5568 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER Qing-Chang Zhong, Senior Member, IEEE, and Yu Zeng

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1 5568 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 0, OCTOBER 204 Control of Inverters Via a Virtual Capacitor to Achieve Capacitive Output Impedance Qing-Chang Zhong, Senior Member, IEEE, and Yu Zeng Abstract Mainstream inverters have inductive output impedance at low frequencies such inverters are called L-inverters). In this paper, a control strategy is proposed to make the output impedance of an inverter capacitive at low frequencies such inverters are called C-inverters). The proposed control strategy involves the feedback of the inductor current through an integrator, which is actually the impedance of a virtual capacitor. The gain of the integrator or the virtual capacitance is first selected to guarantee the stability of the current feedback loop and then optimized to minimize the total harmonic distortion THD) of the output voltage. Moreover, some guidelines are developed to facilitate the selection of the filter components for C-inverters. Simulation and experimental results are provided to demonstrate the feasibility and excellent performance of C-inverters, with the filter parameters of the test rig selected according to the guidelines developed. It is shown that, with the same hardware, the lowest voltage THD is obtained when the inverter is designed to be a C-inverter. A by product of this study is that, as long as the current ripples are kept within the desired range, the filter inductor should be chosen as small as possible in order to reduce voltage harmonics. This helps reduce the size, weight, and volume of the inductor and improve the power density of the inverter. Index Terms Inverters with capacitive output impedance C-inverters), inverters with inductive output impedance Linverters), inverters with resistive output impedance R-inverters), power quality, total harmonic distortion THD). I. INTRODUCTION ENERGY and sustainability are now on the top agenda of many governments. Smart grids have become one of the main enablers to address energy and sustainability issues. Renewable energy, distributed generation, hybrid electrical vehicles, more-electric aircraft, all-electric ships, smart grids etc. will become more and more popular. DC/AC converters, also called inverters, play a common role in these applications to convert a dc source into an ac source. Arguably, the integration of renewable and distributed energy sources, energy stor- Manuscript received February, 203; revised May 5, 203, September, 203, and October 26, 203; accepted November 2, 203. Date of current version May 30, 204. This work was supported by the EPSRC, U.K. under Grants EP/J00333/ and EP/J0558X/. Some preliminary results were presented at the 37th Annual Conference of the IEEE Industrial Electronics Society, Melbourne, Australia, November 20. Recommended for publication by Associate Editor Dr. A. M. Trzynadlowski. Q.-C. Zhong is with the Deparment of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield S 3JD, U.K., and with the China Electric Power Research Institute CEPRI), Beijing, China zhongqc@ieee.org). Y. Zeng is with the Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield S 3JD, U.K. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TPEL age, and demand-side resources into smart grids is the largest new frontier for smart grid advancements [], [2]. Inverters are also widely used in uninterruptible power supplies, induction heating, high-voltage dc transmission, variable-frequency drives, electric vehicle drives, air conditioning, vehicle-to-grid etc. and, hence, have become a common key device for many energy-related applications. How to control the inverters is critical for these applications. There are several important control problems associated with inverters. For example, how to make sure that the total harmonic distortion THD) of the inverter voltage remains within a certain range when the loads are nonlinear and the grid voltage, if present, is distorted; how to make sure that the output voltage of an inverter is maintained within a certain range; how to share loads proportionally according to their power ratings when inverters are operated in parallel; how to make sure that inverters can be operated in the grid-connected mode and the standalone mode and how to minimize the transient dynamics when the operation mode is changed [3]; how to connect inverters to the grid in a grid-friendly manner so that the impact on the grid is minimized [4], [5]; and how to minimize the total microgrid operating cost [6], etc. There have been a lot of research activities on these problems, from one aspect to another, and a systematic treatment of the control problems related to inverters in renewable energy and smart grid integration can be found in []. The voltage THD can be improved by using deadbeat or hysteresis controllers [7], [8], selective harmonic elimination pulsewidth modulation strategies [9], and repetitive controllers [0] [6] [7], [8], injecting harmonic voltages [9], [20], introducing a voltage feedback loop [2] etc. Another way is to investigate the role of the output impedance as it is known that the output filter also contributes to the output voltage quality [22] [25]. It is well known that mainstream inverters have inductive output impedance at low frequencies because of the filter inductor. Moreover, the output impedance of an inverter can also change with the control strategy adopted [26] [30]. The general understanding is that inverters with resistive output impedance are better than inverters with inductive output impedance because resistive output impedance makes the compensation of harmonics easier. Some questions pop up immediately. For example: ) Is it possible to have inverters with capacitive output impedance? 2) If so, what are the advantages, if any? 3) If so, how to achieve parallel operation for such inverters? The preliminary results presented in [3] have shown that an inverter can be designed to have capacitive output impedance. This concept has been further developed in [32] to implement active capacitors that are accurate and stable with respect to the IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 ZHONG AND ZENG: CONTROL OF INVERTERS VIA A VIRTUAL CAPACITOR TO ACHIEVE CAPACITIVE OUTPUT IMPEDANCE 5569 change of environmental factors, e.g., temperature and humidity. In order to facilitate the presentation, inverters with inductive, resistive, and capacitive output impedance are called L-, R-, and C-inverters, respectively. In this paper, a simple but effective control strategy is proposed to design the output impedance of an inverter to be capacitive, following [], [3]. Then, the control parameter i.e., the output capacitance) is designed to guarantee the stability and, furthermore, optimized to minimize the THD of the output voltage. Moreover, detailed analyses are carried out to provide guidelines for selecting the filter components for C-inverters. Note that the typically-needed voltage loop to track a voltage reference [26], [27], [33] is not adopted, which reduces the number of control parameters and the complexity of the controller. Experimental results are presented to demonstrate the feasibility and performance of C-inverters and the guidelines for the component selection. It is shown that, with the same hardware, the lowest voltage THD is obtained when the inverter is designed to be a C-inverter. Note that the output impedance of an inverter can be defined at different terminals that have different pairs of voltage and current and hence can be different. In this paper, the output impedance of an inverter is defined at the terminal with the output voltage and the filter inductor current. In order to avoid confusion, the output impedance that takes into account the effect of the filter capacitor and the control strategy is called the overall output impedance. At low frequencies, for which the major voltage harmonics are concerned, the overall output impedance is more or less the same as the output impedance without considering the filter capacitor. The rest of the paper is organized as follows. A controller is proposed in Section II to force the output impedance of an inverter to be capacitive and the stability is analyzed. The control parameter is optimized to minimize the voltage THD in Section III and guidelines for selecting the filter components are provided in Section IV. Experimental and simulation results are presented in Section V and VI, followed by conclusions and discussions made in Section VII. II. DESIGN OF C-INVERTERS A. Implementation Fig. a) shows an inverter, which consists of a single-phase H-bridge inverter powered by a dc source, and an LC filter. The control signal u is converted to a PWM signal to drive the H-bridge so that the average of u f over a switching period is thesameasu, i.e., u u f. Different PWM techniques and the associated switching effect play an important role in inverter design [34] [36] but from the control point of view, the PWM block and the H-bridge can be ignored when designing the controller; see, e.g., [37] [40]. In particular, this is true when the switching frequency is high enough. The inverter can be modeled as shown in Fig. b) as the series connection of a voltage reference v r and the output impedance Z o, taking the voltage v o as the output voltage and the current i as the output current. This is equivalent to regarding the filter capacitor as a part of the load [37]. The output impedance Z o is inductive when no Fig.. Single-phase inverter. a) Descriptive circuit. b) Simplified model with terminal voltage v o and terminal current i. Fig. 2. a) b) Controller to make the output impedance of an inverter capacitive. controller is adopted and can be made resistive after introducing the proportional feedback of the filter inductor current, which is often used to dampen oscillations in the system. Here, the controller shown in Fig. 2 is proposed to make the output impedance of an inverter capacitive. The following two equations hold for the closed-loop system consisting of Fig. a) and Fig. 2: u = v r i, and u f =R + sl)i + v o ) where R is the equivalent series resistance of the inductor. It is normally small but not exactly 0. Since the average of u f over a switching period is the same as u, there is approximately) v r i =R + sl)i + v o 2) which leads to v o = v r Z o s) i 3) with the output impedance Z o s) given by Z o s) =R + sl +. 4) As a result, the integrator block is added virtually to the original output impedance of the inverter. This is equivalent to connecting a virtual capacitor inside the inverter) in series

3 5570 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 0, OCTOBER 204 with the filter inductor L. It is worth noting that the original filter capacitor C is still required. Although the virtual capacitance introduced by the feedback changes the output impedance within the bandwidth of the controller, the switching noises are often far beyond the reach of this control and an LC filter is still needed to suppress switching noises. The impact of the control strategy is on the change of the inverter dynamics, with some practical implications discussed in the rest of this section. If the capacitor is chosen small enough, the effect of the inductor R + sl) is not significant and the output impedance can be made nearly purely capacitive around the fundamental frequency, i.e., roughly Z o s). 5) Hence, the virtual capacitor resonates with the filter inductor L at a frequency higher than the fundamental frequency, which is able to reduce the harmonic voltage dropped on the filter inductor caused by the harmonic components of the load current. This allows C-inverters to achieve better voltage quality than R- and L- inverters without additional hardware cost. B. Stability of the Current Loop When the controller is implemented digitally, the effect of computation and PWM conversion can be approximated by a one-step delay e st s, where T s is the sampling period. Hence, the approximate block diagram of the current loop can be derived as shown in Fig. 3a). The corresponding open-loop transfer function is Ls) = sl + R e st s 6) which has a pole at s =0but does not have any unstable poles in the right-half-plane of the s-domain. A typical Nyquist plot of such systems is shown in Fig. 3b). In order to make sure that the system is stable, according to the well-known Nyquist theorem, the plot should not encircle the critical point, 0). Assume that the plot crosses the real axis for the first time at the frequency ω 0, then ω 0 satisfies π 2 atanω 0L R ω 0T s = π. 7) In other words, ω 0 can be found as the first positive number from 0 that satisfies R ω 0 L = tanω 0T s ). 8) At this frequency, the loop gain should be less ω 0 ω 2 0 L 2 +R 2 than. In other words, the loop is stable if <ω 0 ω0 2 C L2 + R 2. 9) o It can be easily seen that 0 <ω 0 < π. 0) 2T s Hence, the current loop is stable if < π ) 2 πl + R 2T s 2T 2 ) s Fig. 3. plot. a) b) The current loop. a) Approximate block diagram. b) Typical Nyquist of which the right-hand side is about π 2T s ) 2 L for small R 0. In other words, the loop is stable if the capacitance or the sampling frequency f s = T s is chosen large enough so that the sampling frequency f s is larger than four times the resonant frequency 2π with L, which can be easily met without any L problem. Note that R is not exactly zero in reality, which helps maintain the stability of the loop. C. DC Offset in the System Because of the presence of the integrator, any dc offset in the current i, e.g., that is caused by the conversion process or faults in the system etc., would lead to a dc offset in the output voltage. In order to avoid this problem, some simple mechanisms can be adopted. For example, the integrator can be reset when the inductor current passes zero if the offset exceeds a given level. Alternatively, the integrator can be slightly modified as +ɛ with a negligible positive number ɛ 0. This is equivalent to putting a large resistor ɛ in parallel with, which does not change the performance at non-dc frequencies. III. OPTIMIZATION OF THE VOLTAGE QUALITY Assume that the output current of the inverter is i = 2Σ h=i h sinhωt + φ h ) 2)

4 ZHONG AND ZENG: CONTROL OF INVERTERS VIA A VIRTUAL CAPACITOR TO ACHIEVE CAPACITIVE OUTPUT IMPEDANCE 557 where ω is the system frequency. Then, the amplitude of the h-th harmonic voltage dropped on the output impedance is 2Ih Z o jhω). Moreover, assume that the voltage reference v r is clean and sinusoidal and is described as v r = 2E sinωt + δ). 3) Then, the fundamental component of the output voltage is v = 2E sinωt + δ) 2I Z o jω) sinωt + φ + θ) 4) = 2V sinωt + β) 5) with V = E 2 + I 2 Z ojω) 2 2EI Z o jω) cosφ + θ δ) 6) ) ω Zo jω) sinφ + θ δ) β = arctan. 7) I Z o jω) cosφ + θ δ) E The sum of all harmonic components in the output voltage is v H = 2Σ h=2i h Z o jhω) sinhωt + φ h + Z o jhω)). 8) It is clear that v and v H do not affect each other. v is determined by the clean reference voltage, the fundamental current and the output impedance at the fundamental frequency. v H is determined by the harmonic current components and the output impedance at the harmonic frequencies. According to the definition of THD, the THD of the output voltage is Σ h=2 THD = I2 h Z ojhω) 2 00%. 9) V Hence, the THD is mainly affected by the output impedance at harmonic frequencies. As a result, it is feasible to optimize the design of the output impedance at harmonic frequencies to minimize the THD of the output voltage. For the C-inverter designed in the previous section, according to 4), there is Z o jhω ) 2 = R 2 + hω L ) 2 hω 20) where ω is the rated angular system frequency. In order to minimize the THD of the output voltage, the virtual capacitor should be chosen to minimize Σ h=2i 2 h Z o jhω ) 2 2) because the fundamental component V can be assumed to be almost constant. This is equivalent to minσ h=2i 2 h hω L ) 2 hω 22) where i h = I h I is the normalized h-th harmonic current I h with respect to the fundamental current I. Depending on the distribution of the harmonic current components, different strategies can be obtained. Assume that the harmonic current is negligible for the harmonics higher than the N-th order with an odd number N). Then, can be found via solving 22). Define f )=Σ N h=2i 2 h hω L ) 2 hω. 23) Then, needs to satisfy df ) =2Σ N dc h=2i 2 h hω L ) o hω hω Co 2 =0 24) which is equivalent to Hence Σ N h=2i 2 hl hω ) 2 )=0. 25) Σ N h=2i 2 hl = ω ) 2 Σ N i 2 h h=2 h 2 26) and the optimal capacitance can be solved as Σ N i 2 h h=2 h = 2 ω ) 2 L Σ N 27) h=2 i2 h which is applicable for any current i with a known harmonic profile. The corresponding f ) is f min )=Σ N h=2i 2 h hω L ω L h Σ N h=2 i2 h Σ N i 2 h h=2 h 2 ) 2 ) 2 =ω L) 2 Σ N h=2i 2 h h Σ N h=2 i2 h. 28) h Σ N h=2 Hence, the THD of v o is in proportion to the inductance L of the inverter LC filter. A small L does not only reduce the cost, size, weight, and volume of the inductor but also improves the voltage quality. However, a small L leads to a high di for the dt switches and large current ripples. See the guidelines of selecting the components in the next section for details. Moreover, since L,asmallLleads to a small gain for the integrator, which is good for the stability of the current loop. If the distribution of the harmonic components is not known, then it can be assumed that the even harmonics are zero, which is normally the case, and the odd harmonics are equally distributed. As a result, the optimal can be chosen, according to 27), as = = i 2 h h 2 Σ h=3, 5, 7,..., N h 2 ω ) 2 L Σ N h=3, 5, 7,..., N 29) ω ) 2 L This can be written as = ω ) 2 L N )/2 Σ h=3, 5, 7,..., N h 2. 30) N )/ ) N 2 3)

5 5572 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 0, OCTOBER 204 Fig. 4. The gain factor Original inductor 4 6 3rd only 8 3rd and 5th 0 2 5th only ω/ω * The gain factors to meet different criteria. where N )/2 is the number of terms in the summation. The corresponding f ) is f min )=ω L) 2 Σ h=3, 5, 7,..., N h N )/2 h Σ h=3, 5, 7,..., N h 2 ) 2. 32) If a single h-th harmonic component is concerned, then the optimal is = hω ) 2 L. 33) This forces the impedance at the h-th harmonic frequency close to 0 and hence no voltage at this frequency is caused, assuming R =0. According to the stability analysis carried out in the previous section, the current loop is stable in this case if hω ) 2 L< π 2T s ) 2 L, or in other words, if f s > 4hf, where f = ω 2π is the rated system frequency. A. Special Case I: To Minimize the Third and Fifth Harmonic Components In most cases, it is enough to consider the third and fifth harmonics only. This gives the optimal capacitance 7 = 225ω ) 2 L. 34) As a result, the output impedance is Z o jω)=r + j ωl ) 35) ω ω = R + jω L ω 225 ω ). 36) 7 ω ω ω The gain factor ω ω of the imaginary part with respect to the normalized frequency ω ω is shown in Fig. 4. It changes from negative to positive at around ω ω = At the fundamental frequency, i.e., when ω = ω, the output impedance is Z o = R j ω L j2.23 ω L. 37) It is capacitive as expected because R is normally smaller than ω L. B. Special Case II: To Minimize the Third Harmonic Component In this case, the optimal is = 3ω ) 2 38) L and the corresponding impedance is Z o jω)=r + j ωl ) 39) ω ) ω = R + jω L ω 9ω. 40) ω The gain factor ω ω 9ω ω of the imaginary part with respect to the normalized frequency ω ω is also shown in Fig. 4. It changes from negative to positive at ω =3ω. At the fundamental frequency, i.e., when ω = ω, the output impedance is Z o = R j8ω L j8ω L 4) which is capacitive as well. C. Special Case III: To Minimize the Fifth Harmonic Component In this case, the optimal is = 5ω ) 2 L and the corresponding impedance is Z o jω)=r + j ωl ) ω ω = R + jω L ω 25ω ω 42) 43) ). 44) The gain factor ω ω ω of the imaginary part with respect to the normalized frequency ω ω is also shown in Fig. 4. It changes from negative to positive at ω =5ω. At the fundamental frequency, i.e., when ω = ω, the output impedance is Z o = R j24ω L j24ω L. 45) This is capacitive as well. 25ω IV. COMPONENT SELECTION A. Selection of the Filter Inductor L As discovered in the previous section, the smaller the filter inductor, the smaller the output impedance and the better the voltage quality. Thus, it is better to have a small output inductor than a big one. This leaves the selection of the filter inductor to meet the requirement on the allowed current ripples only. According to [23], it is recommended that the current ripples should satisfy 0.5 ΔI ) I ref with ΔI = U dc 4Lf s 47)

6 ZHONG AND ZENG: CONTROL OF INVERTERS VIA A VIRTUAL CAPACITOR TO ACHIEVE CAPACITIVE OUTPUT IMPEDANCE 5573 where ΔI is the inductor current ripple and I ref is the rated peak current at the fundamental frequency. Thus, the inductor should be chosen to satisfy 5U dc L 5U dc. 48) 8f s I ref 3f s I ref This could be applied to analyze the impact on the dc-bus voltage. For example, assume that L is selected to achieve the maximum current ripple of 0.4I ref. Moreover, assume that the peak of the h-th harmonic current reaches 50% of I ref. Then, the voltage drop of the h-th harmonic current on the inductor is hω 5U dc 8f s I ref I ref 2 = 5hω 6f s U dc. In other words, the maximum increase of the required dc- bus voltage is 5hω 6f s 00%. For h =5, f s =0kHz and ω = 00π rad/sec, this is 4.9% so it is not demanding at all and there is no need to take any special action when determining the dc-bus voltage. B. Selection of the Filter Capacitor C The main function of the LC filter is to attenuate the harmonics generated by the PWM conversion and the H-bridge via reproducing the control signal u, especially the harmonics around the switching frequency f s. When there is no load, the transfer function between u f and v o is Hs) = s 2 LC +. 49) Indeed, the virtual capacitor does not change the role of the LC filter in suppressing the switching noises because the actual output voltage u f generated by the inverter is still passed through the LC filter. The cut-off frequency f c can be found from Hj2πf c ) = 2πf c ) 2 LC = 50) 2 as 2+ f c = 2π 5) LC which is about.5 times of the resonant frequency 2π. Since LC it is very close to the resonant frequency, it is reasonable to use the resonant frequency when selecting the components. The overall output impedance Zs) after taking into account the filter capacitor C is Zs) = Z os) sc Z o s)+ sc At low frequencies, there is = Z o s) scz o s)+. 52) Zs) Z o s) =R + sl + 53) and at high frequencies, there is Zs) sc. 54) This actually verifies that the definition of the output impedance Z o without considering the filter capacitor C does not materially affect the analysis at low frequencies. Defining the output Fig. 5. Overall output impedance of an L-inverter and a C-inverter after taking into account the filter capacitor C. impedance at the terminal with the output voltage and the filter inductor current is simply to facilitate the presentation. For conventional inverters, which are mainly L-inverters, Zs) is inductive at low frequencies. Hence, the overall output impedance Zs) changes its type from inductive to capacitive at the resonant frequency. However, according to 52), the overall output impedance Zs) for the C-inverters designed above is Zs) = sl + R + s 2 LC + scr + C +. 55) It is capacitive at both low frequencies ) and high frequencies sc ). In order to better demonstrate this, the Bode plots of the overall output impedances of typical L- and C-inverters are shown in Fig. 5. The output impedance of the C-inverter is capacitive over a wide range of both low and high frequencies and is inductive only over a small range of mid-frequencies. There is a series resonance between L and, in addition to the parallel resonance between L and C, which is slightly changed because of. The output impedance of the L-inverter is inductive for low frequencies up to the resonant frequency of the filter and capacitive for the frequencies above. The optimization of the voltage quality discussed in the previous subsection is achieved via tuning the series resonance between L and. Since the load current i o may include a large amount of harmonic components, especially when the load is nonlinear, the parallel resonance between L, C, and should be considered when designing the filter. According to 55), the parallel resonant frequency f r can be obtained as f r = C + Co = 2π LC 2π LC C +. 56)

7 5574 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 0, OCTOBER 204 With the same L and C, the resonance frequency f r of C- inverters is higher than, but very close to, that of the corresponding L-inverter or R-inverters, which is 2π, because LC is often much larger than C. In order to avoid amplifying some harmonic current components, the resonance frequency f r is recommended to be chosen between ten times the line frequency ω and half of the switching frequency f s [23]. Hence, f r is often far away from the harmonics to be eliminated by designing. Indeed, if is designed to eliminate the h-th harmonic, then according to 56), there is f r = 2π L Co C That is, the resonant frequency is hω += 2π C C Co C +. 57) +times the harmonic frequency hω under control. If +> 3, then f r > 3hω 2π and it is over nine times the system frequency ω even for h =3. Hence, it is recommended to select f r to satisfy 3hω 2π f r 2 f s 58) that is to select the parallel resonant frequency between three times of the harmonic frequency under control and half of the switching frequency. Accordingly, it is recommended to select the filter capacitor C to satisfy or, equivalently, 3hω 2π hω 2π Co C + 2 f s πf s hω ) 2 C 8. 59) V. SIMULATION RESULTS Simulations were carried out with a single-phase inverter powered by a 350-V dc voltage supply. The switching frequency is 0 khz and the system frequency is 50 Hz. The rated output voltage is 230 V and the rated peak current is chosen as 40 A. Thus, the rated apparent power of the inverter is 6.5 kva. The load is a full-bridge rectifier loaded with an LC filter 2.2 mh, 50 μf) and a resistor R L =30Ω. An extra load consisting of a 200 Ω resistor and a 22 mh inductor in series is connected at t =2s, and disconnected at t =9s to test the transient response of C-inverters, R-inverters, and L-inverters. The inverter reference voltage was generated by the robust droop controller proposed in [3], which is shown in Fig. 6 for convenience. As can be seen from Fig. 6, at the steady state, there is K e E V o )=n i P i where V o is the RMS value of the output voltage. As a result, the RMS output voltage is V o = E n i K e P i which shows that the output voltage is regulated and the voltage error could be maintained small via choosing a large K e. Hence, there is no need to have an extra voltage loop to regulate the Fig. 6. The robust droop controller for C-inverters [3] to generate the voltage reference v r. instantaneous output voltage. The parameters of the robust droop controller were chosen as n i = , m i = , and K e =0, according to [3]. According to 48), the filter inductor should be chosen between 0.55 mh and.46 mh. To make the output voltage THD small, the inductor is chosen as 0.55 mh. The virtual capacitor is chosen to be 400 μf to reduce the third and fifth harmonics. According to 59), the filter capacitor C should satisfy.84 μf C 74 μf 60) from which the filter capacitor was selected as C =20μF. The simulation results of the C-inverter, together with those of an L-inverter and a R-inverter with K i =4, are shown in Fig. 7. The C-inverter achieves lowest output voltage THD among the three types of inverters. When the extra load of a 200 Ω resistor and a 22 mh inductor in series is connected or disconnected, all the three type of inverter are able to respond fast and reach the steady state quickly and smoothly. It can be seen that the transient response of the C-inverter is better than the other two. VI. EXPERIMENTAL VALIDATION Experiments were carried out with a single-phase inverter powered by a 80-V dc voltage supply, which was obtained from a nonregulated diode rectifier. The switching frequency and the system frequency are the same with the ones used in the simulation, respectively. The rated output voltage is 0 V and the rated peak current is 8 A. The load is a full-bridge rectifier loaded with an LC filter 2.2 mh, 50 μf) and a resistor R L = 200 Ω. The inverter reference voltage was also generated by the robust droop controller [3] shown in Fig. 6, and the parameters of the robust droop controller were chosen as n i = , m i = , and K e =0. According to 48), the filter inductor should be chosen between.4 and 3.75 mh. The inductor 2.2 mh on board the inverter falls into this range. Three different cases with the virtual capacitor chosen to reduce the third harmonic, the fifth harmonic, and both the third and the fifth harmonics, respectively, were tested. The corresponding virtual capacitance is 52 μf, 84 μf, and 348 μf, respectively. According to 59), the

8 ZHONG AND ZENG: CONTROL OF INVERTERS VIA A VIRTUAL CAPACITOR TO ACHIEVE CAPACITIVE OUTPUT IMPEDANCE 5575 Fig. 7. Simulation results with the extra load consisting of a 200 Ω resistor and a 22 mh inductor in series connected at t =2sand disconnected at t =9s: C-inverter with = 400 μf to reduce the third and the fifth harmonics left column), R-inverter with K i =4middle column) and L-inverter right column). a) Active power. b) Reactive power. c) Frequency. d) Output voltage RMS V o. e) Output voltage v o. f) THD of output Voltage v o. g) Inductor current i. filter capacitor C should satisfy 0.46 μf C 23 μf. 6) The filter capacitor C =0μF on board the inverter falls into this range. The corresponding resonant frequency is 3 Hz for the case with h =5and 083 Hz for the case with h =3, which leaves enough room for a normal switching frequency, e.g., 5 khz. The experimental results are shown in Fig. 8, together with those from an R-inverter with Z o =4Ωand an L-inverter designed according to the current feedback controller proposed in [37] with K i =4 and K i =0, respectively, for comparison. When the inverter was designed to have capacitive output impedance to reduce the effect of the third and the fifth harmonics, the third harmonic was reduced by about 50% from the case of the L-inverter and by about 65% from the case of the R-inverter, and the fifth harmonic was reduced by about

9 5576 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 0, OCTOBER 204 Fig. 8. Experimental results: output voltage v o and inductor current i left column), harmonic distribution of v o right column). a) C-inverter with = 348 μf to reduce the third and thefifth harmonics. b) C-inverter with = 52 μf to reduce the third harmonic. c) C-inverter with = 84 μf to reduce the fifth harmonic. d) R-inverter with K i =4. e) L-inverter.

10 ZHONG AND ZENG: CONTROL OF INVERTERS VIA A VIRTUAL CAPACITOR TO ACHIEVE CAPACITIVE OUTPUT IMPEDANCE % and 8%, respectively. The THD was reduced by about 40% and 50%, respectively. When the inverter was designed to have capacitive output impedance to minimize the effect of the third harmonic, the third harmonic was reduced by 63% from the case of the L-inverter and by 74% from the case of the R- inverter, respectively. The THD was reduced by about 36% and by 47%, respectively.when the inverter was designed to have capacitive output impedance to minimize the effect of the fifth harmonic, the fifth harmonic was reduced by 4% from the case of the L-inverter and by 3% from the case of the R-inverter, respectively. The THD was reduced by about 37% and 48%,respectively. Apparently, C-inverters performed much better than the R- and L-inverters. Moreover, the THD is the lowest when is designed to optimize the third and fifth harmonics than to optimize these two separately. This is because the major harmonic components of the load current are the third and the fifth harmonics, as can be seen from Fig. 8e). The recorded average RMS values of the output voltage are 09.7 V for the R-inverter, 0.2 V for the L-inverter, and 09.8 V for the C-inverters, which shows the excellent voltage regulation capability of the robust droop control strategy. This is true regardless of the virtual capacitance concept. VII. CONCLUSION AND DISCUSSIONS It has been shown that it is feasible to force the output impedance of an inverter to be capacitive over a wide range of both low and high frequencies although it normally has an inductor connected to the inverter bridge. Such inverters are called C-inverters. One simple but effective approach is to form an inductor current feedback through an integrator, of which the time constant is the desired output capacitance. This is a virtual capacitor, so there is no limit on the current rating and can be applied to any power level. The capacitance can be selected to guarantee the stability of the current loop and an algorithm is proposed to optimize the value of the output capacitance so that the THD of the output voltage is minimized. Detailed guidelines have been provided to place the relevant frequencies properly so that the filter components can be determined. Extensive experimental results have shown that the THD of an inverter can be reduced when it is designed to have capacitive output impedance, with comparison to an inverter having resistive or inductive output impedance. Moreover, no visible dc offsets are seen from the experimental results. One by product of this study is that the filter inductor should be chosen small in order to reduce voltage harmonics and the criterion is reduced to meet the current ripples allowed on the inductor. A small inductor helps reduce the size, weight, and volume of the passive components needed. Since the C-inverter concept is completely new, some issues should be further investigated, in particular, for grid-connected applications. For example, because of the introduction of a capacitor into the output impedance, a natural question is whether this would lead to possible resonance with the rest of the system such as the line, loads, etc.). This may not be an issue because in flexible ac transmission systems FACTS), capacitors have been physically connected in series with transmission lines to improve the line capacity. Another question is whether this will affect the current quality for grid-connected applications. It has been found that C-inverters offer the lowest output voltage THD among R-, L-, and C-inverters with the same hardware. Further investigations should be carried out to explore other advantages and applications of C-inverters. 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Power Electron., vol. 25, no. 2, pp , Dec Qing-Chang Zhong M 03 SM 04) received the Ph.D. degree in control and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000, and the Ph.D. degree in control theory and power engineering awarded the Best Doctoral Thesis Prize) from Imperial College London, London, U.K., in He is currently the Chair Professor in Control and Systems Engineering with the Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, U.K. He is a Distinguished Lecturer of IEEE Power Electronics Society and is invited to represent the U.K. at the European Control Association. From , he spent a six-month sabbatical at the Cymer Center for Control Systems and Dynamics, University of California, San Diego, CA, USA, and an eight-month sabbatical at the Center for Power Electronics Systems, Virginia Tech, Blacksburg, VA, USA. He co-)authored three research monographs: Control of Power Inverters in Renewable Energy and Smart Grid Integration Wiley-IEEE Press, 203), Robust Control of Time-Delay Systems Springer-Verlag, 2006), Control of Integral Processes with Dead Time Springer-Verlag, 200). He also serves as an Associate Editor for IEEE Transactions on Power Electronics, IEEE Access, and the Conference Editorial Board of the IEEE Control Systems Society. His fourth research monograph entitled Completely Autonomous Power Systems CAPS): Next Generation Smart Grids is scheduled to appear in 205. He is the architect of the next-generation smart grid based on the synchronization mechanism of synchronous machines and a Specialist recognized by the State Grid Corporation of China, a Fellow of the Institution of Engineering and Technology, the Vice-Chair of IFAC TC 6.3 Power and Energy Systems) and was a Senior Research Fellow of the Royal Academy of Engineering/Leverhulme Trust, U.K ). His research focuses on advanced control theory and its applications in various sectors, which include power electronics, renewable energy and smart grid integration, electric drives and electric vehicles, robust and H-infinity control, time-delay systems, process control, and mechatronics. Dr. Zhong, jointly with G. Weiss, invented the synchronverter technology to operate inverters to mimic synchronous generators, which was awarded Highly Commended at the 2009 IET Innovation Awards. Yu Zeng received the B.Eng. degree in automation from Central South University, Changsha, China, in She is currently working toward the Ph.D. degree from the Department of Automatic Control and Systems Engineering, the University of Sheffield, Sheffield, U.K. Her research interests include control of power electronic systems, microgrids, and distributed generation, in particular, the parallel operation of inverters.