IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014
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Robust Single-Loop Direct Current Control of LCL-Filtered Converter-Based DG Units in Grid-Connected and Autonomous Microgrid Modes Alireza Kahrobaeian, Student Member, IEEE, and Yasser Abdel-Rady I. Mohamed, Senior Member, IEEE
Abstract—This paper presents a robust direct single-loop current control scheme based on structured singular value (μ) minimization approach for induced-capacitor-inductor (LCL)-filtered distributed generation converters in grid-connected and isolated microgrid modes. Unlike the conventional H∞ -based approach, the proposed interface maintains perturbed system stability, under wide range of grid (or microgrid) impedance variation, without the application of any additional damping loops. Moreover, the performance of the perturbed system in terms of grid-voltage and harmonic disturbance rejection can be improved significantly by the adopted method. This is due to the less conservative nature of the μ-synthesis-based solution as it takes advantage of the additional structure introduced to the uncertainty block by the performance criteria. The salient features of the proposed controller are 1) single-loop direct current control of LCL-filtered converters with inherent damping of the LCL filter resonance without any need for additional damping loops; 2) robust stability and active damping performances by mitigating the LCL resonance under wide range of grid (or microgrid) impedance variation; 3) improving the performance of the current controller by removing its dependency on the grid-voltage feed-forward loop by providing high disturbance rejection feature against fundamental and harmonic voltage disturbances; 4) computationally efficient fixed-order structure with minimum sensor requirements (only grid-side currents are needed for feedback control). A comparative theoretical analysis, timedomain simulation results, and experimental test results are presented to show the effectiveness of the proposed control scheme. Index Terms—Distributed generation, grid-connected and isolated micro-grids, LCL filter, PWM inverters, robust control, stability.
I. INTRODUCTION HE energy sector is moving to the era of smart grids [1]–[3]. Effective integration of distributed and renewable energy resources is essential requirement to achieve the smart grid vision. Renewable distributed generation (DG) units, such as photovoltaic (PV) arrays and wind turbines, are often
T
Manuscript received April 8, 2013; revised July 19, 2013 and October 11, 2013; accepted November 29, 2013. Date of current version May 30, 2014. Recommended for publication by Associate Editor M. Liserre. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 Canada (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2013.2294876
operated in the current control mode in order to control both the power factor and harmonic content of the injected current. The current control mode is suitable for grid-connected applications and it can be used in isolated microgrids with appropriate energy management strategy. Currently, there is a growing interest in using inductor–capacitor–inductor (LCL) ac-side filters in grid-connected inverters, which yield better attenuation of switching harmonics. Accordingly, an LCL-filtered inverter is suitable for high power DG applications, which are characterized by low switching frequencies (few kHz) [1]–[5]. However, low- and high-frequency instabilities can occur due to the resonance frequency introduced by the LCL filter and interactions between the current controller and grid impedance [1]- [5]. The associated undamped resonance peak also limits the open loop gain; thus compromising the controller’s bandwidth yielding to very slow response times [1]. Further, without resonance damping, the resonance mode can be excited during network or converter disturbance. Different damping solutions are proposed in order to remove the LCL filter resonance where the active methods are preferred over the passive ones due to their minimized power losses [6]–[18]. Active resonance damping can generally be achieved by utilizing multiloop control system or full-state feedback control. The inner feedback loops can be from the inverter-side inductor current [8], [9], the filter capacitor current [10], [11] or the filter capacitor voltage [12], [13]. Full-state feedback control is adopted in [14] and [15] to provide resonance damping and achieve complete system controllability. However, these methods require additional number of sensors which might not be considered as an optimized solution. Other resonance damping techniques include the use of special filter and/or sensor arrangement such as split capacitor or two-sensor feedback control system [16], [17]. A filter-based approach is presented in [18] where a notch filter is added to damp resonance peak. Single-loop control structure without using extra damping loops is investigated in [4], [19]–[21]. The stability of the singleloop LCL-interfaced system for different control bandwidth frequencies is investigated in [4]. The analysis suggests that the presence of the LCL resonance peak limits the control bandwidth of the PI controller. Therefore, the closed-loop system dynamics should be chosen to be slower in order to avoid instability. It is shown that stable system operation can be achieved by a careful choice of filter parameters using PI controller [19]. The study in [20] confirms the necessity of small loop gains
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in order to avoid instability. Note that the limited controller dynamics in [4], [19]–[21] makes the adoption of harmonic compensation [3], which is a necessary feature especially in active filter applications, impossible. Moreover, the effect of line parameter uncertainty over system stability has not been addressed in the preceding works reporting single-loop structure. This is especially important considering the fact that even in the multiloop active damping structure, when adopting harmonic compensators, the line inductance uncertainty can easily shift the open-loop cross-over frequency; therefore bringing the frequencies of the harmonic compensators outside the bandwidth of the system leading to instability [1], [3]. A single-loop frequency-domain stabilizing controller is provided in [21] that follows norm minimization method through convex optimization similar to H∞ . Although the proposed controller is proven to stabilize the nominal plant by introducing inherent active damping, the robust stability is not guaranteed for a high range of system uncertainty as plant uncertainties are not considered in the design procedure. Moreover, no control effort is added to reject the harmonic disturbances. Note that deadbeat and predictive current controllers which provide fast dynamics also are highly dependent on system modeling which makes them sensitive toward parameter variation and demand robust or adaptive control design [3]. Motivated by the aforementioned limitations, this paper aims at addressing the following objectives using one single-loop controller: 1) Single loop stabilization with inherent LCL resonance peak damping. 2) Stable system operation under wide range of grid (or microgrid) impedance variation. It should be noted that one important aspect of current-controlled operation of DG units is their performance in an isolated microgrid system where they should continue their stable operation while the voltage is maintained via other dispatchable units [25], [26]. However, the stable operation of such a current-controlled unit can be compromised when connected to a microgrid system due to the lack of “stiff grid” concept. The possible voltage variations as well as higher lines uncertainties in a microgrid operating regime, calls for the adoption of more robust interfaces that not only can work in grid-connected mode but also are able to maintain system stability when connected to a microgrid with higher line and voltage uncertainties. 3) Tracking and selective harmonic compensation feature as it should provide high disturbance rejection performance against fundamental and harmonic voltage disturbances. It should be noted that in order to meet all the aforementioned objectives the proposed controller should achieve an acceptable tradeoff between the controller bandwidth and the stable parameter uncertainty range. Robust control based on structured singular value μ minimization [28]–[30] is adopted in this paper it provides less tradeoff as compared to the conventional H∞ control. Robust H∞ control is reported in many converter-based applications [22]–[24]. H∞ -norm ( • ∞ ) minimization approach is reported in [22] to stabilize perturbed LCL-interfaced DG units where the line parametric uncertainty is modeled as a lumped-
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014
sum unstructured block. However, the suggested scheme fails to provide a single-loop structure and still requires the adoption of the inner current active damping loop. Moreover, the proposed interface fails to provide harmonic filtering feature which is necessary for active filter applications and therefore depends on grid voltage feedforward technique for rejecting the gridside voltage disturbances. Therefore, motivated by the aforementioned limitations, this paper presents an improved direct single-loop current control scheme based on structured singular value μ minimization approach. Unlike the conventional H∞ based approach, the proposed interface is shown to be able to maintain perturbed system stability without the application of any additional damping loops. Moreover, the performance of the perturbed system in terms of grid-voltage and harmonic disturbance rejection can be improved significantly by the adopted method. This is due to the less conservative nature of the μsynthesis-based solution as it takes advantage of the additional structure introduced to the uncertainty block by the performance criteria [29]. In this method, the unstructured uncertainty block adopted in H∞ approach is replaced by a structured one as a fictitious uncertainty block associated with the performance criteria is added. It will be shown that the solution achieved by this method can significantly enhance the robust performance of the system as compared to the H∞ -based method. Mathematical and comparative analyses are provided to show the advantages of the proposed μ-synthesis approach over the conventional H∞ controller in maintaining robust stability as well as the performance. The performance of the proposed controller in mitigating fundamental and harmonic disturbances is investigated in both grid-connected and microgrid regimes using time-domain simulation and experimental studies. The remainder of this paper is organized as follows. System configuration and problem statement are presented in Section II. The proposed control scheme is presented in Section III. Simulation and experimental results are provided to evaluate the effectiveness of the proposed control scheme in Sections IV and V, respectively. Conclusion is drawn in Section VI. II. PROBLEM STATEMENT AND SYSTEM CONFIGURATION Fig. 1 shows the line diagram of the microgrid study system adapted from the IEEE 399 standard for low voltage applications [27]. Without loss of generality only three DG units are assumed to be connected where DG1 is the current-controlled unit (e.g., PV or wind-based unit) and DG2 and DG3 are considered to be the voltage-controlled (i.e., dispatchable) units. A typical microgrid system can operate either in grid-connected or islanded mode depending whether the switch SW is closed or open. In grid connected mode, the demanded power is supplied by both the utility and DG units. However, once being disconnected from the utility, the dispatchable voltage-controlled units (i.e., DG 2,3) are responsible for supplying the total connected load via droop control. In this case, DG1 is considered to inject the maximum power available and therefore can also be considered as a negative load. Fig. 2 shows the conventional control interface of the voltage-controlled units of DG2 and DG3 which consists of 1) droop-controlled power sharing which sets the
KAHROBAEIAN AND MOHAMED: ROBUST SINGLE-LOOP DIRECT CURRENT CONTROL OF LCL-FILTERED CONVERTER-BASED DG UNITS
Fig. 1.
Microgrid study system with voltage- and current-controlled DG units.
Fig. 2.
Dispatchable droop/voltage-controlled DG unit hierarchical interface.
output voltage phase and magnitude; 2) voltage control loop to yield close control characteristics of the output voltage; and 3) inner current loop to control the filter inductor current and actively damping the resonance peak of the output LC filter. In Fig. 2, rf and Lf are the inverter-side resistance and inductance respectively, rc and Lc are the grid-side filter parameters whereas Cf is the filter capacitor, vo is the filter capacitor voltage, io is the injected grid-side current, iinv is the inverter output current, and vinv is the inverter output voltage. The droop control sets the output voltage phase and magnitude by drooping output voltage frequency and magnitude, respectively, via average power measurements as shown in (1) and (2) ω = ωn l − m(Pav − P ∗ ) Vo∗
∗
= Vn l − n(Qav − Q )
(1) (2)
where P ∗ , Q∗ , ωn l , and Vn l are the nominal active and reactive powers, frequency, and voltage set points, respectively, and m and n are the static droop gains. The set points in (1) act as a virtual communication agent for different inverters for autonomous operation. Also, the d-component of the output voltage is used
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in (2); as for the voltage-oriented control, the reference of the output voltage magnitude is aligned with the d-axis of the inverter reference frame. The average active and reactive powers corresponding to the fundamental components are obtained by means of a low-pass filter (LPF) as ωc p (3) Pav = s + ωc ωc q. (4) Qav = s + ωc The static droop gains are selected to achieve correct powersharing performance under different unit ratings according to (5) where S is the rated apparent power, and subscripts i, j denote two microgrid-connected units mi Si = mj Sj , ni Si = nj Sj
∀ i, j.
(5)
The voltage reference provided by the droop controller is applied to the voltage control loop, which in turn regulates the inverter output voltage. More details on the voltage controlled DG interface and their operation in microgrid and grid-connected modes can be found in [25] and [26], respectively. Note that several studies have been reported on the stability of the
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Fig. 3.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014
Block diagram of the current-controlled DG unit interface with LCL filter.
voltage controlled units in microgrid application, however very few research works have been done on the stable operation of current controlled units in isolated microgrid mode. Fig. 3 shows the closed-loop block diagram of the current controlled DG unit (DG1) with LCL filter where ioRef , the reference current, is usually calculated by the higher level dclink voltage control that is in charge of maintaining the dclink voltage according to the DG source power variations. In many applications, the harmonic current of the local load is also added to the reference current in order to compensate for the local load’s harmonics content; resulting in improved grid-side current quality. This provides an excessive feature where the DG unit can operate as a harmonic filter for power quality purposes when needed. However, this requires the current controller to provide high disturbance rejection in the selected harmonics as well as the fundamental one. The additional current feedback loop (from the filter capacitor) which may also be adopted to damp the LCL filter resonance peak is shown in Fig. 3 in dashed line. However, as will be shown in this paper, a direct current control scheme is achievable and it can eliminate the need for this extra loop and the associated sensor and control requirements. Also note that, as shown in Fig. 3, voltage variations at the point of common coupling form an external source of disturbance and thus should be rejected as much as possible. The open-loop dynamics between the output current and the inverter voltage and also the external voltage disturbance can be therefore derived as in (6) where the resonance damping gain Kd has also been considered Gd is
Ginv 1 Lf Cf s2 + Cf (rf + Kd )s + 1 vinv (s)− vPCC (s) io (s) = Γ(s) Γ(s)
(6) 3
2
Γ(s) = Lf Lc Cf s + Cf (Lf rc + Lc (rf + Kd ))s
Fig. 4.
Schematic of the closed-loop system with conventional H ∞ controller.
III. ROBUST CONTROL A. Suboptimal H∞ -Based Robust Control Robust control theory provides a powerful tool to design set of stabilizing controllers that can maintain system stability in presence of parametric uncertainties. Moreover, norm-minimization approach, adopted in robust control design, enables achieving performance criteria such as minimum tracking error and maximized disturbance rejection. Therefore, it seems to be an appealing option to achieve a single-feedback robust current control interface that can provide inherent stabilization as well as the performance objectives. The mixed sensitivity H∞ optimization approach has been adopted in many UPS and DG control applications [21]–[25], where the system uncertainties are modeled as an unstructured but bounded perturbation block. Fig. 4 shows the conventional H∞ optimization problem considering both performance and stability criteria. In Fig. 4, WΔ · Δ denotes the lumped-sum uncertainty block defined by the relative deviation of the perturbed system from its nominal value as in (7) where Ginv Nom and Ginv are the nominal and perturbed transfer functions from Vinv to io , respectively, presented earlier in (6). Note that · ∞ = supω σ[·] and σ[·] stands for the singular value WΔ · Δ∞ = σ
+ ((rf + Kd )rc Cf + Lf + Lc )s + (rf + rc ). Without adopting active damping (i.e., Kd = 0) the resonance frequencies can be seen in both Io /Vinv and Io /VPCC responses in (6) and bring the system to instability. This paper is seeking a control approach that not only eliminates the need for the extra inner current loop by providing inherent damping, but also maintains system stability in presence of high range of line parameter perturbations that might occur during grid connected and/or microgrid operating modes.
Ginv − Ginv Nom Ginv Nom
.
(7)
The weighting WΔ is introduced to normalize Δ (i.e., Δ∞ ≤ 1) and can be obtained based on the worst case relative system uncertainty frequency response. In order to address the tracking and disturbance rejection objectives, the following weighting function is suggested in the form of (8) WPer (s) =
h=1,5,7,11
kh (hω0 )2 s2 + ζ(hω0 )s + (hω0 )2
(8)
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Fig. 5. System relative uncertainty frequency response (dashed) fitted first order W Δ (solid): (a) without active damping (K d = 0) (b) with active damping (K d = 10).
Fig. 6.
(a) Robust stability analysis of K in f without damping (dashed) with damping (solid). (b) Performance of K in f with damping.
where ω0 is the fundamental angular frequency, h is the harmonic number, kh is constant, and ζ is the damping coefficient. Note the resonance mode in (8) provides internal model dynamics at the fundamental and harmonic frequencies to achieve zero steady-state tracking as well as good disturbance rejection at selected harmonics. The suggested weightings make it possible to implement the controller in the stationary reference frame. In Fig. 4, Wu is only added to penalize the controller effort where a small value (0.01) is assigned to fulfill the design convergence condition [29], [30]. In order to apply the robust control theorem, the control structure of Fig. 4 can be rearranged to match the standard 2input, 2-output M-Δ configuration [29], [30]. The H∞ -based robust controller, Kinf can then be designed to fulfill the following mixed-sensitivity optimization problem which follows the robust stability and performance criteria presented in Appendix III M11 (9) M22 < 1 ∞ where M11 models the weighted complementary sensitivity transfer function and M22 is the weighted sensitivity transfer function. The feasibility of designing H∞ -based direct control interface is studied based on (10) where no inner current feedback was applied first (i.e., Kd = 0). The robust stability and nominal performance measures in this case are later compared to the dual-loop structure. Fig. 8(a) and (b) demonstrates the first-order weighting function of WΔ adopted to normalize system deviations in both undamped and damped cases,
respectively. The line-side inductance Lc is assumed to vary up to ten times its nominal values. Note that WΔ should be chosen in a way that (10) holds. Considering the design parameters and the design approach provided in the Appendix III, the suboptimal H∞ controllers for both single-loop and dual-loop cases were designed.
ΔGinv (jω)
∀ω. (10) |WΔ (jω)| ≥ max
Ginv (jω)
Fig. 6(a) compares the robust stability analysis of these two cases based on the small gain theorem presented in Appendix III [30]. It can be seen that without adopting the inner damping loop, the presented H∞ -based controller fails to maintain stability of the perturbed system. However, the robust stability measure of the dual-loop structure suggests that the adoption of the inner damping loop increases the stability margin and therefore can guarantee robust stability of the perturbed system over the modeled uncertainty range and therefore can be considered as an option for robust stable operation. This, however, is achieved with the cost of performance degradation. Fig. 6(b) shows the performance measure of the actively damped H∞ -based controller where, M22 ∞ is greater than 1 implying that the performance criteria cannot be met. Moreover, H∞ -design algorithm fails to achieve robust stability with single-loop structure for the relatively wide range of modeled line perturbations. This suggests that H∞ -based controllers can either be used for a limited range of system uncertainty or more relaxed performance objectives. As it will be discussed in the next section, this can be attributed to the conservative nature of
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Fig. 7.
Standard M-Δ configuration with structured uncertainty block.
Fig. 8. Schematic of the closed-loop system with proposed μ-synthesis controller.
suboptimal-H∞ -based solutions where the structure of the uncertainty block is not accounted for. The designed suboptimalH∞ -based ninth-order controller with inner active damping is discretized using the “Tustin” approximation with 10-kHz sampling time. The resultant controller is shown in (11), as shown at the bottom of the page. B. Direct μ-Synthesis-Based Robust Control According to [29], more information on the structure of the uncertainties provides less conservative solutions as compared to the unstructured case. Considering this extra information not only can increase the stability margin of the perturbed system, it could also improve the performance of the perturbed system. Therefore, some structure can be added to the uncertainty block, if a fictitious normalized uncertainty block ΔP is assumed to be connected to the external inputs and outputs of the system as ˜ can shown in Fig. 7. The new structured uncertainty block of Δ be therefore defined as ˜ = ΔP 0 . Δ (12) 0 Δ Fig. 8 shows the closed-loop scheme with the proposed μsynthesis controller where instead of conventional lump-sum modeling approach, line inductive uncertainty is modeled in
Kinf =
Kμ =
a more detailed way using multiplicative perturbation method. Moreover, a certain structure is added to the uncertainty block by introducing the performance associated block of ΔP as earlier suggested in Fig. 7. Once again, WΔ is the weighting used to normalize the line uncertainty that can vary up to ten times its nominal value. Note that in order to be able to compare the performance of the proposed μ-synthesis controller with the conventional H∞ -based one, same performance weighting function as in (8) is considered for WPer . ˜ By introducing the new structured uncertainty block of Δ, the μ-synthesis approach can now be applied in order to achieve both robust stability and enhanced performance. Note that μ−1 ˜ that makes it is defined as the smallest singular value of Δ singular [28] (more information can be found in the Appendix III). The D-K iteration method, provided by MATLAB robust control toolbox [31], is adopted to compute μ and design the controller. Based on the weighted schemes provided in Fig. 8 and using the design parameters provided in Appendix III, the reduced order μ-synthesis controller Kμ (s) is designed. Note that the proposed controller which is based on structured singular analysis provides a direct current control solution where no inner current control is needed for plant stabilization purposes. Fig. 9 shows the robust stability and performance analysis of the proposed controller. Fig. 9(a) proves that the less conservative solution achieved through this method maintains perturbed system stability as M11 ∞ is under unity. In addition, as can be seen from Fig. 9 (b), the performance of the system is also significantly improved where the sensitivity transfer function is well bounded by the suggested performance criteria (i.e., M22 ∞ < 1). Comparing Fig. 9 with Fig. 6 reveals two important facts about the advantages of the proposed robust control strategy over the counterpart conventional H∞ -based methods: 1) Adding structure to the uncertainty block and using structured singular vale μ analysis results in a direct stabilizing current controller that unlike the conventional methods does not require extra inner-loop damping method to maintain perturbed system stability for relatively wide range of line uncertainties. 2) Comparing the performance analysis in Fig. 9(b) with Fig. 6(b) suggests that even with active damping, robust stability in dual-loop H∞ -based controllers is achieved at the cost of compromised performance. However, the proposed direct control method is well capable of optimizing both stability and performance criteria at the same time.
−0.01378z 9 +0.08736z 8 −0.2238z 7 +0.2713z 6 −0.08614z 5 −0.1845z 4 +0.2723z 3 − 0.1678z 2 + 0.05145z − 0.00639 z 9 − 8.693z 8 + 33.84z 7 − 77.44z 6 + 114.8z 5 − 114.3z 4 + 76.48z 3 − 33.14z 2 + 8.441z − 0.9626 (11)
6.625z 12 − 70.32z 11 +329.8z 10 − 885.5z 9 +1451z 8 − 1343z 7 +274.3z 6 +1026z 5 − 1543z 4 +1147z 3 − 504.4z 2 + 125.6z − 13.77 z 12 − 8.587z 11 +32.95z 10 − 74.06z 9 +107.1z 8 − 102.7z 7 +63.55z 6 − 21.95z 5 +0.4777z 4 +3.649z 3 − 1.985z 2 +0.5414z − 0.0682
(13)
KAHROBAEIAN AND MOHAMED: ROBUST SINGLE-LOOP DIRECT CURRENT CONTROL OF LCL-FILTERED CONVERTER-BASED DG UNITS
Fig. 9.
Fig. 10.
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(a) Robust stability analysis of K μ without damping. (b) Performance of K μ without damping.
Root loci of the discretized system with (a) Conventional single loop P-HERES control (b) Proposed single-loop K μ control.
The designed controller is discretized using the “Tustin” approximation with 10-kHz sampling time. The resultant controller is shown in (13), as shown at the bottom of the previous page. Also the root locus of the proposed controller is shown in Fig. 10 where it is compared to the conventional single-loop proportional-resonant controller with harmonic compensators (P-HRES) presented in Appendix II. Fig. 10 confirms the effectiveness of the proposed controller in damping the filter resonance mode using only one current sensor. Note that in the conventional single-loop methods presented in [4], [19], and [20], stability margins are limited and therefore the loop gain has to be reduced in order to maintain the stability. However, the proposed method enables the current controller to achieve higher dynamics since unlike the conventional single loop controllers, loop gain is not limited by the undamped resonance mode. The response time of the proposed controller is compared to the conventional case in Fig. 11. As can be seen, the proposed method provides better dynamics. It should be noted that this dynamics can be even improved more if the parameter uncertainty range considered in the design procedure is reduced due to the inherent tradeoff between the robust stability and control performance in the robust control design. In order to have a better appreciation of the advantages of the proposed μ-synthesis single loop control over the conventional
dual-loop H∞ -based one, the frequency response of the sensitivity transfer functions in both cases are presented in Fig. 12(a) and (b). Once again, it should be noted that WPer is suggested to provide minimum tracking error and maximized disturbance rejection for the fundamental disturbances and selected (5th, 7th, 11th) harmonics and is assumed to be the same in both μ-synthesis and suboptimal H∞ designs in order to have a fair comparison. As can be seen from Fig. 12(a), the proposed single-loop μ-synthesis controller is well capable of minimizing the output tracking error in the fundamental and selected harmonics. However, as Fig. 12(b) reveals, the performance of the conventional dual-loop H∞ -based controller is compromised. This implies that although the robust stability is achieved as shown in Fig. 6(a) by adopting the inner damping loop, the desired performance is not maintained. Higher sensitivity transfer function in Fig. 12(b) suggests slower dynamics and compromised disturbance rejection of the conventional H∞ -based method as compared to the proposed scheme. As suggested in Fig. 12, the proposed controller is expected to provide higher disturnbance rejection as compared to the conventional H∞ -based robust algorithm. This can be verified by comparing the closed-loop frequency responses from the external voltage disturbance vPCC to the output current io in Fig. 3 for the two cases. The responses are compared in Fig. 13
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Fig. 11.
System’s response times with (a) Conventional single loop P-HERES control (b) Proposed single loop K μ control.
Fig. 12.
Frequency response of system sensitivity transfer function with (a) Single-loop K μ controller. (b) Dual-loop K in f controller.
C. Control Design for Lower Switching Frequencies
Fig. 13. Closed-loop frequency response from the external voltage disturbance to the output current with single-loop K μ controller (solid) with Dual-loop K in f controller (dashed).
where it can be seen that the higher disturbance rejection feature provided by the proposed controller makes the system more robust against external voltage disturbances both in the fundamental frequency and the selected harmonics, therefore eliminating the need to adopt any feed-forward compensators, and reducing the sensor requirements to the minimum (only the grid-side currents are used for feedback control).
In order to achieve faster response dynamics, a 10-kHz switching frequency is adopted in this design. However, the same design method can be easily followed to design μsynthesis-based robust controllers for lower switching frequency applications. The application of the proposed method for 3-kHz switching frequency is shown here as an example. However, it should be noted that the LCL filter parameters should be updated for the new switching frequency based on the following two criteria. First, in order to effectively damp the oscillatory modes using a digital damping controller, the sampling frequency fs should satisfy the Nyquist criterion with respect to the parallel resonance frequency. Therefore, the inequality in (14) should be satisfied for different system parameters [3]
1 Lf + Lc . (14) fs > π Cf Lf Lc Second, it should be noted that as a rule-of-thumb the filter resonance frequency is usually selected around (1/10)th of the switching frequency to ensure high attenuation of the switching harmonics. Therefore, the filter specifications for lower switching frequencies are different. In order to evaluate the performance of the proposed controller using 3-kHz switching
KAHROBAEIAN AND MOHAMED: ROBUST SINGLE-LOOP DIRECT CURRENT CONTROL OF LCL-FILTERED CONVERTER-BASED DG UNITS
Fig. 14.
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Root loci of single loop μ-based controlled system (a) fsw = 3 kHz, (b) fsw = 10 kHz.
frequency, the LCL filter parameters are updated to provide a resonance frequency of 475 Hz to ensure both good attenuation of the switching harmonics and also reasonable bandwidth. The following parameters satisfy the aforementioned filter design criteria: Lf = 2 mH, Lc = 1.2 mH, Cf = 150 μF. Fig. 14 compares the root-loci plots of the designed system, for 10 and 3 kHz, respectively. The proposed controller can damp the filter resonance in both cases while maintaining robust performance under uncertainty in the grid-side inductance. IV. SIMULATION RESULTS To evaluate the performance of the proposed control scheme under both grid-connected and microgrid operating modes, the study system shown in Fig. 1, adopted from the IEEE standard 399 [27] for low voltage application, is implemented for timedomain simulation under MATLAB/Simulink environment. The nominal system parameters are given in Appendix I. The performance of the proposed μ-synthesis current controller is studied in both grid-connected (when the switch SW is ON) or microgrid regime (when the switch SW is OFF). In order to have a better appreciation of the harmonic compensation capability of the proposed controller, it is assumed that a nonlinear dioderectifier type load can also be locally connected to DG1 via switch S1. The performance of the proposed μ-synthesis direct controller in maintaining system stability as well as harmonic compensation is compared to the conventional dual-loop H∞ schemes in different operating conditions and typical system parameters. To study the tracking performance as well as the robust stability of the proposed controller in grid connected mode, Fig. 15 shows DG1 current response as the grid inductance Lg is increased from 0.1 to 4.5 mH at t = 0.6 s to emulate weak grid condition. The local nonlinear load is assumed to be disconnected at this point. The performance of the robustly designed dual-loop H∞ and μ-synthesis controllers is also presented in Fig. 15(a) and (b), respectively. As can be seen, both controllers are well capable of maintaining system stability in weak grid condition. However, it should be noted that in the H∞
Fig. 15. Output current of DG1 when L g is increased from 0.1 mH to 4.5 mH (a) with H ∞ dual loop control (b) with proposed direct μ-synthesis control.
controller, an inner damping loop is adopted whereas the μsynthesis control provides a direct control solution eliminating the need for extra current sensors and multiloop control tuning. Moreover, as Fig. 15(a) shows, the response time of the H∞ controller to the sudden parametric change is higher as compared to the μ-synthesis indicating its slower dynamics (i.e., limited bandwidth). One of the demanding features in current-controlled units is their ability to compensate for current harmonics caused by the connection of local nonlinear loads. Without compensating for the local nonlinear loads, the harmonic content will be injected to the utility grid which affects the overall system power quality indices. The undesirable local harmonics can be compensated from the utility side if the same harmonic contents are generated by the DG unit. This can be achieved by adding the local load current to the reference as shown in Fig. 3. Note that in this case, the DG unit performs as a harmonic filter cancelling out the local nonlinear load harmonics. In order to appreciate the
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Fig. 17. Compensated line current (a) with dual-loop H ∞ control (b) with proposed single-loop μ-control.
Fig. 16. Control performance without compensating the local load nonlinear current (a) Nonlinear local load current (b) DG output current (io ) (c) Line current (IL in e ).
importance of this feature, first, the local current is not added to the reference current (position “1” in Fig. 3). Fig. 16(a)–(c) shows the local nonlinear load current as well as the DG output current and the line current, respectively, when the current reference is first set to be 0.85 p.u. As can be seen, the connection of the nonlinear load at t = 0.6 s results in highly distorted current being injected to the grid. The line current quality can be significantly improved by injecting the same local load harmonics via the DG unit in order to cancel them out from the line current. In order to achieve this objective, the controller should however, provide good harmonic compensation feature (i.e., low sensitivity transfer function) at the selected harmonics. The harmonic compensation ability of the H∞ -based and μ-synthesis controllers are therefore evaluated and compared by injecting the local current to the reference signal (position “2” in Fig. 3). Fig. 17(a) and (b) shows the compensated line currents when dual-loop H∞ and proposed direct μ-synthesis controllers are adopted, respectively. Fig. 17 reveals the higher harmonic compensation capability of the μ-synthesis controller as the current THD is significantly improved as compared to the H∞ case. It should be noted that the enhanced line current quality is achieved by the filtering act of the DG unit; therefore, its output current io is expected to contain the local load harmonic contents in order to filter them out from the line current. Fig. 18 demonstrates the DG output current that has been generated by the μ-synthesis direct current controlled DG unit in order to compensate for the local load harmonics. In order to evaluate the performance of the system during transition from grid-connected to microgrid mode, an intentional
Fig. 18. DG output current generated to compensate for the local nonlinear load with μ-control.
islanding event is created at t = 1 s by switching SW OFF. The local nonlinear load is assumed to be disconnected in this case and the current reference is selected to be 0.88 p.u. resulting in 7-kW output power generated by DG1 in both grid connected and islanded operating modes. However, the droop-voltage controlled units (i.e., DG 2,3) are expected to regulate their output power to match with the local power demand as the system switches to the microgrid operational mode. Fig. 19 (a) shows the generated power by each DG unit whereas Fig. 19(b) and (c) presents the current responses when conventional dual-loop H∞ , proposed direct μ-synthesis are adopted, respectively. Note that, by disconnecting the microgrid system of Fig. 1 from the grid, the effective line inductance seen by DG1 will be changed yielding to unstable operation. Fig. 19 suggests that both dualloop H∞ and direct μ-synthesis controllers are well capable of maintaining stability as verified by the robust stability analysis earlier in Figs. 6 and 9. It should be noted however that comparing Fig. 19(b) and (c) suggests the slower dynamics of the dual-loop H∞ controller as compared to the proposed direct μ-synthesis-based interface during the transition period. The lack of stiff grid in microgrid operating mode can make the system’s voltage prone to variations and disturbances. Such voltage uncertainty at the point of common coupling, as discussed earlier in Fig. 3, acts as an external source of disturbance on the current controlled units affecting their output current. Fig. 20(a) and (b) shows the performance of the dual-loop H∞ and μ-synthesis controllers, respectively, in rejecting a 10% voltage sag event created by the distant DG unit (i.e., DG2) at
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Fig. 21. Closed-loop output current response when the distorted voltage is applied (a) Dual-loop H ∞ control (b) Proposed single-loop μ-control.
Fig. 19. Microgrid system response due to the islanding event (a) power responses (b) DG1 output current with dual-loop H ∞ control (c) DG1 output current with proposed single-loop μ-control.
Fig. 22. Comparative current harmonic analysis in presence of voltage harmonics for conventional dual-loop H ∞ and single-loop μ-controllers.
Fig. 20. Closed-loop output current response when a 10% voltage sag is applied (a) dual-loop H ∞ control (b) Proposed single-loop μ-control.
t = 0.4 s. Fig. 20 reveals the better disturbance rejection capability of the proposed controller as compared to the conventional H∞ case. It can be seen that the faster dynamics provided by the proposed single-loop controller enables it to remarkably reduce transient current overshoots as compared to its dual-loop H∞ based counterpart. However, it should be noted that the voltage disturbances are not limited to the fundamental frequency. The presence of nonlinear loads especially in a microgrid system can introduce voltage harmonic distortions due to the lack of stiff grid concept and therefore the current controlled units’ ability in rejecting selected voltage harmonics from their output current is of outmost importance for microgrid stable operational mode. In order to investigate the voltage harmonic rejection feature of
the proposed controller, 0.6% fifth and 0.2% seventh harmonic distortions are introduced to the microgrid voltage. The output current of DG1 in the presence of the voltage harmonic content is compared with the dual-loop H∞ and μ-synthesis controllers as shown in Fig. 21(a) and (b), respectively. Comparing the output current THD in both cases reveals that the proposed μ-synthesis controller is demonstrating better voltage harmonic rejection as compared to its counterpart. This agrees with the analysis provided earlier in Fig. 13. The fifth and seventh harmonic content of the currents of Fig. 21 are compared in Fig. 22 in order to provide better appreciation of the enhanced harmonic disturbance rejection of the proposed controller compared to the conventional one. As can be seen, the fifth and seventh harmonics (% of fundamental) are reduced significantly by the adoption of the μ-synthesis controller which is the reason for the improved current THD from 4.35% to 0.35%. V. EXPERIMENTAL RESULTS To validate the presented analysis and proposed control design, a 1.0 kW, 208 V, 60 Hz, laboratory-scale grid-connected converter system, shown in Fig. 23, is used. A semistack IGBT voltage-source converter is used to interface the DG unit to the utility grid through the output LCL filter. The dSpace1104
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Fig. 23.
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View of the experimental setup.
control system is used to implement the proposed control scheme in real time. The pulse width modulation algorithm is implemented on the slave processor (TMS320F240-DSP) of the dSPACE controller. The dSPACE1104 interfacing board is equipped with eight digital-to-analog channels (DAC) and eight analog-to-digital channels (ADC) to interface the measured signals to/from the control system. The software code is generated by the Real-Time-WorkShop under a MATLAB/Simulink environment. A variable resistive load box is used as the output load. The sampling/switching frequency is 10 kHz. The current and voltage sensors used are HASS 50-S and LEM V 25-400, respectively. The LCL filter parameters are Lf = 1.2 mH, C = 50 μF, and Lc = 0.5 mH. The converter switching frequency indicates that the proposed controller can be implemented under high switching frequencies. Since the sharp IGBT commutation spikes may impair the current acquisition process, the synchronous sampling technique with a symmetric PWM module is adopted. With this method, the sampling is performed at the beginning of each modulation cycle. Only two phases current are fed-back as the neutral is isolated. A three-phase dq-PLL is adopted for grid synchronization. The proposed controller is compared to the conventional P-HRES controller and singleand dual-loop H∞ controllers. The design parameters of the controllers are given in Appendices II and III, respectively. Fig. 24 (a) shows the output current response with the proposed direct μ-synthesis controller along with the voltage waveform at the point of common coupling. The voltage THD is 2.5% and its harmonic spectrum is also presented in Fig. 24(a). The presence of low-order voltage harmonics makes it possible to evaluate the harmonic disturbance rejection capability of the proposed controller and compare it with the conventional H∞ and P-HRES controllers. Fig. 24(b) and (c) presents the current and voltage responses when the conventional dual-loop H∞ and P-HRES controllers are adopted, respectively. Table I presents the voltage and current low-order harmonic content as well as their THD. Comparing the current quality of the output current in presence of voltage harmonic distortion for different control schemes reveals the improved performance of the μ-synthesis controller in rejecting voltage harmonics. Note that H∞ controller fails to provide good disturbance rejection, and therefore the current THD is the highest. This can be attributed to the fact that in H∞ minimization algorithm, robust stability is only achieved (to provide robust performance under grid impedance
Fig. 24. Output Current and voltage waveforms with (a) proposed single-loop μ-control (b) dual-loop H ∞ control (c) dual-loop P-HRES control. TABLE I COMPARATIVE CURRENT THD AND HARMONIC CONTENTS IN PRESENCE OF DIFFERENT CONTROL ALGORITHMS WHEN GRID VOLTAGE THD IS 2.5%
variation) at the cost of compromised performance as it was discussed earlier in Section III. In order to evaluate the robust stability of the proposed controller, the grid-side inductance is increased up to 240% of its nominal value as Lc is replaced with a 1.2 mH inductor. Fig. 25(a) and (b) shows the output current of the perturbed system with μ-synthesis and the conventional dual-loop H∞ control, respectively. As can be seen, both controllers are well capable of preserving system stability of the perturbed system. However, once again it should be noted that for H∞ control, this is achieved at the cost of extra current sensors and an inner damping loop as well as the compromised performance as suggested earlier in Fig. 24 and Table I. The performance of the dual-loop P-HRES controller is also presented in Fig. 25(c) where it can be seen that introducing higher grid-side inductance can significantly reduce the stability margins yielding to oscillatory system performance as the harmonic compensators fall out of the crossover frequency. Fig. 25(c) clearly shows that even with adoption of inner damping loops, proportional resonant controller is incapable of maintaining system stability when low-order harmonic compensators are also added.
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Fig. 27. Current response of the proposed μ-synthesis controller (solid) to a tracking step change (dashed).
VI. CONCLUSION
Fig. 25. Perturbed system output current with (a) proposed single-loop μ-control (b) dual-loop H ∞ control (c) dual-loop P-HRES control.
Fig. 26. DG Current response when L c is increased from 0.5 to 1.2 mH with the single-loop H ∞ control.
In order to show the importance of adopting an inner damping loop in the H∞ -based design approach, Fig. 26 shows the performance of the single-loop H∞ control when Kd = 0 and the line inductance is increased to 1.2 mH. As can be seen in Fig. 26, prior to changing the line inductance, the controller is well capable of maintaining the nominal stability by providing inherent damping as suggested in [21]. However, it cannot provide robust stability as expected. Fig. 27 shows the tracking response of the proposed controller as the reference has been increased at t = 1.135 s. As can be seen the controller shows fast dynamics in respond to the reference step-change indicating the high bandwidth performance of the close-loop system with direct current control and inherent damping of LCL filter resonance.
This paper has presented a robust single-loop direct current control strategy for LCL-filtered distributed generation converters that can be used in both grid-connected and islanded microgrid modes. The proposed method is synthesized based on structured singular value μ minimization to maintain stable operation for a wide range of grid-side parametric uncertainty. Unlike the conventional H∞ -based approach, which requires additional inner damping loop, the less conservative nature of μ-synthesis-based solution eliminates the need for dual-loop control structures yielding faster dynamics and less number of current sensors. Moreover, introducing the performance criteria in the uncertainty block structure significantly improves system performance as compared to the conventional H∞ -based controllers. The high disturbance rejection feature of the proposed scheme at the fundamental and selected harmonics removes the dependency of the current control loop on grid-voltage feedforward and also makes it possible for the DG unit to operate as a harmonic filter when connected to nonlinear loads. The proposed controller ensures robust stability and performance under grid impedance variation and/or network reconfiguration in islanded microgrid operation; therefore, it enhances the stability and performance of distributed generation units. APPENDIX I The parameters of the test system shown in Fig. 1 and converter control parameters are given as follows: DG1: 8kVA, 480 V(LL), 60 Hz, Lf = 1.2 mH, Cf = 50 μF, rf = 0.1 Ω, Lc = 0.5 mH, rc = 0.1 Ω, fsw = 10 kHz. DG2,DG3: 8 kVA, 480 V(LL), 60 Hz, Lf = 0.6 mH, Cf = 50 μF, rf = 0.1 Ω, fsw = 5 kHz, ωc = 30 rad/s, Kpv = 0.5, Kiv = 4e3, Kpi = 0.6, Kii = 100, H = 0.75, fn l = 60.5 Hz, m2 = m3 = 2e-5, n2 = n3 = 1.3e-5. L2: 0.17+.92j%, L3: 0.1+0.46j%, L4: 1.2+2.36j%, L5: 0.8+.42j%, (Sbase :8kVA,Vbase :0.48kV). APPENDIX II Proportional resonant controller with harmonic compensators (P-HERES). One of the well-established controllers in the literature that provides both good tracking and disturbance rejection in the stationary reference-frame is the proportional-resonant controller
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problem
Fig. 28.
Standard M-Δ configuration.
with harmonic compensators (P-HRES) defined in (A-1) where fifth, seventh, and eleventh harmonics are considered for better disturbance rejection Kih ξs (A-1) KP HRES = KP + s2 + ξs + (hω0 ) h=1,5,7,11
where ω0 is the fundamental angular frequency, KP , Kih are constants, ζ is the damping coefficient, and h is the harmonic order. The adopted design parameters in this paper are KP-HRES (with damping): Kd = 10, ω0 = 120π, KP = 4, Kih (h = 1, 5, 7, 11) = 500, ζ = 1.5, fsw = 10 kHz, KP-HRES (without damping): Kd = 0, ω0 = 120π, KP = .05, Kih (h = 1, 5, 7, 11) = 100, ζ = 1.5, fsw = 10 kHz. APPENDIX III A. Suboptimal H∞-Based Robust Control Theorem Any control structure can be rearranged to match the closedloop format represented in Fig. 28 where matrix M shows the relation between system inputs and outputs. The variables w, z, v, and y, in Fig. 28 are vector signals, where w denotes the exogenous inputs including reference command and voltage disturbance; z denotes the weighted error signal; v and y are the input and output signals of the uncertainty block. Based on the 2-input, 2-output structure of Fig. 28, (A-2) can be concluded as the relationship between input and output vectors y v M11 (s) M12 (s) × . (A-2) = w z M21 (s) M22 (s) M (s)
Based on the small-gain theory [29], the robust stability of the closed loop system of Fig. 28 can be studied according to small gain theorem. Small gain theorem: Let M11 (s) be stable, for all Δ, Δ∞ < 1, the perturbed system of Fig. 28 is robustly stable if M11 ∞ < 1. Therefore, in order to maximize the perturbed system stability range, the stabilizing controller should be designed in a way that M11 ∞ is minimized. In order to satisfy the performance criteria, the effect of the exogenous input w over output z should also be minimized in a way that the associated transfer function (also known as sensitivity transfer function) lies below 1/Wp er where the performance weighting. This can be translated to minimizing M22 ∞ to be under unity. Therefore, using a suboptimal H∞ minimization algorithm (e.g., using MATLAB robust control toolbox [31]), the H∞ -based robust controller Kinf can be designed to fulfill the following mixed-sensitivity optimization
M11 M22 < 1. ∞
(A-3)
However, it should be noted that the robust stability and performance criteria have opposing effects which makes the aforementioned mixed-H∞ optimization a challenging task. (M22 is the weighted sensitivity transfer function while M11 models the weighted complementary sensitivity transfer function). The design parameters of the H∞-based robust controller adopted in this paper are Kinf (without damping): Kd = 0, ω0 = 120π, WΔ = (7.512s + 4.996)/(s + 2690), WPer : K1 = 0.2, K5 = 0.05, K7 = K11 = .01, ζ = 0.01. Kinf (with damping): Kd = 10, ω0 = 120π, WΔ = (0.2684s + 0.00733)/(s + 367.8), WPer : K1 = 0.2, K5 = 0.05, K7 = K11 = 0.01, ζ = 0.01. B. μ-Synthesis-Based Robust Control Theorem Definition 1: When M (s) is an interconnected transfer matrix ˜ is as in Fig. 7, the structured singular value with respect to Δ defined as in (A-4) μΔ˜ (M (s)) = sup μΔ˜ (M (jω)) ω ∈R
(A-4)
˜ ˜ where μ−1 ˜ (M ) is the smallest singular value of Δ (i.e., σ(Δ)) Δ ˜ = 0. Note that μ introduces a less that makes det(I − M Δ) conservative measure for stability analysis of the perturbed system which also considers performance criteria. The less conservative nature of μ as in (A-5) is proved in [28], [29] and indicates that the stability condition can be achieved easily when dealing with structured uncertainties. Note that the equality only holds when Δ is unstructured μΔ˜ (M ) ≤ M ∞ .
(A-5)
The adopted design parameters of the μ-synthesis-based robust controller proposed in this paper are Kμ : Kd = 0, ω0 = 120π, WΔ = (0.5802s + 0.00047)/(s + 132), WPer : K1 = 0.2, K5 = 0.1, K7 = K11 = 0.05, ζ = 0.01. REFERENCES [1] M. Liserre, R. Teodorescu, and F. Blaabjerg, “Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values,” IEEE Trans. Power Electron., vol. 21, no. 1, pp. 263–272, Jan. 2009. [2] I. J. Gabe, V. F. Montagner, and H. Pinheiro, “Design and implementation of a robust current controller for VSI connected to the grid through an LCL filter,” IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1444–1452, Jun. 2009. [3] Y. A.-R. I. Mohamed, “Suppression of low- and high-frequency instabilities and grid-induced disturbances in distributed generation inverters,” IEEE Trans. Power Electron., vol. 26, no. 12, pp. 3790–3803, Dec. 2011. [4] J. Dannehl, C. Wessels, and F. W. Fuchs, “Limitations of voltage-oriented PI current control of grid-connected PWM rectifiers with LCL filters,” IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 380–388, Feb. 2009. [5] R. Teodorescu, F. Blaabjerg, M. Liserre, and P. Loh, “Proportional resonant controllers and filters for grid-connected voltage-source converters,” Proc. IEE Electric Power Appl., vol. 153, pp. 750–762, Sep. 2006. [6] M. Bierho_ and F. Fuchs, “Active damping for three-phase PWM rectifiers with high-order line-side filters,” IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 371–379, Feb. 2009.
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[7] J. Dannehl, F. Fuchs, S. Hansen, and P. Thgersen, “Investigation of active damping approaches for PI-based current control of grid- connected pulse width modulation converters with LCL filters,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1509–1517, Jul./Aug. 2010. [8] Y. Tang, P. Loh, P. Wang, F. Choo, and F. Gao, “Exploring inherent damping characteristic of LCL-filters for three-phase grid-connected voltage source inverters,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1433– 1443, Mar. 2011. [9] G. Q. Shen, D. H. Xu, L. P. Cao, and X. C. Zhu, “An improved control strategy for grid-connected voltage source inverters with an LCL filter,” IEEE Trans. Power Electron., vol. 23, no. 4, pp. 1899–1906, Jul. 2008. [10] W. Xuehua, R. Xinbo, L. Shangwei, and C. K. Tse, “Full feedforward of grid voltage for grid-connected inverter with LCL filter to suppress current distortion due to grid voltage harmonics,” IEEE Trans. Power Electron., vol. 25, no. 12, pp. 3119–3127, Dec. 2010. [11] E. Twining and D. G. Holmes, “Grid current regulation of a three-phase voltage source inverter with an LCL input filter,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 888–895, May 2003. [12] M. Malinowski and S. Bemet, “A simple voltage sensorless active damping scheme for three-phase PWM converters with an LCL filter,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1876–1880, Apr. 2008. [13] I. J. Gabe, V. F. Montagner, and H. Pinheiro, “Design and implementation of a robust current controller for VSI connected to the grid through an LCL filter,” IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1444–1452, Jun. 2009. [14] E. Wu and P. W. Lehn, “Digital current control of a voltage source converter with active damping of LCL resonance,” IEEE Trans. Power Electron., vol. 21, no. 5, pp. 1364–1373, Sep. 2006. [15] J. Dannehl, F. W. Fuchs, and P. B. Thogersen, “PI state space current control of grid-connected PWM converters with LCL filters,” IEEE Trans. Power Electron., vol. 25, no. 9, pp. 2320–2330, Sep. 2010. [16] G. Shen, D. Xu, L. Cao, and X. Zhu, “An improved control strategy for grid-connected voltage source inverters with an LCL filter,” IEEE Trans. Power Electron., vol. 23, no. 4, pp. 1899–1906, Jul 2008. [17] G. Shen, D. Xu, L. Cao, and X. Zhu, “A new feedback method for PR current control of LCL-filter-based grid-connected inverter,” IEEE Trans. Ind. Electron., vol. 57, no. 6, pp. 2033–2041, Jun. 2010. [18] J. Dannehl, M. Liserre, and F. Fuchs, “Filter-based active damping of voltage source converters with LCL filter,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3623–3633, Aug. 2011. [19] R. Teodorescu, F. Blaabjerg, M. Liserre, and A. Dell’Aquila, “A stable three-phase LCL-filter based active rectifier without damping,” in Proc. 38th IAS Annu. Meeting Ind. Appl. Conf. Rec., 2003, vol. 3, pp. 1552–1557. [20] J. Yin, S. Duan, and B. Liu, “Stability analysis of grid-connected inverter with LCL filter adopting a digital single-loop controller with inherent damping characteristic,” IEEE Trans. Ind. Inf., vol. 9, no. 2, pp. 1104– 1112, May 2013. [21] B. Bahrani, M. Vasiladiotis, and A. Rufer, “High-order vector control of grid-connected voltage source converters with LCL-filters,” IEEE Trans. Ind. Electron., to be published. [22] S. Yang, Q. Lei, F. Z. Peng, and Z. Qian, “A robust control scheme for gridconnected voltage-source inverters,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 202–212, Jan. 2011. [23] S. Lee, S.-J. Chiang, and J.-M. Chang, “H∞ loop-shaping controller designs for the single-phase UPS inverters,” IEEE Trans. Power Electron., vol. 16, no. 4, pp. 473–481, Jul. 2001. [24] M. Aten and H. Werner, “Robust multivariable control design for HVDC back to back schemes,” Proc. Inst. Electr. Eng., vol. 150, no. 6, pp. 761– 767, Nov. 2003.
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[25] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis and testing of autonomous operation of an inverter-based microgrid,” IEEE Trans. Power Electron., vol. 22, no. 2, pp. 613–625, Mar. 2007. [26] A. Kahrobaeian and Y. A.-R. I. Mohamed, “Interactive distributed generation interface for flexible micro-grid operation in smart distribution systems,” IEEE Trans. Sustainable Energy, vol. 3, no. 2, pp. 295–305, 2012. [27] B. Bahrani, M. Vasiladiotis, and A. Rufer, “High-order vector control of grid-connected voltage source converters with LCL-filters.” IEEE Trans. Ind. Electron., in press, 2013. DOI: 10.1109/TIE.2013.2276442. [28] J. C. Doyle, “Structured uncertainty in control system design,” in Proc. 24th IEEE Conf. Decision Control, Dec. 1985, pp. 260–265. [29] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design. New York: Wiley, 1996. [30] F.M. Callier and C.A. Desoer, Linear System Theory.. New York: Springer-Verlag, 1991. [31] Robust Control Toolbox User’s Guide, tMath Work Inc., Natick, MA, USA, Mar. 2005.
Alireza Kahrobaeian (S’12) received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Tehran, Tehran, Iran, in 2007 and 2010, respectively. He is currently working toward the Ph.D. degree at the University of Alberta, Edmonton, AB, Canada. His research interests include control and stability analysis of micro-grids and smart grid systems.
Yasser Abdel-Rady I. Mohamed (M’06–SM’11) was born in Cairo, Egypt, on November 25, 1977. He received the B.Sc. (with honors) and M.Sc. degrees in electrical engineering from Ain Shams University, Cairo, in 2000 and 2004, respectively, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2008. He is currently with the Department of Electrical and Computer Engineering, University of Alberta, AB, Canada, as an Associate Professor. His research interests include dynamics and controls of power converters; distributed and renewable generation, and microgrids, modeling, analysis and control of smart grids, electric machines, and motor drives. Dr. Mohamed is an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. He is also a Guest Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Special Section on “Distributed Generation and Micro-grids”. His biography is listed in Marque’s Who is Who in the World. He is a registered Professional Engineer in the Province of Alberta.
