 Open Access
 Total Downloads : 184
 Authors : Vu Tran, Mufeed Mahd
 Paper ID : IJERTV2IS120306
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 31122013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Control Design for GridConnected Invertersvia Lyapunov Approach
Vu Tran
Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, USA
Mufeed MahD
Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, USA
Abstract
This paper develops a Lyapunov approach to design for grid connected inverter. The control objective is to create the output current to track a reference currentin proportional with the fundamental harmonic of the grid voltage.
The grid voltage is described as the outputs of an autonomous linear oscillatory system, andits harmonics can be estimated via an observer. A state space description for the whole system is obtained from the state of the inverter circuit and that of the oscillatory system for the grid voltage. Based on the state space description, a Lyapunov approach is applied to design a statefeedback controller with minimal tracking error. The design problem is cast into an optimization problem, which can be eectively solved with linear matrix inequality (LMI) toolbox in Matlab. The eectiveness of the Lyapunov approach is validated via SimPower simulation.

Introduction
As the demand for power is increasing signicantly, renewable energy sources have recently received a lot of attention as an alternative way of generating directly electricity. Using renewable energy systems can eliminate harmful emissions from polluting the environment while also oering inexhaustible resources of primary energy. There are many sources of renewable energy, such as solar energy, wind turbines, water turbines, and geothermal energy. Most of the renewable energy technologies produce direct current (DC) power and hence inverters are required to convert the DC to the alternating current (AC) power.
Standalone (island) and gridconnected are two kinds of inverters. These two types have several similarities, but are dierent in terms of control
function. A standalone inverter is used in o grid applications. The generated power from renewable energy is delivered to loads, or can be stored in batteries. That systems requires complexity and high maintenance, such that rechargeable batteries. That also increases the size and cost for the system. Grid connected inverters overcome this limitation. For grid connected inverters, they must follow the voltage and the frequency characteristics of the utility generated power presented on the distribution line. We consider grid connected inverters in this paper, as its main advantage is that no battery is required for storing the energy from renewable sources, which reduces the size and cost of the system. Moreover, it is easier to create a portable inverter due to the compact size of the system.
Investigations of dierent congurations and control methods for gridconnected inverters are being focused in recent years. A comprehensive review of singlephase gridconnected inverters [1] has covered some of the standards that inverters for grid applications must be fullled. It also provided a classication of the inverters regarding the stage (single, dual stage, and multistring inverter), transformers and types of Interconnections, and types of grid interfaces (linecommutated currentsource inverter and selfcommutated voltagesource inverter).
Reducing the harmonics is a main problem which needs much eort in power inverters these days. The IEEE 929 standard stated that the Total Harmonic Distortion (TDH) of voltage and current should be lower than 5% in normal operation. Harmonics are not desirable because they cause overheating, decreased voltampere capacity, increased losses, distorted voltage and current waveform, etc. Several researches have been proposed to reduce the voltage THD of inverters. For example, the repetitive control theory has been successfully applied to PWM inverters [2][9], active lters [10][12], deadbeat control [13], [14], to reduce THD. Harmonic droop control technique [15] is also presented. Repetitive control has an excellent
ability in eliminating periodic disturbances, however, in practical; this technique is limited in slow dynamics, poor tracking accuracy, and poor performance to non periodic disturbances. Deadbeat and slidingmode controls have excellent dynamic performance in control of output voltage, but these techniques suer from complexity, sensitivity, and steadystate errors In order to eliminate the current distortion, some current control methods are proposed, such that proportional resonant controller and multiresonant controllers in [16], active power lters in [12], [17].
A promising control techniques in gridconnected inverter is output current tracking. The inverters current polarity must be controlled to match the voltage polarity of the grid. Various synchronization methods are summarized in [18], [19] and [20]. The current hysteretic comparison control method, timing control of current instantaneous comparison method and the triangle wave comparison control method of timing tracking current are proposed in [21]. In [22], algorithms of current decoupling are derived for performing the reactive power control of grid connected inverter. Through zerocrossing detecting circuit in [23] and [24], the inverter is controlled so as

State space description and control objective

Openloop description for the circuit
Figure 1. Equivalent circuit for a gridconnected inverter
Fig. 1 describes the equivalent circuit of a grid connected inverter, where vg is the grid voltage. The input voltage u is actually the output of the transistor bridge driven by a pulsewidthmodulated (PWM) signal.
Let the duty cycle of the PWM signal be d and the DC voltage supply be VS. Under the assumption of ideal switching, u = (2d 1)VS. Thus u can be
to generate the output current in phase with the grid voltage. A current control employing internal model
considered as the input.
principle in [25] is proposed to suppress the harmonic currents injected into the grid. Although most existing
Let the state of the circuit as = 1
controllers give satisfactory results, the theory behind the dynamics and performances is not clearly described.
Dene
0
1
1
For a more clearly understanding of the whole
1 +
systems dynamics, a systematic state space approach is
=
developed in this paper. The design problem will be proposed using advanced nonlinear control system theory and linear matrix inequality (LMI) optimization technique, as in [26][30]. The problem of tracking
1
0
+
0
current error with minimal tracking error will be casted into Lyapunov framework, which ensures stability and the total harmonics distortion (THD) requirements.
1
= =
0
The circuit can be described as
0
1
The paper is organized as follows. Section II describes the open loop description for the inverter circuit; following is the statespace description for the grid voltage and an observer, and control objective. Section III reviews the main tool to be used in this paper Lyapunov approach to evaluate the tracking error. Section IV casts the problem of tracking error into Lyapunov framework and converts the design problem into the LMI optimization. Section V uses SimPower in MATLAB to simulate and verify the
= + + (1) whereu is the ontrol input and vg can be considered as an external disturbance.

Statespace description for the grid voltage and an observer
The grid voltage is periodic with frequency 50Hz or 60Hz. The frequency may subject to some
perturbation but can be measured. Let the fundamental frequency be (rad/second). According to [31], the
results. Section VI concludes the paper.
0
grid voltage vg(t) can be written as:
=
=
= 1 sin 0 +
(2)
The magnitude bkand the phase kfor each
harmonic can be evaluated with a bank of resonantfilters [16], [32], [33], or a composite observer [34], [35]. The resonant filters are described with transfer functions, while the composite observers are described via statespace equations. They are all based on the internal model principle in [36], [37]. Here we adopt the main ideas in [35] to describe vg via state space equations and then construct an observer to estimate the state. The advantage of using the state space description is that the dynamics of the whole system can be simply described by stacking up the state equation for vg and that for the circuit, i.e., (1). The resulting state equation for the whole system makes it very convenient to study the interaction between the grid voltage and the dynamics of LCL filter. Furthermore, it facilitates analysis of system performance via advanced tools developed in recent years, such as the Lyapunov approach and the linear matrixinequality (LMI) based optimization.
the pair (g,Sg), is observable. Thus an observer can be constructed to estimate the state wg, and hence (bk,k) for all k. Let the state of the observer be wz. We have
= ( )
where L is the observer gain which can be designed via various approaches.
If the frequency 0 for the observer is exactly the same as the frequency of the grid voltage, then the observer error wz(t) wg(t) will go to 0 asymptotically and we can use the estimated state wz for various purposes. If the grid frequency is subject to perturbation, this frequency can be measured on line and used for the observer. Due to robustness, the same gain L should be stabilizing for a certain range of 0 and Sg(0). The discretetime version of the observer is usually used in practice. More details can be found in [35]. With the estimated state wz, the rst harmonic of vg is estimated as C1wz.
Let
Define
0 = 0
0
0
0
0
0

The control objective
Ideally, we would like to feed the grid a sinusoidal current ig, which is in phase with vg. The magnitude of
0 0 0 0
0 20 0 0
=
ig can be varied depending on the need of the grid and the local energy storage devices. This objective can be stated as a reference tracking problem where the
0 0
0 0
( 1)0 0
0 0
reference for the grid current is given as
, = 1 () (4)
wherer is a positive number that can be changed. It
where all 0s in the above matrix are 2 by 2 blocks. Also dene
g=[1 0 1 0 Â·Â·Â· 1 0]
Then vg is the output of the following autonomous linear oscillatory system:
= , = (3)
would be more convenient to introduce another exosystem:
= 0 , , = 1 0 (5)
2
wherewrR . The condition (4) can be satised if wr(0)
= rwg1(0). Since 1 = 01 , we have wr(t)= rwg1(t).
2N
wherew R
A feature of the matrix S
is that S +
There is some redundancy introducing (5). The purpose
g . g g
T T T
Sg = 0. Because of this, we have wg(t) wg(t)= wg(0)
2
wg(0) = wg(0) for all t. This kind of statespace
descriptions for periodic signals has been widely used in the output regulation literature for tracking periodic references or rejection of periodic disturbances [26] [28], [36], where the linear system (3) wasreferred to as the exogenous system, or simply, exosystem.
2N
The state wgR can be decomposed as
is to make it easier to handle r.
The control objective is to minimize the magnitude of the tracking error at steady state.
e(t)= ig(t) ig,ref(t) (6)
Note that ig can be considered as an output to the inverter system (1): ig=[0 0 1]xc. The two exosystems
(3) and (5) can be combined to obtain a 2(N + 1) order system.
Dene
=
=
1
2
2
wherewgk R , k = 1, …, N. By the structure of g, we
and
have =
1 0
th
(). Thek harmonic is
=
0 =
0 0
exactly
=1
() =
( + )
0 0
1
1 0
sin 0
. Let C1 be
2 = 0 0 1 0
a 1 by 2N row vector whose rst element is one and the rest are all zero. Then the rst harmonic, denoted vg1, is
1 = 1 ()
It is easy to verify that the system (3), in particular,
Then
= , = 1 , , = 2 (7) This system describes all the dynamics of the grid voltage and the reference current. Combined with the
statespace description of the LCL lter, the dynamics of the whole system can be described. It is clear that the exosystem (7) evolves all by itself and is driven by its initial condition w(0). Recall that wconsists of the states for all the harmonics of vg and for ig,ref , in particular,
1
state space description for the whole system, which can be easily obtained by combining the statespace equation (1) for the circuit and the statespace equation
(7) for the vg and ig,ref. Since ig=[0 0 1]xc, if we let C =[0 0 1 2], then
= 1 + (12)
2
0 0
=
=
To reduce the tracking error, we apply a simple
T
Since S + S = 0, we have
= 0 0 = 0 2 (8)
state feedback
= 1 + 2 = 1 2
(13)
= 0 0 = 2 (9)
1
1
= 0 0 = 2 2 (10)
Substitute the feedback law in (13) into (12), we have the closedloop system
2 th
= + 1 1 + 2
(14)
for all t. Thus wgk(0)
represents the power of the k
0
harmonics of v
harmonics of v
2
g and w(0) the total power of the
lus the power of the reference current.
= (15)
g p
The condition wr(t)= rwg1(t) implies that ig,refis proportional to the rst harmonic of vg. In terms of the combined state w, this can be written as
1 2+2 = 2 2+1 (11)
As long as A + BK1 is stable, the solution for the above system will be bounded and for any initial condition, the solution will approach a steady state
oscillation. Since A + BK1 is stable, the eect of the initial condition of x (0) ill vanish asymptotically.
This condition will be used as a constraint in an c w
optimization problem to be formulated.


Evaluation of the tracking error using Lyapunov approach

Evaluation of tracking error for general exosystem
From (7), both the grid voltage and the reference current are driven by an autonomous linear system. According to [38], the state variable of an autonomous system contains all the information that determines its
Thus the steady state oscillation, in particular, the tracking error, depends only on the initial condition w(0). Hence, a gain from the norm of the initial
condition (0) = (0)(0)to the magnitude of e at steady state can be dened. The main result of [26] was applied to estimate this gain via a quadratic
Lyapunov function , = .
Here we summarize the main result of the
Lyapunov approach when applied to the linear system (14). Denote
1 , 2 = + 1 1 + 2
future behavior, it can be eectively used to correct the
Theorem: for > 0, if
0
e exist a sitive denite
dynamic behavior of the whole system. In the case of
ther po
T (2N+5)Ã—(2N+5)
the inverter circuit, the state w can be used to minimize the tracking error of the grid current (6).
The state of the whole system is a combination of
matrix P =P R that
T
, and a number > 0 such
the circuit state xc and the exosystem state w. They can be either measured or estimated via an observer, thus state feedback is feasible. If the tracking error can be eectively evaluated via a certain performance measure, the next step would be minimizing this performance measure via a certain optimization algorithm.
The Lyapunov approach developed in [26] can be applied for this kind of problems. It deals with more general systems (nonlinear, timevarying) with periodic excitations, which could be disturbance or reference. The objective is to evaluate the magnitude of certain output at steady state, which could be the tracking error.
To apply the Lyapunov approach, we rst need the
C C P (16)
1 , 2 + 1 , 2
0 0 (17)
0 2+2
where I2N+2 is an identity matrix of dimension 2N + 2, then for any initial condition xc(0) and w(0), xc(t) and e(t) will converge to a bounded set. Moreover, the tracking error e at steady state is bounded by
1
() 2 (0) .
The number satisfying Theorem is called a bound on the steady state gain from w(0) to the tracking error e. For given K1, K2, this steady state gain can be evaluated by minimizing satisfying the LMI constraints (16) and (17), by using the LMI toolbox in Matlab.
Here we note that the norm of the initial condition,
In terms of , this constraint can be written as
2
w(0) ,
is closely related to the magnitude of vg
and
0 (20)
ig,ref . Furthermore
(0) 2 =
2 + 2 2 = 2 + 2 2
whereWrm is given by
0
0 0 0
=1
1
=1
3
0 2 0 0
2
Thus w(0)
represents the total power of all the
= 0
2
0
0 0
harmonics of vg plus the power of the conference ig,ref .

Improved evaluation using structural information
0 0 0 2
3) Phase of reference current condition: The reference current is in phase with the first harmonic vg1. This implies that the state wr is proportional to the state
In practice, the THD of the grid voltage is below a
certain level and the magnitude of the reference current
wg1. Let
,1
= ,2 , 1
1,1
= 1,2 .
is within a given range. In this section, such structural information will be eectively utilized to improve the evaluation of the tracking error. Specically, the
Then ,11,2 = ,21,1 . In terms of the whole state, this is equivalent to
= 0 (21)
structural information will be exactly expressed in terms of quadratic inequalities and incorporated in the Lyapunov approach to obtain less restrictive constraints, thus reducing the minimal value of for the optimization problem.
where is given by
03 0
0 02
0 0
0 0 1
= 0 0
1 0
0 1
0 1

THD bound condition: Consider the grid voltage expressed in (2). The THD value is
2 1
0 1 0
02(1) 0
0 02
=2 2
= 1
Now we can use the quadratic inequalities (19)
(21) to improve the evaluation of the tracking error.
Suppose that a known bound on the THD is .
Consider the closed loop system (14). Suppose that
Then we have
0 the THD of the grid voltage is less than 0 and ig,ref(t)= rvg1(t) with r rmax. For >0, if there exist a positive
T (2N+5)Ã—(2N+5)
2 22
denite matrix P=P R
, and number , 1,2
0 1
=2
0, 3 R such that
T
Recall from (9) that 2 = for all t, the above inequality can be expressed as
2 2 1 (18)
C C P (22)
1 , 2 + 1 , 2
0 0
=
0 1
0
In terms of the combined state , this inequality
2+2
1 2 3 (23)
can be expressed as
0 (19)
whereI2N+2 is a(2N+2) identity matrix, then for any initial condition xc(0) and w(0), xc(t) and e(t) will
converge to a bounded set. Moreover, the tracking error
where
03 0 0 0
e at steady state is bounded by
2 1 2 2 2 2 1
= 0 0 2
0 0
0 0
0 0
2(1) 0 0 02
() 2 1 + + + 1 2
For given feedback gain K1,K2, the magnitude of the tracking error e can be evaluated by solving an
and 0p denotes a pÃ—p 0 block and other 0s have compatible dimensions.

Magnitude of reference current condition: The reference current is proportional tothe rst harmonic vg1and is set as i2,ref(t)=rvg1(t). Suppose that r is
optimization problem with matrix inequality
constraints.The constraint (23) is less restrictive than the corresponding condition (17) due to the additional parameters 1,2,3 in the terms 1WT HD, 2Wrm and
3Wrp, which result from the structural information.
bounded by rmax. Then we have 2 .
1
1



Design of statefeedback law via LMI based optimization
The analysis problem in the previous section can be readily turned into a design problem by considering K1,K2as additional optimization parameters. Putting everything together, we have the following optimization problem
only the rst 5 harmonics are kept. The resulting vg
used for simulation is
vg(t) = 7.9554sin(0t 0.4868) + 0.0084sin(20t 0.525)
+ 0.0299sin(30t + 2.67) + 0.0032sin(40t 1.1385) + 0.1911sin(50t + 0.3363)
The reference current is ig,ref= r7.9554sin(0t 0.4868), with r is the proportional factor. The output
s.t a)
,1,2, ,1,2,3
T
C C P
(24)
current ig will be created in proportion with ig,ref.
By choosing different weighting for K, and solving the optimization problem, we obtained the
b) 1 , 2 + 1 , 2
0 0
0 2+2
1 2 3
c) P> 0, 1,2, > 0
When K1 andK2are considered as optimization parameter, the above optimization problem has bilinear termsin constraint b). We may use the pathfollowing method as used in [39] to find the optimal or sub optimalsolution.

Computation and simulation results
The Simulinkmodel was constructed for the inverter using SimPower in Matlab. The model is included a transistor bridge controlled by PWM signals. The parameters for the circuit in Fig. 1 are givenas follows: Lf = 150H, Lg = 450H, Cf= 22F, Rf = 0.02, Rg = 0.02, Rd = 1.
The switching frequency is 20KHz and the DC voltage supply is 12V. The state feedback is
feedbackgain K.
K = [0.2005 7.1971 72.8533 0.2944
0.0012 0.2942 0.0023 0.2940 0.0035 0.2936 0.0046
0.2932 0.0058 1.3568 0.0545]
Figure 3a. Grid current ig and tracking error with r=0.2
1+ 1
1+ 1
processedby a low pass filter 1
3000
before sent to drive
the transistor bridge.
The grid voltage vg was measured via a transformer and its harmonic components were analyzedinMatlab.
Figure 2.Hamonic components of grid voltage
Fig. 2 is the grid voltage harmonic components. Its fundamental frequency is 0 = 60Hz. The total harmonic distortion (THD) is 2.434%. For simulation,
Figure 3b. Grid current ig and tracking error with r=0.3
Fig. 3a and 3b show the tracking performance by the PWM model with r=0.2 and 0.3 respectively. As expected, the output current ig tracks the reference igref, which is proportional with the first harmonic of grid voltage vg, ig igref = rvg1. With r=0.2 and r=0.3, the magnitude of ig will be about 1.6 and 2.4,and the THD value for ig is 0.9369% and 6798%respectively.The
magnitude of tracking error at stable state is about 0.08(A). Fig. 4a and 4b show the harmonic components of the current output ig.
Figure 4a. Harmonic components of the current output igwith r=0.2
Figure 4b. Harmonic components of the current output igwith r=0.3

Conclusions
This paper developed a Lyapunov approach to modelling and design the statefeedback control for gridconnected inverters. The method was applicable to inverters connected to the grid via a transformer. The model ensures the internal stability and makes ecient use of harmonic information of the grid voltage, and the magnitude/phase of the reference current. The eectiveness of this design was then validated by SimPower simulation. The results show the robustness of the design, the output of the inverter can be fed to the grid with low THD.

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