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Full Feedforward of Grid Voltage for Discrete State Feedback Controlled Grid-
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DOI: 10.1109/TPEL.2012.2190524

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4234 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

Full Feedforward of Grid Voltage for Discrete State

Feedback Controlled Grid-Connected Inverter
With LCL Filter
Mingyu Xue, Yu Zhang, Member, IEEE, Yong Kang, Yongxian Yi, Shuming Li, and Fangrui Liu

Abstract—Due to multifeedback of state variables, a discrete try, where renewable energy generation is of particular concern.
state-space controller offers outstanding control bandwidth as well To pursue green power as well as to meet the stringent har-
as control stability for popular LCL-type grid-connected inverters, monic standard, conventional L filter is replaced by the LCL
while the grid current is still vulnerable to the grid-voltage har-
monics. The feedforward of grid voltage, usually employed in con- filter, for better harmonic elimination capability. However, this
tinuous controllers to solve such a problem, is too complicated to be brings challenges to suppress the LCL resonance.
applied for the discrete state-space controller. By means of contin- Due to the additional power losses, passive damping [2] has
uous transformation, a full grid-voltage feedforward (GVFF) de- been gradually substituted by the active ones. The capacitor
coupling strategy is successfully proposed in this paper to make the voltage lead-lag compensation method proposed in [3] and [4]
feedforward possible for the discrete state-space controller. Based
on the transfer function analysis and comparison of discretized and requires fine tuning to compensate the phase at the turn point of
continuous system, comprehensive verification is also provided to LCL filter and leads to a higher overshoot. A strategy of injecting
verify the effectiveness of the derivation of the GVFF path. Sub- oscillation in antiphase is proposed in [5]. The capacitor current
sequently, the robustness analysis of the proposed strategy to the feedback methods are presented in [6]–[9], of which, the method
grid impedance is also performed. Moreover, a grid-voltage es- in [6] is based on the idea of dual loops, and methods in [7]
timator instead of the measured voltage is employed for the full
GVFF, which not only retains the important information of grid and [9] are based on the idea of virtual resistor. The weighted-
voltage but also eliminates the influence of accompanied noises. average-current feedback control method presented in [10] is to
The distinct features of the proposed feedforward controller plus obtain the equivalent control plant of first order. However, the
implementation strategy are the super steady waveform, dynamic reference and feedback are not of the same quantity, thereby ne-
response, and robustness to the variation of grid impedance. Be- cessitating compensation. A notch filter is employed in cascade
sides, the complexity of the algorithm is moderate and the compu-
tational burden is not significantly increased. Finally, simulation in [11], whereas complicated inheritance method is required to
and experimental results are provided to verify the feasibility and achieve adaptiveness and robustness. Among all of these strate-
validity of the proposed strategy. gies, the capacitor current feedback is prevalent.
Index Terms—Continuous transformation, discrete state feed- The LCL-type inverter works well in most conditions with
back, feedforward, LCL filter, robust. active damping; however, the stability fades under conditions
of low resonance frequency and weak grid with relatively large
impedance [3], [12]. Furthermore, condition like low-voltage
I. INTRODUCTION ride through puts more stress on the dynamics. It indicates that to
OLTAGE-SOURCE pulsewidth modulation (PWM) grid- stabilize the system is insufficient. From the pole point of view,
V connected converters [1] are widely used in modern indus- the potential instability results from the imperfect pole place-
ment, due to shortage of freedom degrees. Thus, the bandwidth
and stability margin are in contradiction. As is well known,
Manuscript received September 16, 2011; revised November 11, 2011 and higher bandwidth brings faster response, while larger stability
January 26, 2012; accepted February 23, 2012. Date of current version May margin introduces less overshoot. The multifeedback is, there-
31, 2012. This work was supported by the Natural Science Foundation of
China under Award 51007026, Award 50837003, and Award 50877032, and by fore, necessary to obtain adequate freedom degrees for ideal
the Key Program of Delta Power Electronics Science and Education Develop- pole placement. In [13]–[15], the authors proposed some sys-
ment Plan (DREK2010002). Recommended for publication by Associate Editor tem design methods based on discrete state-space theory, which
M. Ponce-Silva.
M. Xue, Y. Zhang, Y. Kang, and Y. Yi are with the State Key Labora- are able to achieve high bandwidth and suitable stability mar-
tory of Advanced Electromagnetic Engineering and Technology, Huazhong gin simultaneously. Another merit is that the design is standard
University of Science and Technology, Wuhan 430074, China (e-mail: and straightforward, thereby getting rid of any trial and error.
mysure1984@gmail.com; zyu1126@mail.hust.edu.cn; ykang@mail.hust.edu.
cn; cnyyx@sina.com). However, such a controller takes charge of reference tracking
S. Li is with the Zhicheng Champion Ltd., Guangdong 523718, China only, while the coupling of grid-voltage current is actually more
(e-mail: lsm@zhicheng-champion.com). complicated due to the multifeedback.
F. Liu is with the State Key Laboratory of Advanced Electromagnetic En-
gineering and Technology, Huazhong University of Science and Technology, The proportional resonant plus harmonic compensator con-
Wuhan 430074, China, and also with Ryerson University, Toronto, ON M5B troller (PR+HC) [16] is good at suppressing the grid background
2K3, Canada (e-mail: fangruihust@163.com). harmonics; however, its main drawbacks are the following: 1)
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org. the computational burden increases with the number of harmon-
Digital Object Identifier 10.1109/TPEL.2012.2190524 ics concerned; 2) only harmonic within the bandwidth is able

0885-8993/$31.00 © 2012 IEEE

XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4235

to be dealt with; and 3) the intended dynamics alters, usually

slows down. All these give rise to grid-voltage decoupling by
feedforward, which is to incorporate the variation of the grid
voltage in the control through an additional feedforward path.
Single proportional grid-voltage feedforward (GVFF) is usually
enough for L-type grid interface, but not for LCL’s, where the
disturbance path is of high order, requiring the GVFF path to
be of the same order. An approximate full GVFF with converter
Fig. 1. Single-phase grid-connected inverter with LCL filter.
current controlled is provided in [17], and a startup impedance
specification by GVFF is presented in [18]. It is noticeable that
the full GVFF method proposed in [9] offers attractive grid rejec- introduced by the LPF will seriously ruin the accuracy. Thus,
tion ability, whereas it is effective for the case of active damping it is so challenging to design an accurate but noise-immune
with capacitor-current feedback only. Moreover, system delay is differentiator that the sensed grid voltage is not suitable to be
neglected and stiff grid is assumed as well. A more generalized adopted directly. A “clean” grid voltage can then be restored
full GVFF method for discrete state-space controller remains for substitution. As one possible solution, the Kalman filter [14]
undeveloped yet. can be employed to filter the grid voltage, except that the model
The system expressed by the shift operator z is so complicated of each grid harmonic voltage is required; thus, the number
that cumbers the establishment of full GVFF path. In addition, of concerned harmonic voltage is limited by the computational
the differentiator in z-plane is not as evident as that in s-domain, burden. In addition, Rodriguez et al. employ the adaptive band-
where a single s is intuitively interpreted as d/dt. A further pass filter (ABPF) for an advanced grid synchronization [20],
recognition is that a digital setup runs obeying its physical na- while the ABPFs can, at the same time, serve to estimate the grid
ture only, no matter whether discrete or continuous controller is voltage. However, it encounters the same problem of computa-
applied. All make it rational to transform the discrete controlled tional burden as with the Kalman filter. Thus, unless such a grid
system to be continuous, which is for the first time proposed and synchronization method has been adopted already, the ABPFs
accomplished in this paper, by mapping the presentation of the should not be practical. In contrast, the grid-voltage estimator
inverter-grid formed system, from discrete state space into its (GVE) provided in [21] is able to inherently capture a clean
continuous counterpart. The main benefit of this method, namely grid voltage that highly resembles the actual one, with fixed and
continuous transformation, lies in its generality: the system em- low computational burden. It is thus promising to implement
ploying any type of discrete controller—active damping based the full GVFF with GVE instead of the measured one, hence
on filter or based on multiloop [19], inverter current controlled to allow pure differentiator to ensure accuracy. Moreover, this
or grid current controlled, rectifier or inverter, and even with ac- paper develops a phase compensator for GVE to further improve
tive damping or with passive damping, the grid-voltage-current the performance.
coupling can be unveiled explicitly, while a modification of the System modeling and discrete state-space control strategy
controller simply implies finding another one. Based on that, are presented in Section II, followed by the corresponding con-
this paper successfully proposes a full GVFF decoupling strat- tinuous transformation as well as the establishment of the full
egy for discrete state feedback controller, particularly when grid GVFF path in Section III. Then, analysis of stability with GVFF,
current is directly injected into the grid. Such full GVFF path as well as the robust but accuracy implementation, is performed
represents an extension to the existing ones. in Section IV. Finally, simulation and experimental results are
Although the state-space controller has reserved considerable provided in Sections V and VI, respectively, to verify the validity
stability margin for grid impedance variation, the unsuitable im- of the proposed strategy.
plement of GVFF will on the other way lead to degradation of
current control performance or loss of stability. A deep insight
unveils that the major cause is the accompanied noise when II. SYSTEM MODELING AND DISCRETE STATE-SPACE CONTROL
performing the GVFF. Such noise includes the common-mode A single-phase grid-connected inverter is shown in Fig. 1
noise that can be reduced rather than eliminated by common- where the dc link can be simplified as a battery with constant
mode filter, and the sensitive information of grid current, which voltage of Vdc and the grid internal impedance as Lg and Rg are
is induced by grid impedance and then is reflected in the sensed considered as well. From the relationship between the system
grid voltage, hence is absorbed in the main control loop through input and output, the following modeling can be carried out.
the GVFF to affect the dynamics of the main control loop. Con-
sidering the GVFF strategy involves two parts as grid voltage
and path, which contains multiordered differentiators, it is nec- A. Modeling in s-Domain
essary for either a noise immune but accuracy realization of Take the state vector as x = [ I1 VC f I2 ]T , the input vec-
path, or a “clean” grid voltage. tor as u = [ Vi Vg ]T (Vi is the inverter output voltage that
With respect to the path realization, to the authors’ knowl- can be derived from the duty cycle), and the output vector as
edge, a common solution is to employ a first-order low-pass fil- C = [ 0 0 1 ]—grid current is set as control object. Provided
ter (LPF) for each order of differentiator, while the sensed grid that the variables in the following equations are normalized, the
voltage is kept as another part of GVFF. However, the phase lag continuous state-space equation can be derived by Kirchhoff’s
4236 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

Fig. 3. Control block diagram for discrete state feedback.

where
G = eA T = L−1 (sI − A)−1
T

Fig. 2. State-space model of a single-phase grid-connected inverter with LCL H = [ H1 | H2 ] = G(t)dtB.

filter. (a) In continuous domain. (b) In discrete domain. 0

Here, L−1 is the inverse Laplace transform. The symbolic ma-

trixes are presented in Appendix A.
Similarly, the inverter input voltage and its command in
current law as z-domain are governed by the following equation:
∗
ẋ = Ax + Bu Vi (k) = Vi (k + 1). (4)
y = Cx (1) Taken Vi as another state variable [13], the whole system can
be modeled as (5) and the corresponding plant is presented in
where Fig. 2(b)
⎡ ⎤   
R1 1
− − 0 x(k + 1) G H1 x(k)
⎢ L1 L1 ⎥ =
⎢ ⎥ Vi (k + 1) 0|1×3 0 Vi (k)
⎢ 1 1 ⎥
A=⎢
⎢ 0 − ⎥,  
⎢ Cf Cf ⎥
⎥ 0|3×1 ∗ H2
⎣ + Vi (k) + Vg (k). (5)
1 R2 ⎦ 1 0
0 −
L2 L2
⎡ 1 ⎤ C. Discrete State Feedback Control
0
 ⎢ L1 ⎥ The expanded state vector is defined as x = [ x Vi ]T . As
| ⎢
B2 = ⎢ 0 ⎥
B = B1 0 ⎥. depicted in Fig. 3, all the states are feedback through the vec-
⎣ 1 ⎦
tor Kf = [ KI 1 KV C f KI 2 KV i ]. A Kalman observer is
0 −
L2 employed to reduce sensors required, while x̂ is the estimated
state vector. Here, GdPI is the discrete PI controller expressed by
The inverter output voltage Vi follows its command Vi∗
KP + KI /(1 − z −1 ), and Gdf f is the discretized GVFF path to
through a system delay Td (sum of one sample delay T and
be addressed in the next section. Taking into account the orders
PWM transport delay 0.5T ), giving
introduced by one sample delay as well as PI controller, the
Vi = Vi∗ e−sT d . (2) system is clearly of fifth-order.
Pole placement can be accomplished either by linear
The continuous generalized control plant derived from (1) and quadratic regulate or by direct pole placement. Since system
(2) is presented in Fig. 2(a). performance is highly dependent on the poles, the direct one,
which is straightforwardly oriented to the final behavior, is pre-
B. Modeling in z-Plane ferred in this paper, while the relative issues have been tackled
in the author’s previous work [22].
Due to small sampling interval T , the grid voltage Vg can The remainder of this paper is dedicated to the GVFF strategy
be considered constant in each interval. Thus, the system state- design. A flowchart of the GVFF design procedure is illustrated
space equation can be obtained by means of zero-order hold in Fig. 4, where the numbers indicate the design order. The
as derivation of discrete plant is achieved by Phases 1–3, while the
following steps are focusing on mapping the discrete controlled
x(k + 1) = Gx(k) + Hu(k)
system into the continuous domain to obtain the GVFF path. It
y(k) = Cx(k) (3) is worth mentioning that although directly acquiring the GVFF
XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4237

plant shown in Fig. 2(a), we obtain

Vi∗ = GsPI (I2∗ − I2 ) − Kf x + Gsf f Vg (8)
where GsPI = KP + Ki /s, Ki = KI /T , Gsf f is the continuous
counterpart of Gdf f , and the continuous indicator (s) has been
omitted for simplicity. Noting Kf the feedback gain vector of
the original state variables, i.e., Kf = [ KI 1 KV C f KI 2 ],
substituting (2) into (8) results in
GsP I (I2∗ − I2 ) − Kf x + Gsf f Vg
Vi = . (9)
esT d + KV i
Notice the difference between Kf x and Kf x, as the former
one is the expanded state feedback, while the latter one is the
original state feedback. Equation (9) is then substituted into (1)
and the whole continuous closed-loop transfer function can be
Fig. 4. Developing GVFF for discrete state feedback controller. obtained as
∗
C{B1 GsPI I2 + [B2 (esT d + KV i ) + B1 Gsf f ]Vg }
path based on the continuous state feedback controller might I2 =
(sI − A)(esT d + KV i ) + B1 (GsPI C + Kf )
serves as a simple way, it unavoidably suffers from stability
problem and is unfeasible for practical application, which will 1
= {(KP s + Ki )I2∗ − [Cf L1 (KV i + esT d )s2
be addressed in the following section. D
+ Cf KI 1 s + (KV C f + KV i + esT d ) − Gsf f ]sVg } (10)
III. PROPOSED CONTINUOUS TRANSFORMATION AND FULL
GVFF PATH where I is the identity matrix and D is the characteristic poly-
nomial
The grid voltage, usually rich of various harmonics, is such
a mass disturbance on the inverter that seriously distorts the D = Cf L1 L2 (KV i + esT d )s4 + Cf KI 1 L2 s3
grid current if not suppressed effectively. The inverter output
impedance is defined to describe the grid rejection capability as + [(L1 + L2 )(esT d + KV i ) + KV C f L2 ]s2

Vg (s) D(s) Vg (z) D(z) + (KI 1 + KI 2 + KP )s + Ki .

Ro (s) = = or Ro (z) = = (6)
I2 (s) F (s) I2 (z) F (z) Hence, the GVFF path can be derived by equating the second
where D(s)/D(z) is the characteristic polynomial determined term of the left side of (10) to zero
by the main loop of system, while F (s)/F (z) is the disturbance Gsf f = Cf L1 (KV i + esT d )s2 + Cf KI 1 s
path.
The larger the impedance, the stronger the grid-voltage re- + (KV C f + KV i + esT d ) (11)
jection capability. The output impedance can be significantly
improved either by maximizing the gain of D(s)/D(z) at fre- and its alternate form by first-order Taylor expanding as
quencies concerned or by minimizing the gain of F (s)/F (z). Gsf f =Cf L1 Td s3 + Cf L1 (1 + KV i )s2
The function of full GVFF addressed here is dedicated to re-
ducing the gain of F (s). Considering the high complexity in + (Td + Cf KI 1 )s + KV C f + KV i + 1 (12)
z-plane, a continuous transformation is expected to facilitate where the coefficients are clear as long as the discrete state
the establishment of the full GVFF path. feedback controller has been set.

A. Continuous Transformation and Full GVFF Path B. Verification

In most of reference paper, the GVFF path is obtained in All numerical evaluations hereinafter are based on the pa-
the continuous domain, while it is discretized and implemented rameters listed in Table I. The direct pole placement controller
in the digital controller to pursue the performance achieved is designed. Subsequently, the comparison of the system be-
in the continuous domain. Similarly, the discrete system can fore and after transformation is comprehensively investigated
be transformed to a continuous one to derive the full GVFF in terms of open-loop frequency response, step response, and
path. Referring to Fig. 3, the inverter voltage command can be output impedance to evaluate the validity of the proposed GVFF
obtained strategy.
Vi∗ (k) = GdP I [I2∗ (k) − I2 (k)] − Kf x (k) + Gdf f Vg (k). (7) The poles are specified following the guideline presented
in [22]
This is the so-called discrete control function in Phase 5 in √ 2
Fig. 4. By mapping it into the generalized continuous control p2,3 = e(−ζ ±j 1−ζ )ω n T
4238 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

TABLE I and its continuous counterpart below the high-frequency region,

SETUP PARAMETERS
i.e., lower than 1/5fs , verifying the validity of continuous trans-
formation. It is worth mentioning that a similar verification for
another continuous modeling method using Pade approximation
for esT d has also been performed in [23].
For a continuous system, an infinite output impedance is the-
oretically possible providing that the GVFF path of (11) can
be ideally achieved. However, it is not practical for digital im-
plementation. Considering that the noninteger sample of esT d is
difficult to realize, the Taylor expansion is employed hereinafter.
Without losing generality, the differentiators are discretized by
pure backward difference, i.e., s → (1 − z −1 )/T . The resulting
discrete output impedance is depicted in Fig. 7, where one can
see the proportional part of GVFF improves Ro mainly in the
low- and medium-frequency range (comparing lines 2 and 3);
the first-order part boosts Ro further, at the cost of smaller Ro in
the high-frequency range (comparing lines 3 and 4); the second-
order part acts in a similar way of the first-order one (comparing
lines 4 and 5); however, the third-order part contributes almost
nothing but even smaller Ro at high frequency (comparing lines
5 and 1). For this reason, the third-order part is abandoned, oth-
erwise the stability degrades significantly. As a result, the final
output impedance Ro is depicted as line 5.

C. Evaluation of Continuous and Discrete State Feedback

Although the state feedback controller can be designed in the
continuous domain, there is remarkable discrepancy between
these two design approaches. The resulted continuous pole-zero
map with Taylor expanding is shown in Fig. 8, with the poles
evaluated as

s = { 25416 −3241 ± j17300 −3644 ± j3173 }.

Fig. 5. Discrete pole-zero map with grid impedance varying (frequency unit: It is noticed that none of these poles are mapping with
Hz). L g : 0 −→ 2.5mH. Other zeros lying in the negative real axes outside the
unit circle are not provided.
their discrete counterparts through esT , so is the PI zero. A
deep insight shows that the continuous complex poles feature
 ωn = 4832 rad/s, ζ = 0.754 and ωn = 17953 rad/s, ζ = 0.184,
2π respectively, comparing with ωn = 10660 rad/s, ζ = 0.707 and
z1 = 1 − 0.15 [1 − real(p2 )]
ωn T ωn = ∞, ζ = 1 for the discrete ones. Besides, the intrinsic ze-
ros, which are dependent on the sampling frequency, have dis-
p1 = 0.9z1 , p4 = p5 = 0 (13)
appeared in the continuous pole-zero map. In other words, the
where the damping ratio ζ is usually set as 0.707, and the angular continuous design is blind to these zeros such that it may lead
natural resonance frequency ωn is set as min(0.5ωr , 0.1 · 2πfs ); to problems, especially when inverter current is controlled, for
hence, ωr is the angular filter resonance frequency. The result- such case features the intrinsic zeros lying on the unit circle
ing pole-zero map is shown in Fig. 5 and the scaled control within the low frequency region hence requiring to be damped
parameters are with poles [15]. From a practical point of view, the discrete
state feedback should be adopted as benchmark, as it is well
[ KP KI ] = [ 8.8197 2.0220 ] accordance with the nature of digital control such that promis-
ing performance as expected. On the other hand, the discrep-
Kf = [ 13.7919 − 1.2618 − 7.5489 0.9594 ] .
ancy manifested previously totally ambiguities the guideline
for continuous pole placement, hence makes it unfeasible. Un-
The continuous closed-loop transfer function containing esT d fortunately, a persuasive explanation for such phenomena has
here is not numerically convergent; however, the step response not been available yet. However, with respect to GVFF, it is
is still unveiled thanks to the Taylor’ expanding form. Fig. 6 emphasized that the continuous transformation as well as the
demonstrates the accordance between the discrete performance full GVFF path is still valid.
XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4239

Fig. 6. Comparison between discrete performance and its continuous counterpart. (a) Open-loop frequency response. (b) Step response. (c) Output impedance
without GVFF. z–discrete one; s1 –continuous one with es T d ; s2 –continuous one with Taylor expanding.

IV. STABILITY ANALYSIS AND THE PROPOSED ROBUST

IMPLEMENTATION FOR FULL GVFF
Referring to (12), the GVFF path contains multiordered dif-
ferentiators that exhibit a significant challenge in the feedfor-
ward path design, especially for the cases of weak grid and
heavy noise. This section is dedicated to addressing this issue.

A. Stability Analysis With GVFF

It can be concluded from (12) that the full GVFF path is
independent of PI controller, grid current feedback gain, and the
grid-side inductor. This means that the grid impedance has no
influence on the full GVFF path, which is of particular concern.
However, the grid impedance does jeopardize the effectiveness,
through the other part of GVFF—grid voltage sensed at the
common coupling point (PCC). Noting Gs f f the practiced GVFF
path, (10) can be reformulated by replacing Vg with (Lg s +
Fig. 7. Output impedance with full GVFF in discrete domain. 1: R o under Rg )I2 + Vs , as detailed in Fig. 1
full GVFF; 2: R o under no GVFF; 3: R o under proportional part of GVFF; 4:
R o under proportional and first-order parts of GVFF; 5: R o under proportional, GsP I I2∗ + (−Gsf f + Gs
f f )Vs
first-order and second-order parts of GVFF. I2 = . (14)
D/s + (Gsf f − Gsff )(Lg s + Rg )

Unless Gsf f = Gsf f , which would not happen in practical, or the

sensitive information of I2 induced and magnified by the grid
impedance will inevitably flow into the main loop, hence alter
the stability.
Equation (14) is continuous and only serves to qualitative
analysis. The following case studies are carried out in the dis-
crete domain, using pole-zero maps, to determine the grid com-
patibility of such inverter. First, pure differentiator is employed
as benchmark. Because this case is highly susceptible to noise,
an LPF with expression of 1/(τ s + 1) is employed for each or-
der of differentiator. Without loss of generality, τ is selected as
1 μs; hence, the modified differentiator is discretized by means
of zero-pole matched.
The resulting pole trajectories are shown in Fig. 9. It is seen
that the system turns to be seventh-order due to the GVFF.
When differentiator is purely realized, grid impedance of less
than 0.8 mH is in tolerance (as indicated by the bold line).
However, such compatibility region is considerably enlarged
once the LPF is introduced (as indicated by the dash line). It can
Fig. 8. Continuous pole-zero map of system with discrete state feedback be concluded from abundant investigation that the compatibility
control (frequency unit: Hz). of grid impedance follows such a way: the more accuracy of
4240 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

Fig. 11. GVE with phase compensator (z −N ). G 3 and H 3 are the third-row
elements in matrixes G and H , respectively.

B. Proposed Robust Implementation for Full GVFF

Fig. 9. Poles vary with the grid impedance, when full GVFF is applied (fre-
It is so challenging to design an accurate but noise-immune
quency unit: Hz). The trajectory of zeros are not provided for clarity. L g :0 −→ differentiator that the sensed grid voltage is not suitable for
0.8 mH. GVFF any longer. A “clean” grid voltage can then be restored
for substitution. Bolsens et al. [14] employ another Kalman ob-
server to estimate the grid voltage. However, the computational
burden increases sharply with the harmonics concerned. Then,
the GVE employed for sensorless operation [21], [24] is more
popular, for it is able to inherently capture a grid voltage that
highly resembles the actual one but noiseless. The schematic is
shown in Fig. 11, manifesting itself built up by the integration of
error between actual grid current and its estimated evolution us-
ing the grid-voltage estimation. Since the Kalman observer for
states has been developed (see Fig. 3), the increased complexity
is fixed and low.
However, unlike the sensorless operation, the grid-voltage
transducer is still kept for grid synchronization as well as for
state rebuilding by Kalman observer. Although the voltage trans-
ducer introduces additional cost, the system robustness can be
guaranteed, either in scenario of islanding detection, or in the
startup phase, which otherwise features overshoot current oc-
curring due to the convergence time of voltage estimation [24].
Fig. 10. Output impedance with LPF in discrete domain. 1—τ = 0 μs, 2—
A selection rule of the integral gain is presented in [21], ex-
τ = 10 μs, 3—τ = 100 μs. pressed as λ < 2L1 (L1 + L2 )/T 2 . However, one should keep
the gain small enough to avoid deteriorating the whole stability,
though conditioned by the accuracy of estimate grid voltage.
An extra procedure of z −N , namely phase compensator, is
differentiator, the higher requirement for grid-voltage sensing, proposed to eliminate the phase error between the estimated grid
i.e., either noise or sensing delay should be as small as possible. voltage and the actual one. Such phase error results from the fi-
Of course, such compromise is almost impossible in practice. nite gain of the integer (λ), in a similar fashion as closed-loop
Otherwise, less accuracy results in more compatibility of grid control with PI controller tracks the sinus reference but results
impedance, however, at the expense of output impedance, as in nonzero steady error. On the other hand, z −N is to obtain ad-
depicted in Fig. 10 (the output impedance is still defined as vance phase by delaying the estimated grid voltage by less than
Vg (z)/I2 (z), regardless of Lg as well as Vs ). Moreover, as the one cycle, as the grid voltage evolves periodically. Moreover,
grid impedance is usually unknown, the cutoff frequency of the one-cycle-delay characteristic also avoids ringing that may
LPF should be set low enough. In a word, the obstacle lies in be induced by the feedforward path during dynamic operation.
the coordination of accuracy and robustness. Thus, such a GVE can be interpreted as an expanded Kalman
XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4241

Fig. 13. Simulation result of GVE with phase compensator. Weak grid as-
sumed (L g = L 2 , R g = R 2 ). 1—the actual grid voltage; 2—the output of
GVE without phase compensator; 3—the output of GVE with phase compen-
sator. The zoomed distort of GVE is transferred from the pass grid cycle at the
Fig. 12. Simulation grid current under stiff/weak grid. (a) Stiff grid assumed. trigger moment by the phase compensator.
1—without GVFF, 2—with sensed grid voltage plus pure differentiator for
GVFF. (b) Stiff grid assumed. Estimate grid voltage without phase compen-
sation for full GVFF. (c) Stiff grid assumed. Estimate grid voltage with phase
compensation for full GVFF. (d) Weak grid assumed, where the grid impedance
is the same as grid inductor. Estimate grid voltage with phase compensation for
full GVFF. The marked exhibits the inertial effect of phase compensator.

filter for the grid voltage, except for requiring the inverter model
rather than the grid-voltage model.
Fig. 14. Simulation result under grid swag mode.
V. SIMULATION RESULTS
A MATLAB/Simulink model is developed for simulation grid voltage at the trigger moment would transfer to the next
where the solver is chosen as variable step discrete with step current cycle, as marked in Fig. 12(d). This results from the
of 0.2 μs. The inverter parameters listed in Table I are em- one-cycle-delay characteristic of the phase compensator, hence
ployed for both simulation and experiments. In order to test the can be recognized as inertia, while a zoomed view can also be
grid-voltage rejection capability, the third, fifth, and seventh har- found in Fig. 13. Notice that an improperly high integral gain
monics with content of 4.82%, 4.18%, and 3.54%, respectively, of GVE as well as high grid impedance would enhance this ef-
are introduced in the grid voltage. The dynamic performance is ficacy. However, with proper integral gain and moderate rate of
tested by switching the load from empty to rated value at the current reference, it should normally not be troublesome, even
peak and vice versa. in weak grid.
The simulation results under stiff/weak grid are shown in The proposed strategy is also tested with step change of the
Fig. 12, without current-track-error limiter inserted in the signal grid voltage. When it is suddenly decreased by 15% and then
path. Fig. 12(a)–(c) assumes stiff grid impedance. The compar- recovered after 3.5 line cycles, the waveforms are shown in
ison between the current waveforms shown in Fig. 12(a) fur- Fig. 14. It can be seen that the grid current responds to the
ther verifies the validity of GVFF path established previously. voltage variation quickly in an inverse direction and gets to
The comparison of current waveforms shown in Fig. 12(b) and the steady state within 3 line cycles. Due to the inertia of GVE
(c), respectively, necessities the phase compensator. The cur- through GVFF, the grid current exhibits one more current spikes
rent waveform in Fig. 12(d), where weak grid is assumed, i.e., after the step change of the grid voltage.
grid impedance is the same as grid inductor, demonstrates that A local RLC load with quality factor of 2.5 and resonant
the system is robust to the variation of grid impedance, though frequency of 50 Hz is chosen to test the islanding behavior of
with somewhat disorder during startup (lasts for near one cy- the proposed strategy. When the grid voltage is disconnected
cle), resulting from the initiation of phase compensator in GVE, at 1.4 s, the PCC voltage, frequency, and the output current
as illustrated in Fig. 13. If necessary, such disorder can be re- waveforms are shown in Fig. 15, where the current I2 is scaled
moved in the initial current cycle, either by using the sensed grid by a factor of 30 to make it visible. As shown in Fig. 15(a),
voltage for the proportional part of GVFF, or by using GVE out- due the close match of the local load and the inverter output
put before the phase compensator. Besides, the distortion in the active power, the amplitude of the PCC voltage varies little. In
4242 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

Fig. 15. Simulation results under islanding detection mode. V P C C and I2

are presented in volts and amperes, respectively. (a) Without active islanding
method. (b) With active islanding method (APS).
Fig. 16. Single phase grid-connected inverter prototype.

addition, the PCC frequency can still be seen within the range A. Steady Operation Under Normal Grid
of 49.5–50.5 Hz. Notice that the frequency spike is the transient Normal grid is assumed first, i.e., the grid impedance is rep-
response of the PLL to the change of PCC voltage. This means resented by the transformer impedance only.
that the outage of the utility may not be identified so that active Fig. 17(a) shows the results in case of GVFF with LPF, which
islanding detection strategies are still required, e.g., auto phase is discretized by the mean of pole-zero matched. However, the
shift (APS) [25], by which islanding is successfully detected result without any feedforward of grid voltage is not provided,
within about 4 line cycles, as shown in Fig. 15(b). Thus, it since the inverter is triggered by the overcurrent once PWM is
exhibits that with the proposed control strategy, active islanding activated, due to the small startup impedance. The comparison
is still necessary to ensure the safety operation of the inverter. among grid current waveforms as well as the corresponding total
harmonic distortion (THD) shown in Fig. 17(b) demonstrates
the necessity of full GVFF. Although the smaller τ of LPF, the
stronger grid rejection, the current waveform 4 demonstrates
VI. EXPERIMENTAL RESULTS
that the stability will be weaker, where extra current distortion as
A 1-kW prototype is built to verify the proposed method, as marked, together with audible noise, is induced by applying LPF
shown in Fig. 16. The dc bus is formed by a dc source. The con- with τ = 20 μs instead of τ = 50 μs, though the grid current
trol chip is 32-bit fixed-point 100-MHz TMS320F2808. With the THD is reduced. So, it is hard to coordinate the accuracy and
help of Kalman observer, only the grid current, the grid voltage, robustness with LPF for GVFF.
and the dc bus are sensed. An isolated 3-kVA 1:1 transformer The result in the case of PR+HC controller instead of PI
is connected between the grid and PCC to emulate the practical controller is also provided in Fig. 18(a). The PR+HC controller
environment, with parasitic parameters left undetected. is expressed as
The control schematic is shown in Figs. 3 and 11. The sam-
ple action is taken at the top and bottom of the carrier signal. 2ζ  ωg s
GPR+HC (s) = KP + Ki
Current reference generation as well as grid synchronization s2 + 2ζ  ωg s + ωg 2
is realized by filtering the modulus-scaled sampled grid voltage 
Ki 2ζ  nωg s
with an ABPF. The measurement noise of grid current is set to 50 + (15)
mA (monitored). The process noise in terms of inverter voltage 3 n =3,5,7 s2 + 2ζ  nωg s + (nωg )2
(contributed by process noise, etc.) and grid voltage are set to
be 0.5 V (estimate) and 3.5 V (monitored), respectively. Finally, where the gain is the same with the PI controller; hence, ζ  is
the feedback vector of Kalman observer is numerically obtained the damping factor of the quadratic generalized integer, which
using MATLAB. The Kalman observer, GVE, full GVFF, and is set 0.025 here. The current waveform together with its THD
state feedback are realized by a small set of C code arrays and shown in Fig. 18(b) demonstrates the necessity of harmonic
the total computing time of the proposed current controller is compensators with order up to eleventh. However, as stated
about 6 μs, comparing with the interrupt interval of 50 μs. Be- in [26], the higher the order, the smaller the phase margin.
sides, a current-track-error limiter is inserted in the signal path Besides, the increasing computational burden also remains an
to obtain a limitation to 1.5 A. issue, for each harmonic compensator takes 1.6μs to run.
XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4243

Fig. 17. Experimental grid current waveform of steady operation, using LPF
Fig. 18. Experimental grid current waveform of steady operation, normal grid
for GVFF, normal grid assumed 1—with proportional component of GVFF,
assumed. 1—PR+HC; 2—full GVFF using GVE without phase compensator;
τ = 50 μs; 2—with proportional and first-order components of GVFF, τ =
3—full GVFF using GVE with phase compensator.
50 μs; 3—with full GVFF, τ = 50 μs; 4—with full GVFF, τ = 20 μs. (a) Grid
current waveform. (b) Grid current harmonic magnitude percent of the rated
fundamental amplitude.
selective harmonic compensation be effective, ζ  should not be
too big [26]. On the other hand, in the case of full GVFF with
In contrast, the results in case of full GVFF with GVE are GVE (with phase compensator), there is current spike as shown
presented in Fig. 18(a). It clearly manifests the high performance in Fig. 19(c), which is in accordance with the simulation re-
offered by full GVFF with GVE; hence, the phase compensator sults, as caused by the inertia of the phase compensator. Such
serves to improve the power quality further, as indicated by spike fades with time under the condition that either the current-
Fig. 18(b). Specifically, the phase compensator functions not reference rate or the grid impedance is moderate. This means
only to make the estimate grid voltage coincident with the actual that the previous grid-voltage spike induced by the previous cur-
one, but also to compensate the deviation between the estimate rent change is not able to produce an equivalent current spike,
states and the actual states, by providing more lead phase. In this so that the impact on the PCC voltage is not as strong as the
paper, λ and N are set as 2.5 and 393, respectively; hence, the previous one. With this interactivity going on, there comes no
time constant of analogous LPF used for sampling grid voltage spike eventually.
is set as 12 μs.
C. Dynamic Operation Under Weak Grid
B. Dynamic Operation Under Normal Grid Next, to testify the robustness against the grid impedance
Then, still under normal grid, Fig. 19(a) shows the dynamic variation, weak grid is emulated by inserting another inductor
result in case of full GVFF with LPF (τ = 50 μs), which features same as L2 between the inverter output and the transformer. The
smooth and rapid transient response. In contrast, the transient result in case of full GVFF with LPF (τ = 50 μs) is shown in
response in case of PR+HC controller is much dependent on Fig. 20(a), which is almost identical with that under normal grid
the damping factor ζ  , i.e., the bigger the ζ  , the faster the re- [referring Fig. 19(a)]. It happens similarly in case of PR+HC
sponse. However, in order to get a low bandwidth so that the controller, as shown in Fig. 20(b). However, in case of GVFF
4244 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

Fig. 19. Experimental result of empty/rated load switch. Normal grid assumed. Fig. 20. Experimental result of empty/rated load switch. Weak grid assumed.
(a) Full GVFF with LPF (τ = 50 μs). (b) PR+HC controller. (c) Full GVFF with (a) Full GVFF with LPF (τ = 50 μs). (b) PR+HC controller. (c) Full GVFF
GVE (with phase compensator). with GVE (with phase compensator).
XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4245

with GVE (with phase compensator), there is more current dis- APPENDIX B
tortion, which is again caused by the inertia effect of the phase
FULL GVFF PATHS FOR OTHER OPERATIONS
compensator, hence enhanced by the grid impedance. Notice
that the weaker performance of GVE (without phase compen- It is also common to utilize the predictive nature of the es-
sator) under weak grid is also blamed, as reflected by the distor- timated quantities to compensate for total system delays. The
tion at the negative current peak. It can thus be concluded that following is dedicated to establishing the corresponding full
the super steady current waveform offered by full GVFF with GVFF path.
GVE is somewhat at the cost of dynamic performance under In a first step, we obtain the discrete control function
weak grid, though is still acceptable.
Vi (k) = GdPI [I2∗ (k) − Iˆ2 (k)] − Kf x̂(k) + Gdf f Vg (k − 1).
(16)
VII. CONCLUSION
Here, the feedback gain vector is redefined as Kf =
The discrete state feedback control for grid-connected in-
[ KI 1 KV C f KI 2 ]. One should also notice that Vg (k − 1)
verter with LCL filter offers excellent reference tracking ability,
results from the fact that the delay compensation deals with
while the coupling of grid voltage and current introduced by the
the main loop only, while in the feedforward path, there is still
multifeedback makes the voltage feedforward rather difficult.
delay.
A full feedforward of grid voltage for discrete state feedback
Then, mapping (16) into continuous domain [see Fig. 2(a)]
controlled grid-connected inverter with LCL filter is proposed
results in the continuous control function
in this paper by transforming the discrete controlled system into
its continuous counterpart to derive the GVFF path. Such path
represents an extension to the existing ones. The transforma- Vi = GsPI (I2∗ − I2 ) − Kf x + Gsf f Vg e−sT d . (17)
tion process and the corresponding theoretical verification are
provided as well. Moreover, the robustness analysis of the pro- Next, substitution of (17) into (1) results in the whole contin-
posed strategy to the grid impedance is also performed. In the uous transfer function as
feedforward path, high-order differentiators are required that are
sensitive to the grid impedance and the noise. An improved GVE C[B1 GsPI I2∗ + (B1 Gsf f e−sT d + B2 )Vg ]
I2 =
is employed to successfully solve such problems. Therefore, the sI − A + B1 (GsPI C + Kf )
proposed strategy enables the grid-connected inverter to have 1
super steady waveform, dynamic waveform and robustness to = {(KP s + Ki )I2 ∗ − (Cf L1 s2 + Cf KI 1 s
D
the variation of grid impedance and grid-voltage harmonics,
while the complexity remains moderate and the computational + KV C f + 1 − Gsf f e−sT d )sVg } (18)
burden is low. Simulation and experimental results verified the
feasibility and validity of the proposed strategy. where

D = Cf L1 L2 s4 + Cf KI 1 L2 s3 + (L1 + L2 + KV C f L2 )s2
APPENDIX A
+ (KP + KI 1 + KI 2 )s + Ki .
SYMBOLIC MATRIXES
Regardless of the parasitic resistances, the discrete state trans- Finally, the full GVFF path is derived from (18) as
fer matrix is expressed by
Gsf f = (Cf L1 s2 + Cf KI 1 s + KV C f + 1)esT d (19)
⎡ L + L cos ω T sin ωr T L2 (1 − cos ωr T ) ⎤
1 2 r
− and its Taylor expanding form as
⎢ L1 + L2 ωr L1 L1 + L2 ⎥
⎢ ⎥
⎢ sin ωr T sin ωr T ⎥ Gsf f = Cf L1 Td s3 + Cf (L1 + KI 1 Td )s2
G=⎢
⎢ cos ωr T − ⎥
⎥
⎢ ωr Cf ωr Cf ⎥
⎣ L (1 − cos ω T ) + [(KV C f + 1)Td + Cf KI 1 ]s + KV C f + 1. (20)
1 r sin ωr T L + L cos ω T ⎦
2 1 r
L1 + L2 ωr L2 L1 + L2 It is worthy to point out that (19) is the generalization of the
GVFF path presented in [9], which is specific for the case with
and the input matrix is expressed by
capacitor current feedback for damping, and esT d disregarded,
⎡ ⎤ i.e., Kf = [ KI 1 0 −KI 1 ].
L2 sin ωr T sin ωr T
T+ −T + For other cases, e.g., inverter current is set as control object,
⎢ ωr L1 ωr ⎥
1 ⎢ ⎥ one only needs to alter the output vector C in (1), while the
H= ⎢ L2 (1 − cos ωr T ) L1 (1 − cos ωr T ) ⎥
L1 + L2 ⎢
⎣
⎥
⎦ remaining manipulation is almost the same. Moreover, if active
sin ωr T L1 sin ωr T damping is filter based [19], the full GVFF path is simplified as
T− −T −
ωr ωr L2 (grid current controlled)

where ωr = LL11L+L 2
. Gsf f = Cf L1 Td s3 + Cf L1 s2 + Td s + 1. (21)
2 Cf
4246 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 10, OCTOBER 2012

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generalized integrators for current control of active power filters with zero (HUST), Wuhan, China, in 1988, 1991, and 1994,
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[20] P. Rodriguez, A. Luna, I. Etxeberria, J. R. Hermoso, and R. Teodorescu, He is currently working toward the M.S. degree in
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XUE et al.: FULL FEEDFORWARD OF GRID VOLTAGE FOR DISCRETE STATE FEEDBACK CONTROLLED GRID-CONNECTED INVERTER 4247

Shuming Li was born in Hunan, China, in 1980. He Fangrui Liu received the B.Eng. degree in electrical
received the B.S. degree (network) in electrical en- engineering from the Huazhong University of Sci-
gineering from the Huazhong University of Science ence and Technology, Wuhan, China, in 2002, and
and Technology, Wuhan, China. the Ph.D. degree from Nanyang Technological Uni-
Since 1999, he has worked as an Electronic Engi- versity, Singapore, in 2006.
neer in Zhicheng Champion Ltd., Guangdong, China, He joined the College of Electrical and Electronic
and has been a Principal Designer for several gener- Engineering, Huazhong University of Science and
ations of photovoltaic inverter and UPS. He is cur- Technology in September 2006, where he has been a
rently the Project Manager in the R&D Department. Lecturer since September 2008. He is also a Postdoc-
His research interest and professional specialities in- toral Fellow with the Department of Electrical and
clude the low-cost, highly reliable power electronic Computer Engineering, Ryerson University, Canada.
converters. His research includes power converters, ac motor drives and renewable energy
resources.

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