.01Ω C = 20μF Vdc=400V 10kHz 380V 50Hz
In order to obtain maximum active power delivery from the inverter to the grid, the output current reference i2ref should keep in phase with grid voltage us to obtain unity-power-factor (UPF) condition. In the following simulations, the grid voltage is give by (27)
where Um = 311V is the amplitude of the grid voltage. Meanwhile, the current reference is
i2 ref = I m sin ωt
80
Current(A) / Voltage(V)
60
Current(A) / Voltage(V)
0.08
0.05
0.06
0.07
0.08
Current(A) Current(A)
Time(s)
Current(A)
Current(A)
-40 0.12
0.14
0.15
0.16
0.17
0.18
THD = 0.14%
-20 -40 0.12
Current(A)
Current(A)
i2
0
0.13
0.14
0.15
0.16
0.17
0.16
0.17
0.18
i2
0.2us
20 0 -20 -40
0.08
Fig. 5. The inverter output current and grid current with i2 ref = 40sin ωt .
0.14
i2ref
0 -20 0.06
0.08
0.1
0.12
0.14
0.16
Output current
20
i2
0 -20 -40 0.04
0.18
Time(s)
0.12
0.1
Reference current
40 20
40
40 20
0.15
40
-40 0.04
THD = 4.23% 0.13
0.14
Fig. 8. The grid voltage and current with a suddden chang of i2ref at the time 0.1s.
i1
0
0.13
Time(s)
40
-20
-20 -40
-80 0.06
Fig. 4. The inverter output current and grid current with i2 ref = 20sin ωt 20
i2
0.2us
-60
THD = 0.14% 0.04
0.08
0
60
i2 0.03
0.07
20
80
20
-20 0.02
0.06
Fig. 7. The grid voltage and current with i2 ref = 20sin ωt .
i1
0
0.05
Time(s)
THD = 8.46% 0.07
0.04
-60
20
0.06
0.03
40
-80 0.12
Fig. 8 and Fig. 9 show the transient behavior when the reference current i2ref suddenly changes from 20sin ωt to 40sin ωt at the time t = 0.1s. It can be seen that the grid current tracking its reference well even in the transition operation condition, which means that the system has excellent dynamic performance.
0.05
-40
Fig. 6. The grid voltage and current with i2ref = 20sinωt.
Simulation results in steady state with the reference current i2 ref = 20sin ωt and i2 ref = 40sin ωt are shown in Fig. 6 and Fig.7, respectively. It can be seen from these simulation results that the UPF condition is obtained and the grid current track the current reference with good accuracy. Thus the proposed method has excellent steady-state performance.
0.04
0 -20
Time(s)
Fig.4 and Fig. 5 show the inverter output current i1 and grid current i2 with Im = 20A and Im = 40A, respectively. It can be seen that the switching harmonics in the inverter output current i1 is greatly attenuated by the LCL-filter, and the grid current is nearly sinusoidal.
0.03
20
-80 0.02
where Im is the amplitude of the reference current.
-20 0.02
i2
40
-60
(28)
0
0.2us
60
Current(A) / Voltage(V)
us = U m sin ωt
80
0.06
0.08
0.1 Time(s)
0.12
0.14
0.16
Fig. 9. The grid current and its reference with a suddden chang of i2ref at the time 0.1s.
375
VI.
CONCLUSION
An LQR controller for a single-phase grid-connected inverter with LCL-filter has been proposed. Based on the small-signal model of the inverter, the design procedure of the LQR controller is detailed presented in this paper. The excellent performance of the inverter system in both steady state and transient has been verified through simulations on Matlab/Simulink platform. The LQR controller for the inverter has proven to achieve a nearly sinusoidal waveform of the grid current and a UPF condition to guarantee the maximum active power delivery form the inverter to the grid. REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
F. Blaabjerg, Z. Chen, and S. B. Kjaer, “Power electronics as efficient interface in dispersed power generation systems,” IEEE Trans. Power Electron., vol. 19, no. 5, pp. 1184–1194, Sep. 2004. J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz and et al., “Powerelectronic systems for the grid integration of renewable energy sources: a survey,” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1002–1016, Aug. 2006. Y. Xue, L. Chang, S. B. Kjaer, J. Bordonau and T. Shimizu, “Topologies of single-phase inverters for small distributed power generators: an overview,” IEEE Trans. Power Electron., vol. 19, no. 5, pp. 1305–1314, Sep. 2004. 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. E. Twining and D. G. Holmes, “Grid current regulation of a three-phase voltage source inverter with an LCL input filter,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 888–895, May 2003. F. Liu, Y. Zhou, S. Duan and et al., “Parameter design of a two-currentloop controller used in a grid-connected inverter system with LCL filter,” IEEE Trans. Ind. Electron., vol. 56, no. 11, pp. 4483–4491, Nov. 2009. G. Shen, D. Xu, L. Cao, and X. Zhu, “An improved control strategy for grid-connected voltage source inverters with a LCL filter,” IEEE Trans. Power Electron., vol. 23, no. 4, pp. 1899–1906, Jul. 2008. G. Shen, X. Zhu, J. Zhang and D. Xu, “A new feedback method for PR current control of LCL-filter-based grid-connected inverter,” IEEE Trans. Ind. Electron., vol. 57, no. 6, pp. 2033–2041, Jun. 2010. D. S. Naidu, Optimal Control Systems. New York: CRC Press, 2002.
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