To track the sinusoidal current under stationary frame and suppress the effects of loworder grid harmonics, the multiresonant quasiproportional plus resonant (PR) controller has been extensively used for digitally controlled LCLtype pulsewidth modulation (PWM) converters with capacitor–current–feedback active damping. However, designing the controller is difficult because of its high order and large number of parameters. Moreover, the computation and PWM delays of the digitally controlled system significantly affect damping performance. In this study, the delay effect is analyzed by using the Nyquist diagrams and the system stability constraint condition can be obtained based on the Nyquist stability criterion. Moreover, impact analysis of the control parameters on the current loop performance, that is, steadystate error and stability margin, identifies that different control parameters play different decisive roles in current loop performance. Based on the analysis, a simplified controller design method based on the system specifications is proposed. Following the method, two design examples are given, and the experimental results verify the practicability and feasibility of the proposed design method.
I. INTRODUCTION
With the development of renewable energy and smart grid, energy storage systems (ESSs) have become increasingly interesting. ESSs could smooth the output power and decouple energy generation from demand
[1]
. As an interface between storage elements and the power grid, a voltage source pulsewidth modulation (PWM) converter plays an important role in the singlestage and multistage power conversion systems (PCS) for ESSs
[1]
,
[2]
. To smooth the injected currents, the conventional L filter is replaced by the LCL filter because of its better harmonic attenuation ability
[3]

[10]
. However, given the resonance hazard of the LCL filter, damping solutions are required to stabilize the system.
Two main methods are used to dampen resonance, namely, passive damping and active damping. However, active damping is more well known than passive damping because no additional power loss occurs
[3]

[10]
. Among the various active damping solutions, capacitor–current–feedback active damping is selected in this study because of its effectiveness, simple implementation, and extensive application
[6]

[10]
. Capacitor–current–feedback active damping is equivalent to a virtual resistor connected in parallel with the filter capacitor
[5]
. This conclusion is drawn by excluding the delay effect.
However, computation and PWM delays occur in the digitally controlled system. The computation delay is the interval between the sampling instant and duty ratio update instant. The PWM delay is caused by the zeroorder hold effect, which keeps the duty ratio constant after it has been updated
[10]

[13]
. Given the delay effect, the capacitor–current–feedback active damping is equivalent to a variable virtual impedance, which consists of a resistor connected in parallel with a reactor, rather than a virtual resistor. When the virtual resistor is negative, two unstable poles will be generated in the grid current loop
[10]
. As a result, the resonance peak should not be dampened to less than 0 dB to ensure system stability
[8]
. Thus, the capacitor–current–feedback gain should be selected with extreme caution.
In addition to system stability, highquality injected power is another essential object in the control of the LCLtype PWM converter. Thus, the selection and design of the current controller is crucial. The stationary
α–β
frame is selected in this study to prevent the inconvenient decoupling in the synchronous
d–q
frame
[6]
. To track the sinusoidal current reference and suppress the selected loworder current harmonics, the proportional plus multifrequency resonant (multiresonant proportional plus resonant [PR]) controller has been used extensively
[14]

[19]
. An ideal resonant controller can provide infinite gain to eliminate the steadystate error, but it occurs at the target frequency only. Any perturbation, such as frequency deviation, will lead to a significant reduction in the generated gain
[18]
. However, in fact, the grid frequency is allowed to deviate by ±0.5 Hz. Hence, the performance of the controller will be reduced, especially when applied to weak grids and microgrids where the frequency deviates even worse
[20]
. Moreover, attaining an ideal resonant controller is sometimes impossible because of finite precision in digital systems. To address these issues, the quasiresonant controller is proposed
[12]
,
[19]
,
[21]

[24]
. The quasiresonant controller can provide a sufficiently large gain around the target frequency to reduce its sensitivity to the grid frequency fluctuation and can be attained in digital platforms with a higher accuracy.
However, the design of the quasiresonant controller, especially the multifrequency quasiresonant controller (multiresonant quasiPR controller), is more difficult than that of the ideal resonant controller because the steadystate error should be taken into account in addition to stability and the stability margin. The design of a single quasiPR controller is relatively easy and has been presented in
[9]
,
[12]
. In general, the single quasiPR controller can be designed based on steadystate error, crossover frequency (
f_{cs}
), phase margin (PM), and gain margin (GM) of the system, which have a significant effect on system performance and stability margin. However, these design methods are not applicable for the multiresonant quasiPR controller because calculating the PM of the controller is impossible because of its high order and large number of parameters. In
[21]
, a guideline on designing the multifrequency quasiresonant controller (without the proportional controller) is presented, which considered grid frequency deviation, grid synchronization, grid impedance variation, and transient response. In
[24]
, pole placement is used to determine the controller parameters by properly selecting the poles to guarantee system stability and acceptable performance of the current loop. In
[23]
, the controller parameters are designed separately mainly based on the requirements of the steadystate errors and PM. However, these design methods are inconvenient for engineers, and previous studies do not focus considerable attention on capacitor–current–feedback active damping.
The effect of computation and PWM delays on the active damping performance is analyzed in detail by using Nyquist diagrams. The effect of the controller parameters on the current loop performance with the application of frequency response theory in the continuous domain is also investigated. Then, a simplified practical design method of the multiresonant quasiPR controller and capacitor–current–feedback coefficient is proposed in this study. The paper is organized as follows: In Section II, the average switching model (ASM) of the internal current loop considering the control delay is derived. Based on the derived ASM, the effect of the delay on the active damping performance, which influences the stability constraint condition of the current loop, is investigated by using the Nyquist stability criterion in Section III. In Section IV, the effect of the control parameters on system performance, that is, system stability constraint condition, steadystate error, and stability margin, are investigated. Based on the analysis, a simplified design method is proposed in Section V, and two design examples are conducted step by step by using the proposed method. In Section VI, the effectiveness of the proposed design method is verified by using the experimental results from a prototype of a threephase LCLtype PWM converter. The conclusion is given in Section VII.
II. CONTROL STRATEGY AND MODEL OF THE LCLTYPE PWM CONVERTER
Fig. 1
shows the configuration of a threephase LCLtype gridconnected PWM converter in the stationary
α–β
frame. The LCL filter is composed of
L
_{1}
,
C
, and
L
_{2}
.
C_{dc}
is the direct current (DC) link capacitor. As the equivalent series resistors (ESRs) of
L
_{1}
,
C
, and
L
_{2}
can provide a certain degree of damping and help stabilize the system, the ESRs are omitted in this study to obtain the worst case.
Topology and control strategy of a threephase LCLtype PWM converter in the stationary α–β frame.
As an interface between storage elements and the power grid in twostage PCS, the primary objective of the PWM converter is to exchange power with the grid by controlling the grid current
i
_{2}
directly. To directly control the battery charge–discharge and prolong its service time, the DC link voltage
u_{dc}
is also controlled by the PWM converter. As such, the
d
axis current reference is generated by the outer DC voltage loop. Thus, the
α
axis and
β
axis current references
are obtained by using the reverse Park transformation to the
d
axis and
q
axis current references
To synchronize with the grid voltage
u_{g}
, the phase angle of
u_{g}
is detected through a decoupled double synchronous reference frame phaselocked loop
[25]
. The capacitor current
i_{C}
serves as feedback to damp the LCL filter resonance actively, and
K
is the feedback coefficient. The capacitor–current–feedback signal is subtracted from the output of the current controller. Then, the capacitor–current–feedback signal is normalized with respect to
u_{dc}
/2 to obtain the modulation reference, which is fed to a digital PWM modulator.
As previously mentioned, computation and PWM delays occur in the digitally controlled system. The computation delay is one sampling period
T_{s}
in the synchronous sampling case when sampling is conducted at the beginning of a switching period. The calculated duty ratio is not updated until the next sampling instant. The PWM delay is definitely a half sampling period. Thus, the total delay is one and a half sampling periods (1.5
T_{s}
)
[10]
–
[13]
. The singlephase equivalent ASM of the current loop for the converter in inverter mode is shown in
Fig. 2
. We noted that the antialiasing filter could be removed in the synchronous sampling case
[11]
. Therefore, the grid current
i
_{2}
can be derived as follows:
Singlephase equivalent ASM of the current loop for the digitally controlled LCLtype PWM converter in inverter mode.
where
T
(
s
) is the loop gain of the system and is expressed as follows:
and
G_{g}
(
s
) is expressed as follows:
where
is the resonance angular frequency of the LCL filter.
As shown in Eq. (1), the grid voltage loworder harmonics have a significant effect on the grid current
i
_{2}
. To suppress the effect of the loworder harmonics, the multiresonant quasiPR controller is employed. The transfer function is expressed as follows:
where
h
can take the values 1, 3, 5, 7, …,
m
, with
m
being the highest current harmonic to be attenuated.
III. EFFECT OF THE COMPUTATION AND PWM DELAYS ON THE ACTIVE DAMPING PERFORMANCE
As shown in
Fig. 2
, considering
u_{g}
as the disturbance, the block diagram can be transformed into the standard dualloop structure shown in
Fig. 3
. The loop gain of the active damping loop
T_{ic}
(
s
) can be derived as follows:
Standard dualloop structure of the grid current loop with capacitor current active damping.
As shown in
Fig. 3
, the grid current loop
T
(
s
) has no openloop poles that lie in the right half plane (RHP), except for the active damping loop. That is to say, the number of RHP closedloop poles of the active damping loop determines the number of RHP openloop poles of
T
(
s
). In this study, the Nyquist diagram of
T_{ic}
(
s
) is used to determine the number of RHP closedloop poles by examining the magnitude at the negative real axis crossing frequency
ω_{pc}
. Notably, the crossing points at high frequencies caused by the delay effect affect stability only slightly. As such, the conclusions obtained on the delay effect are summarized as follows. Similar results can be found in
[10]
.

2. IfK>Kcwhenωres<ωs/6, then the crossing point inFig. 4(b) is moved to the left of (−1,j0), that is, Tic(ωpc) > 1, and the Nyquist curve ofTic(s) will make one clockwise encirclement of the point (−1,j0). As such, the active damping loop is unstable with two generated RHP closedloop poles, andT(s) contains two RHP openloop poles.

3. IfK> 0 whenωs/2 ≥ωres≥ωs/6, thenωpc=ωres, Tic(ωpc) = ∞, and the Nyquist curve always encircles the critical point once in the clockwise direction [seeFig. 4(c)]. Thus, the active damping loop is unstable with two RHP closedloop poles, andT(s) contains two RHP openloop poles.
Nyquist diagrams for the positive frequency of the active damping loop with different ω_{res}. (a) Analog control (no delay). (b) ω_{s}/6 > ω_{res}. (c) ω_{s}/6 ≤ ω_{res} ≤ ω_{s}/2. (d) ω_{s}/2 < ω_{res}.
Notably, when
ω_{res}
>
ω_{s}
/2, the Nyquist curve may encircle the critical point [see
Fig. 4
(d)]. However, this case will never occur because
ω_{res}
<
ω_{s}
/2 is required to ensure system controllability
[26]
.
IV. SYSTEM PERFORMANCE ANALYSIS
As shown in Eqs. (2) and (4), the system is of high order and contains many control parameters. Thus, analyzing system performance is difficult. As such, the controller and system model have to be simplified first.
 A. Simplified Controller and System Model
The quasiPR controller shown in Eq. (4) can be rewritten as follows:
where
=
nK_{rh}
/
K_{p}
is the relative resonant gain of the PR controller and n is the number of resonant controllers.
Fig. 5
shows the Bode diagram of the PR controller derived using Eq. (7) with different parameters. The following conclusions can be drawn: (1)
determines the relative gain at the target frequency
ω_{h}
. The gain gradually increases with the increase in
. However, the phase lag introduced by the controller is also increased. (2)
ω_{c}
mainly influences the resonant bandwidth at the target frequency to improve its robustness against the frequency fluctuation. (3)
K_{p}
shifts the magnitude plot up and down and has only a slight effect on the phase plot.
Bode diagram of the multiresonant quasiPR controller (ω_{c} = 3) with different parameters.
Based on Eq. (7), the gain at
ω_{h}
can be obtained as follows:
As shown in
Fig. 5
, the quasiPR controller can be approximated to
K_{p}
at frequencies greater than
ω_{m}
, that is,
Typically, the crossover angular frequency
ω_{cs}
is restricted to a value lesser than
ω_{res}
. Therefore, the LCL filter can be simplified as an L filter when calculating the magnitude at
ω_{cs}
and the frequencies lesser than
ω_{cs}
, which is also applicable for digitally controlled systems
[8]
. As such, the magnitude of
T
(
s
) and
G_{g}
(
s
) at
ω_{cs}
and the frequencies lesser than
ω_{cs}
can be simplified as follows:
Moreover, 
T
(
jω_{cs}
) = 1, combining Eqs. (9) and (10) produces the following equation:
 B. System Stability Constraint Condition Analysis
As analyzed previously, two RHP openloop poles might be generated in
T
(
s
) because of the delay effect. Thus, to ensure system stability, the Nyquist curve for positive frequency has to make one counterclockwise encirclement of the point (−1,
j
_{0}
). In the Nyquist diagrams of
T
(
s
) shown in
Fig. 6
, the negative real axis crossings might occur at
ω_{res}
or at
ω_{res}
and
ω_{s}
/6. Combining Eqs. (2), (9), and (12), the loop gain
T
(
s
) at
ω_{res}
and
ω_{s}
/6 can be obtained as follows:
Nyquist diagrams of the loop gain T(s) with G_{PR}(s) = K_{p}. (a) ω_{res} < ω_{s}/6. (b) ω_{res} ≥ ω_{s}/6.
From Eq. (13), we observed that the Nyquist curve of
T
(
s
) always crosses over the negative real axis at
ω_{res}
for
K
> 0. As shown in Eq. (6),
K_{c}
> 0 for
ω_{res}
<
ω_{s}
/6 and
K_{c}
≤ 0 for
ω_{res}
≥
ω_{s}
/6. Thus, if
ω_{res}
<
ω_{s}
/6, the Nyquist curve crosses over the negative real axis one more time at
ω_{s}
/6 for
K
>
K_{c}
[see
Fig. 6
(a)] and, if
ω_{res}
≥
ω_{s}
/6, the Nyquist curve certainly crosses over the negative real axis at
ω_{s}
/6 for
K
> 0 [see
Fig. 6
(b)]. We noted that, if
ω_{res}
=
ω_{s}
/6, the crossing points at
ω_{res}
and
ω_{s}
/6 coincide with each other.
We assume the magnitude requirements of
T
(
s
) at
ω_{res}
and
ω_{s}
/6 are
M
_{1}
and
M
_{2}
, respectively. Based on the previous analysis, the stability constraint condition on the grid current loop can be derived as follows:

The sampling frequencyfsis typically at least twice that of the switching frequencyfsw. Based on Eq. (16), the curve offcswith the increase infresforM1= 0.707 is depicted inFig. 7. Whenfresis close tofs/6,fcsdecreases significantly. This finding implies thatK
Curves of f_{cs} with the increase in f_{res} for M_{1} = 0.707.

2. IfK>Kcwhenωres<ωs/6, then two RHP openloop poles exist inT(s). Thus, T(jωres) ≤M1< 1 and T(jωs/6) ≥M2> 1 are required. Then, the value range ofKcan be derived as follows:

3. IfK> 0 whenωres>ωs/6, then two RHP openloop poles always exist inT(s), and T(jωres) ≥M1> 1 and T(jωs/6) ≤M2< 1 are required. Then, the value range ofKcan be expressed as follows:

4. Ifωres=ωs/6, then two RHP openloop poles always exist inT(s). However, the Nyquist curve is only tangent to the negative real axis and never crosses over, as shown inFig. 6(b), which means that the system can hardly be stable irrespective ofK.
 C. Steadystate Error Analysis
As shown in Eq. (1), the grid current
i
_{2}
comprises two parts. One part is the command–current component generated by the current reference
. The other part is the voltage–current component generated by the grid voltage
u_{g}
. Based on Eq. (1), the grid current error can be derived as follows:
Considering the role of the controller, if the magnitude of
T
(
s
) at the fundamental angular frequency
ω
_{1}
is sufficiently large, then 1 +
T
(
jω
_{1}
) ≈
T
(
jω
_{1}
). Moreover, as the influence of the filter capacitor is negligible at
ω
_{1}
, considering Eqs. (1), (10), and (11), the fundamental component of
i
_{2}
(
s
) can be approximated as follows:
As the quasiPR controller can provide sufficiently large gain at
ω
_{1}
, the voltage–current component could be attenuated to decrease its value, that is, −
u_{g}
(
jω
_{1}
)/(
K_{p}
+
K
_{r1}
) ≈ 0. Thus, simplification of the steadystate error involves the amplitude error only, not the phase error. Moreover, as shown in Eq. (1), the harmonic currents are generated only by the grid harmonics. Therefore, the steadystate error requirement can be converted to the amplitude error requirements of the current components at
ω
_{1}
and
ω_{h}
which are denoted by
ε_{i}
and
ε_{uh}
, respectively. Based on Eq. (19),
ε_{i}
and
ε_{uh}
are defined as follows:
To ensure that the system is stable,
ω_{h}
should be lesser than
ω_{cs}
. Accordingly, substituting Eqs. (10) and (11) into Eq. (21), the relationship between the gain of the PR controller and steadystate amplitude errors can be approximated as follows:
Considering Eqs. (8) and (12), the relationship between
and the steadystate errors can be calculated as follows:
 D. Stability Margin Analysis
Based on the stability constraint condition analysis discussed previously, we noted that the magnitude requirements
M
_{1}
and
M
_{2}
determine the GM of the system. Therefore, we focus on the PM only. As shown in Eq. (2), the PM is codetermined by the phases of the control object
G_{LCL}
(
s
) and PR controller at
ω_{cs}
. The phase of
G_{LCL}
(
s
) at
ω_{cs}
decreases with the increase in
K
[see
Fig. 8
(a)]. With respect to the PR controller,
Fig. 5
shows that the phase lag caused by the PR controller increases with the increase in ih
.
ω_{c}
is relatively small. Thus, the effect of
ω_{c}
on the PM is disregarded.
K_{p}
has no effect on the phase response, but
K_{p}
affects
ω_{cs}
. Thus, the PM of the system is related to
K
,
, and
K_{p}
.
Bode diagrams of the control objective G_{LCL}(s) with different K (a) and the loop gain T(s) with different K_{p} (b).
As analyzed previously,
K
regulates system stability and
influences steadystate error. Therefore, when system stability and steadystate error have been ascertained, the system phase response could be derived by using Eq. (2) with
K_{p}/n
= 1, and PM is related to
K_{p}
only. As shown in
Fig. 8
(b), the PM is changed with different values of
ω_{cs}
, which is approximately proportional to the value of
K_{p}
. As 
T
(
jω_{cs}
) = 1, substituting Eq. (9) into Eq. (2), the accurate relationship between
K_{p}
and
ω_{cs}
can be derived as follows:
V. DESIGN OF THE CURRENT CONTROLLER AND CAPACITOR–CURRENT–FEEDBACK COEFFICIENT
 A. Design Procedure of the Control Parameters
As analyzed previously, the damping gain
K
mainly influences the system stability, the relative resonant gain ih
mainly regulates the steadystate error, and the proportional gain
K_{p}
mainly affects the PM of the system. Thus, a simplified controller design method based on the specifications of the current loop is proposed as follows:
Step 1
. The specifications of the grid current loop are determined, specifically
ε_{i}
and
ε_{uh}
by the requirements of the steadystate errors at the target frequencies, the PM by the requirements of the dynamic response and robustness, and
ω_{cs}
by the requirement of the dynamic response speed.

As the fundamental voltage amplitude is greater than the harmonic voltages in the grid,εu1should be less thanεuh. In general, PM in the range of (30°, 60°) is required for good dynamic response and robustness. However, when more resonant controllers are used, the PM requirement has to be reduced because PM significantly decreases with the increases inandn. In[27],ωcsis limited to less than 0.3 times that ofωresto ensure systemsufficient PM. However, considering the variation ofωres, which will approachωs/3[10], because of the delay effect, the limit ofωcscould be relaxed, especially forωres<ωs/3. In general,ωcscould be set at approximately 0.45ωres. Moreover,ωcsshould be greater than the highest resonant frequencyωmof the controller and lesser thanωs/10, that is,ωs/10 >ωcs>ωm.
Step 2
.
K
is designed based on the stability requirement.

The value range ofKcan be obtained from Eq. (15), (17), or (18), which depends onωres. We noted thatMx≥ 1.414 (x= 1, 2) forMx> 1 andMx≤ 0.8 forMx< 1 are required to ensure robust system stability. Then, we select a suitable value from the value range ofK, with the compromise of dynamic performance and robustness.
Step 3
.
ω_{c}
is designed based on the deviation range of the grid fundamental frequency.

Based on the definition of bandwidth, the difference of the two frequencies where the gain of the resonant part is equal tois the resonant bandwidth. Suppose that the maximum allowable deviation of the grid fundamental frequency is Δf, thus,ωc= 2πΔfcan be obtained.
Step 4
.
is designed based on the steadystate error requirements.

Based on the requirements of the amplitude steadystate errorsεiandεuh,can be calculated from Eqs. (24) and (25). To ensure a larger system PM,should have a smaller value.
Step 5
.
K_{p}
is designed based on the PM requirement.

After the previous steps, the phase response of the system can be obtained by drawing the Bode plot of the loop gainT(s) from Eqs. (2) and (7), withKp/n= 1. We check the PM at the predesignedωcs. If the PM satisfies the requirement in Step 1,ωcsremains unchanged; if not, a larger PM should be selected to ensure system stability with an acceptableωcs. Then, we calculateKpfrom Eq. (26).
 B. Design Example
Based on a 5 kW prototype in the laboratory, two different filter capacitor values are considered to range the filter resonance frequency. The parameters of the LCLtype PWM converter are given in
TABLE I
. We consider four resonant controllers (
h
= 1, 5, 7, 11). Given that
f_{res}
in Case I is close to
f_{s}
/6,
K
>
K_{c}
is preferred rather than
K
<
K_{c}
. From
[20]
, the maximum deviation of the grid fundamental frequency is approximately 0.5 Hz. As such, Δ
f
is equal to 0.5 Hz in Step 3. The design procedures and results are shown in
Table II
, where GM
_{1}
and GM
_{2}
denote the GM around
ω_{res}
and
ω_{s}
/6, respectively. The parameters of the quasiPR controller corresponding to Eq. (4) are
K_{p}
= 9.6,
K
_{r1}
= 180, and
K_{rh}
= 84 for Case I and
K_{p}
= 7.8,
K
_{r1}
= 146.25, and
K_{rh}
= 68.25 for Case II.
LCL FILTER SYSTEM PARAMETERS
LCL FILTER SYSTEM PARAMETERS
DESIGN PROCEDURE AND RESULTS
DESIGN PROCEDURE AND RESULTS
Fig. 9
shows the Bode diagrams of the grid current loop before and after compensation with different controllers. By comparison, we observed that the phase lag introduced by the controller will shift the −180° crossing point, which could improve system robustness to a certain extent, especially for
f_{res}
close to
f_{s}
/6 when the active damping loop is unstable. Specifically, for Case I, the GM increases from GM
_{1}
= 0.898 dB and GM
_{2}
= −0.782 dB with one resonant controller (
n
= 1) to GM
_{1}
= 1.27 dB and GM
_{2}
= −1.27 dB with four resonant controllers (
n
= 4), but the PM decreases from 33.7° to 31.2°.
Bode diagrams of the grid current loop before and after compensation with different controllers: (a) Case I and (b) Case II.
For Case II, the GM increases slightly from 2.21 dB to 2.27 dB and the PM decreases from 34.1° to 29.3°. Moreover, considering the delay effect, we observed that the LCL resonant frequency deviates from
ω_{res}
. The actual resonant angular frequency
and the actual damping ratio
ξ'
are derived in the Appendix and expressed as follows:
In Eq. (27), by letting
, obtaining sin(1.5
T_{s}ω
) > 0,
f
(
ω
) > 1 for
ω
<
ω_{s}
/3 and sin(1.5
T_{s}ω
) < 0,
f
(
ω
) < 1 for
ω
>
ω_{s}
/3 becomes relatively easy. As a result, with the increase in
K
,
is greater than
ω_{res}
for
ω_{res}
<
ω_{s}
/3 and lesser than
ω_{res}
for
ω_{res}
>
ω_{s}
/3, but never exceeds
ω_{s}
/3. This finding means that
will be close to
ω_{s}
/3 with the increase in
K
. We noted that, for
K
=
K_{c}
, the active damping loop is marginally stable with
ω_{pc}
=
ω_{s}
/6 and has no contribution to the resonance damping,
=
ω_{s}
/6.
In the actual condition,
L
_{1}
and
C
do not significantly change, except for
L
_{2}
(considering the impact of grid impedance).
Fig. 10
shows the Nyquist plots of the grid current loop around the critical point when
L
_{2}
is decreased by 50% or increased by 100%. This finding indicates that the grid current loop remains stable for both cases when
L
_{2}
is decreased by 50% or increased by 100%. Nevertheless, the PM of Case I changes from 23.9° to 33.3° [see
Fig. 10
(a)] and the PM of Case II changes from 26.6° to 2.07° [see
Fig. 10
(b)]. We noted that the GM decreases significantly for Cases I and II, and the active damping loop of Case II becomes unstable when
L
_{2}
is decreased by 50%. Thus, to improve system robustness,
K
should have a larger value in Step 2 when the active damping loop is stable.
Nyquist plots of the grid current loop around the critical point when L_{2} changes from −50% to 50%: (a) Case I and (b) Case II.
VI. EXPERIMENTAL VERIFICATION
A 5 kW prototype has been constructed in the laboratory to verify the effectiveness of the proposed design method. The key parameters of the prototype are listed in
TABLE I
. A
Yy
type galvanic isolation transformer is placed between the LCLtype PWM converter and the grid. The grid voltages and currents are sensed by voltage/current halls. The control algorithm is implemented in a 32bit floatpoint digital signal processor (TMS320F28335). The quasiPR controller is discretized by Tustin transformation. In this study, the capacitor current is indirectly sensed through the difference between
i
_{1}
and
i
_{2}
, and the current waveforms are inverted on the oscilloscope.
Fig. 11
shows the experimental waveforms at no load for Cases I and II, where the current tracking error of
α
axis
e_{iα}
is measured through the analogtodigital conversion interface on the control board. From the spectra of
u_{ga}
, we observed that loworder harmonics exist in the real power grid, and the total harmonic distortion (THD) is 2.293%. The measured steadystate errors normalized with respect to
u_{g}
are listed in
TABLE III
. Considering the current distortion and the effect of dead time, the errors are slightly larger than the actual values of
ε
_{u1}
, which are calculated from Eq. (21) using the designed parameters.
Experimental waveforms at no load for (a) Case I and (b) Case II. From top to bottom: the DC link voltage u_{dc}, the grid voltage u_{ga}, the grid current i_{2a}, the spectrum of u_{ga}, and the current tracking error of αaxis e_{iα}.
MEASURED RESULTS
Fig. 12
shows the experimental waveforms when the DC resistant load is 40 Ω, with power factor (PF) set to 1.0. We observed that the grid current is sinusoidal and the harmonics at the target frequencies (5th, 7th, and 11th) have been well suppressed. Moreover, the resonant peak is not dampened to less than 0 dB, which coincided with the design results. However, considering the leakage inductance of the isolation transformer, the actual resonant frequency
is slightly lesser than the theoretical value shown in
Fig. 9
. The actual resonant frequencies are approximately 1,900 Hz for Case I and 1,650 Hz for Case II. The measured error
e_{iα}
, PF, and harmonic contents at the target frequencies are listed in
TABLE III
. The measured errors are 0.524% for Case I and 0.636% for Case II, which are slightly smaller than that at no load because the command–current error
ε_{i}
at rectifier mode can cancel some of the voltage–current error
ε
_{u1}
, which can be observed in Eq. (19) when the direction of current
i
_{2}
at inverter mode is positive in the actual system. Given that the gain at
ω
_{1}
for Case I is larger than that for Case II, the measured error and PF for Case I are slightly smaller than that for Case II.
Experimental waveforms with a DC resistant load R = 40 Ω and PF set to 1.0 for (a) Case I and (b) Case II. From top to bottom: the DC link voltage u_{dc}, the grid voltage u_{ga}, the grid current i_{2a}, the spectrum of i_{2a}, and the current tracking error of αaxis e_{iα}.
To evaluate dynamic performance, the LCLtype PWM converter operating without the outer voltage loop is used. The DC link voltage is provided by a threephase noncontrolled rectifier.
Fig. 13
shows the transient experimental results when the grid current reference
ranges between 1 A and 5 A for Cases I and II with PF set to 1.0. We observed that the inverters rapidly responded to the reference change and the current tracking error of
α
axis
e_{iα}
is sustained at approximately zero all the time. Nevertheless, oscillation occurs during the current step change because the resonant peaks are not dampened to less than 0 dB, which implies that the actual damping ratios are small because of the delay effect. Based on Eq. (28), the actual damping ratio
ξ'
can be calculated at approximately 0.07 for Case I and 0.01 for Case II.
Transient experimental results at inverter mode when the grid current reference ranges between 1 A and 5 A: (a) Case I and (b) Case II.
VII. CONCLUSIONS
In this study, we analyzed the characteristics and controller design method for the digitally controlled LCLtype PWM converter based on the multiresonant quasiPR controller and capacitor–current–feedback active damping. The effect of the delay on the active damping performance is investigated by using the Nyquist diagrams. If the damping loop is unstable, two RHP openloop poles are generated in the grid current loop, which is codetermined by the LCL resonant frequency (
f_{res}
) and the active damping gain (
K
). Then, the system stability constraint condition can be obtained based on the Nyquist stability criterion. Moreover, impact analysis of the control parameters on the current loop performance identifies that different control parameters play different decisive roles in the current loop performance:
K
mainly influences the system stability, the relative resonant gain mainly regulates the steadystate error, and the proportional gain mainly affects the PM of the system. Based on the analysis, a simplified controller design method based on the system specifications is proposed. The proposed method can obtain the optimum controller, which ensures system stability with high robustness and strong ability to suppress the effect of the grid voltage loworder harmonics. Following the method, two design examples are given and the design results are directly used on a laboratory prototype. The experimental results are consistent with the design specifications. These findings confirm the practicability and operability of the proposed design method.
Acknowledgements
This work was supported by the National Basic Research Program of China under Award No. 2010CB227206.
BIO
Yongcan Lyu was born in Hubei province, China, in 1985. He received his B.E. degree in Automation from Northeastern University, Shenyang, China, in 2007 and his M.S. degree in Thermal Engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2009. He is currently working toward his Ph.D. degree in Electrical Engineering at the School of Electrical and Electronic Engineering, HUST, Wuhan, China. His current research interests include digital control technique and battery energy storage systems.
Hua Lin was born in Wuhan, China, in 1963. She received her B.S. degree in Industrial Automation from Wuhan University of Technology, Wuhan, China, in 1984; her M.S. degree in Electrical Engineering from Naval University of Engineering, Wuhan, China, in 1987; and her Ph.D. degree in Electrical Engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2005. From 1987 to 1999, she was with the Department of Electrical Engineering, Naval University of Engineering, as a lecturer and associate professor. Since 1999, she has been with the School of Electrical and Electronic Engineering, HUST, where she became a full professor in 2005. From October 2010 to April 2011, she has been a visiting scholar with the Center for Advanced Power Systems, Florida State University, Tallahassee, FL, USA. She has been engaged in research and teaching in the field of power electronics and electrical drives. Her research interests include highpower, highperformance AC motor drives, novel power converters, and their control. She has authored or coauthored more than 30 technical papers in journals and conferences. Dr. Lin received the secondgrade National Scientific and Technological Advance Prize of China in 1996 and 2003.
Vazquez S.
,
Lukic S. M.
,
Galvan E.
,
Franquelo L. G.
,
Carrasco J. M.
2010
“Energy storage systems for transport and grid applications,”
IEEE Trans. Ind. Electron.
57
(12)
3881 
3895
DOI : 10.1109/TIE.2010.2076414
Singh B.
,
Singh B. N.
,
Chandra A.
,
AlHaddad K.
,
Pandey A.
,
Kothari D. P.
2004
“A review of threephase improved power quality ACDC converters,”
IEEE Trans. Ind. Electron.
51
(3)
641 
660
DOI : 10.1109/TIE.2004.825341
Tang S.
,
Peng L.
,
Kang Y.
2011
“Active damping method using GridSide current feedback for active power filters with LCL filters,”
Journal of Power Electronics
11
(3)
311 
318
DOI : 10.6113/JPE.2011.11.3.311
Hu G.
,
Chen C.
,
Shanxu D.
2013
“New active damping strategy for LCLFilterBased GridConnected inverters with harmonics compensation,”
Journal of Power Electronics
13
(2)
287 
295
DOI : 10.6113/JPE.2013.13.2.287
He J.
,
Li Y.
2012
“Generalized closedloop control schemes with embedded virtual impedances for voltage source converters with LC or LCL filters,”
IEEE Trans. Power Electron.
27
(4)
1850 
1861
DOI : 10.1109/TPEL.2011.2168427
Zhou L.
,
Yang M.
,
Liu Q.
,
Guo K.
2013
“New control strategy for threephase gridconnected LCL inverters without a phaselocked loop,”
Journal of Power Electronics
13
(3)
487 
496
DOI : 10.6113/JPE.2013.13.3.487
Parker S. G.
,
McGrath B. P.
,
Holmes D. G.
2014
“Regions of active damping control for LCL filters,”
IEEE Trans. Ind. Appl.
50
(1)
424 
432
DOI : 10.1109/TIA.2013.2266892
Bao C.
,
Ruan X.
,
Wang X.
,
Li W.
,
Pan D.
,
Weng K.
2012
“Design of injected grid current regulator and capacitorcurrentfeedback activedamping for LCLtype gridconnected inverter,”
in Proc. IEEE Energy Conversion Congress and Exposition (ECCE)
579 
586
Bao C.
,
Ruan X.
,
Wang X.
,
Li W.
,
Pan D.
,
Weng K.
2014
“Stepbystep controller design for LCLtype gridconnected inverter with capacitor–currentfeedback ActiveDamping,”
IEEE Trans. Power Electron.
29
(3)
1239 
1253
DOI : 10.1109/TPEL.2013.2262378
Pan D.
,
Ruan X.
,
Bao C.
,
Li W.
,
Wang X.
2014
“CapacitorCurrentFeedback active damping with reduced computation delay for improving robustness of LCLType GridConnected inverter,”
IEEE Trans. Power Electr.
29
(7)
3414 
3427
DOI : 10.1109/TPEL.2013.2279206
Buso S.
,
Mattavelli P.
2006
Digital Control in Power Electronics
Morgan & Claypool
17 
66
Holmes D. G.
,
Lipo T. A.
,
McGrath B.
,
Kong W. Y.
2009
“Optimized design of stationary frame three phase AC current regulators,”
IEEE Trans. Power Electron.
11
(24)
2417 
2426
DOI : 10.1109/TPEL.2009.2029548
Zhang X.
,
Spencer J. W.
,
Guerrero J. M.
2013
“Smallsignal modeling of digitally controlled GridConnected inverters with LCL filters,”
IEEE Trans. Ind. Electron.
60
(9)
3752 
3765
DOI : 10.1109/TIE.2012.2204713
Liserre M.
,
Teodorescu R.
,
Blaabjerg F.
2006
“Multiple harmonics control for threephase grid converter systems with the use of PIRES current controller in a rotatiing frame,”
IEEE Trans. Power Electron.
21
(3)
836 
841
DOI : 10.1109/TPEL.2006.875566
Lascu C.
,
Asiminoaei L.
,
Boldea I.
,
Blaabjerg F.
2007
“High performance current controller for selective harmonic compensation in active power filters,”
IEEE Trans. Power Electron.
22
(5)
1826 
1835
DOI : 10.1109/TPEL.2007.904060
Yepes A. G.
,
Freijedo F. D.
,
Lopez O.
,
DovalGandoy J.
2011
“Analysis and design of resonant current controllers for voltagesource converters by means of nyquist diagrams and sensitivity function,”
IEEE Trans. Ind. Electron.
58
(11)
5231 
5250
DOI : 10.1109/TIE.2011.2126535
Yepes A. G.
,
Freijedo F. D.
,
Lopez O.
,
DovalGandoy J.
2011
“High performance digital resonant controllers implemented with two integrators,”
IEEE Trans. Power Electron.
26
(2)
563 
576
DOI : 10.1109/TPEL.2010.2066290
Zmood D. N.
,
Holmes D. G.
2003
“Stationary frame current regulation of PWM inverters with zero steadystate error,”
IEEE Trans. Power Electron.
18
(3)
814 
822
DOI : 10.1109/TPEL.2003.810852
Teodorescu R.
,
Blaabjerg F.
,
Liserre M.
,
Loh P. C.
2006
“Proportionalresonant controllers and filters for gridconnected voltagesource converters,”
IEE Proc. Electric Power Applications
153
(5)
750 
762
DOI : 10.1049/ipepa:20060008
IEEE Recommended practice for utility interface of photovoltaic (PV) systems
2000
IEEE Standard 929
Castilla M.
,
Miret J.
,
Matas J.
,
de Vicuna L. G.
,
Guerrero J. M.
2009
“Control design guidelines for singlephase gridconnected photovoltaic inverters with damped resonant harmonic compensators,”
IEEE Trans. Ind. Electron.
56
(11)
4492 
4501
DOI : 10.1109/TIE.2009.2017820
Castilla M.
,
Miret J.
,
Matas J.
,
de Vicuna L. G.
,
Guerrero J. M.
,
Abeyasekera T.
2008
“Linear current control scheme with series resonant harmonic compensator for singlephase gridconnected photovoltaic inverters,”
IEEE Trans. Ind. Electron.
55
(7)
2724 
2733
DOI : 10.1109/TIE.2008.920585
Huacheng Y.
,
Hua L.
,
Yongcan L.
,
Yong L.
,
Xingwei W.
2013
"A multiresonant PR inner current controller design for reversible PWM rectifier,"
in Proc. IEEE Applied Power Electronics Conference and Exposition (APEC)
316 
320
Li B.
,
Zhang M.
,
Huang L.
,
Hang L.
,
Tolbert L. M.
2013
“A robust multiresonant PR regulator for threephase gridconnected VSI using direct pole placement design strategy,”
in Proc. IEEE Applied Power Electronics Conference and Exposition (APEC)
960 
966
Rodriguez P.
,
Pou J.
,
Bergas J.
,
Candela J. I.
,
Burgos R. P.
,
Boroyevich D.
2007
“Decoupled double synchronous reference frame PLL for power converters control,”
IEEE Trans. Power Electron.
22
(2)
584 
592
DOI : 10.1109/TPEL.2006.890000
Gabe I. J.
,
Montagner V. F.
,
Pinheiro H.
2009
“Design and implementation of a robust current controller for VSI connected to the grid through an LCL filter,”
IEEE Trans. Power Electron.
24
(6)
1444 
1452
DOI : 10.1109/TPEL.2009.2016097
Tang Y.
,
Loh P.
,
Wang P.
,
Choo F.
,
Gao F.
2012
“Exploring inherent damping characteristic of LCLfilters for threephase gridconnected voltage source inverters,”
IEEE Trans. Power Electron.
27
(3)
1433 
1443
DOI : 10.1109/TPEL.2011.2162342