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Analysis of resonance and harmonic amplification for grid‐connected inverters

Analysis of resonance and harmonic amplification for grid‐connected inverters IntroductionThe penetration level of distributed generation system (DGS) based on renewable energy is gradually increasing because of the mature power generation technology and urgent demand for clean and sustainable energy [1, 2]. The grid‐connected inverter is widely used in DGS due to its advantages of potential for full control of both dc link voltage and power factor [3]. To reduce the high‐frequency harmonics caused by grid‐connected inverters, the LCL filter is widely used with the advantage of smaller values of inductance and capacitance [4]. Currently, the active damping methods are usually employed to solve the resonance problem of LCL filter [5, 6]. However, the uncertain grid impedance threatens the stability of LCL‐filtered grid‐connected inverters. The grid impedance which depends on the different operations of the power system can vary over a wide range [7]. When the LCL‐filtered grid‐connected inverter is connected to a weak grid with large grid impedance, the system stability can be affected and harmonic resonance may occur. Consequently, the grid injected current can be rich in harmonics which will significantly damage the system and equipments [8, 9].In the previous works, the impedance‐based stability criterion has been extensively implemented to analyse and discuss the stability of grid‐connected converters. In [10–16], the impedance‐based stability criterion without coupling of the grid impedance and inverters is introduced. It has been concluded that an inverter is stable if the ratio of the grid impedance and the inverter output impedance satisfies the Nyquist stability criterion. In [14], the Nyquist impedance‐based stability criterion is applied for illustrating how the individual responses of interconnected complex sources and loads affect the total closed‐loop stability. The presented Nyquist application indicates how the individual impedance responses contribute to the closed‐loop system poles. Wang et al. [17] analysed a power‐electronics‐based ac system where two active rectifiers and a power factor correction capacitor are connected to the point of common coupling (PCC). It has been pointed out that large grid impedance can induce system instability, and the interactions of paralleled converters may result in unstable oscillations when multiple harmonic compensators are employed. For n‐parallel grid‐connected inverters, the inverters are coupled due to the grid impedance and the equivalent grid impedance for each inverter will become n times bigger. Thus, the impedance characteristics of the system are changed [17–19]. Based on the above stability criteria, some stability enhancement methods have been proposed, such as the impedance shaping method [14], the adaptive active damper method [20], and the passivity‐based method [21] and so on.The stability of grid‐connected inverters can also be analysed based on the inverter current control models that account for the coupling of the grid impedance and inverters. In [22–24], the modelling and control analysis of grid‐connected inverters are presented, and it is shown that the inverters do not behave as expected due to the grid impedance. Generally, the current control bandwidth will reduce remarkably when the inverter is connected to a weak grid with large grid impedance [7]. Therefore, the possible wide range of the grid impedance can lower the stability of the inverters with multiple selective harmonic compensators, when the frequencies of harmonics to be suppressed are close to or above the current control bandwidth [7, 14, 25, 26]. In addition, the resonance damping methods of the LCL filter may fail in grid‐connected inverters. Especially for considering the digital control delay, the phase characteristics of the system become worse and the system stability is threatened. In [27], a digitally controlled LCL‐filtered grid‐connected inverter with the capacitor‐current‐feedback active damping is investigated. It turns out that the LCL filter resonance frequency is changed by a virtual impedance which is caused by the proportional feedback of the capacitor current. If the actual resonance frequency is higher than one‐sixth of the sampling frequency, the virtual impedance will contain a negative resistor component. As a result, the inverter becomes much easier to be unstable if the resonance frequency is moved closer to fs/6. Lu et al. [28] comprehensively analysed the stability and robustness of a grid‐connected inverter with LCL‐filter in the discrete domain, where the grid voltage is fed forward. The critical resonance frequency of LCL filter is identified and two more boundary frequencies are revealed in a weak grid. It has been demonstrated that the feedforward of grid‐voltage would significantly alter the inverter stability in a weak grid.The current‐controlled grid‐connected inverter is usually modelled with a Norton circuit which consists of a current source in parallel connection with an equivalent admittance. The actual grid can be modelled with a Thevenin circuit that consists of a voltage source in series connection with a grid impedance. When the equivalent admittance is matched with the grid impedance, the series resonance can arise taking the grid voltage background harmonics as the voltage source. Similarly, the parallel resonance can arise taking the inverter output current harmonics as the current source [10–12]. In a word, series and parallel resonance phenomena can arise due to the varied grid impedance [29–31]. The harmonics in the grid injected current can be excessive due to the resonances, and the harmonic problem will be more serious in an inverter intensive microgrid [14, 32, 33]. In [34], the frequency‐varying resonances caused by the interaction between the inverters and grid impedance are analysed. By modelling the output admittance of the peak‐current controlled flyback micro‐inverter, Wang et al. [35] introduce a way to analyse the quasi‐resonance issue of the impedance network. It is pointed out that the flyback micro‐inverter has less risk of harmonic resonance compared with the conventional string inverter because the resonant frequencies of the flyback micro‐inverter are normally out of the dominant frequency range. In [36], the resonance issues caused by the submarine transmission cable for an offshore wind farm is investigated. The high shunt capacitance of the submarine cable can cause severe high‐frequency resonance, and a series of considerable resonant peaks are found because of the high‐order LC configuration.Nevertheless, the system stability and resonance of grid‐connected inverters were just investigated separately. Although the relationship between system stability and resonance is analysed in [37], the process is not clear and the argument is inadequate. Thus, their relationship needs to be identified further. In this paper, in order to expediently describe the relationship of system stability and resonance, the resonance is divided into three cases: positive incomplete resonance, complete resonance, and negative incomplete resonance, which exactly correspond to the three stability states: stable, critical stable, and unstable. In the grid‐connected inverter, if all the resonances are positive incomplete resonance, the inverter is stable. If there is no less than one negative incomplete resonance, the inverter is unstable. If the complete resonance occurs, the resonance frequency is the system cutoff frequency and the phase margin is zero at this frequency. In summary, when the stability of the inverter is reduced, the complete resonance will gradually be induced. Moreover, the negative incomplete resonance will arise when the inverter is unstable. In addition, the principle of harmonic amplification of the grid injected current is studied by using the grid voltage background harmonics and the inverter output voltage harmonics as the harmonic sources. It is found that the harmonic amplification is always accompanied by low stability of the system. The lower the stability of the system, the more serious the harmonic amplification. The centre frequency of harmonic distribution is where the system has the lowest stability. Finally, simulation and experimental tests are done to verify the effectiveness of the proposed comprehensive analysis.The remaining part of this paper is organised as follows: in Section 2, the modelling of a single phase grid‐connected inverter is given. Next, the system stability and resonance of grid‐connected inverters are analysed in Section 3 based on the equivalent impedance model and the current control model. Then, the harmonic amplification of the grid injected current is discussed in Section 4. In Sections 5 and 6, the simulation and experimental results, respectively are presented to verify the analysis in this paper. Finally, Section 7 concludes this paper.Description and modelling of single phase grid‐connected inverterFig. 1 shows the specific structure of a single phase LCL‐filtered grid‐connected inverter. The LCL filter consists of an inverter side inductor L1, a filter capacitor C1, and a grid side inductor L2. The grid is considered as an Thevenin circuit that consists of a voltage source ug in series connection with a grid impedance Zg. ur is the inverter output voltage, ig is the grid injected current, and upcc is the voltage at the PCC. The current reference iref is generated using the synchronisation signal sin θ and the current amplitude command Iref. kd is the active damp factor. For the grid‐connected inverter, the dc link voltage can be considered as a constant, namely Udc [24].1Fig.Single phase LCL‐filtered grid‐connected inverterThe digitally controlled inverter has an inherent computation delay and a pulse‐width modulation delay caused by the zero‐order hold effect, also the sampler has to be taken into account [22]. The product of these three elements in the s‐domain is derived as1Gd0(s)=e−Ts⋅s1−e−Ts⋅sTs⋅sIn order to obtain rational transfer functions, the second‐order Padé approximation is applied. Then, Gd0 (s) is approximated as (2). That has been demonstrated the approximation Gd (s) maintains the s‐domain analysis with a fair agreement between simplicity and accuracy [38]2Gd(s)=(1.5Ts⋅s)2−9Ts⋅s+12(1.5Ts⋅s)2+9Ts⋅s+12The current control loop of the inverter is modelled as shown in Fig. 2a, where KPWM is the transfer function of the inverter bridge. The current controller Gc (s) adopts a damped proportional resonant controller which is expressed as3Gc(s)=kP+kR2ξω0ss2+2ξω0s+ω02,where kP is the proportional gain, kR is the resonant gain, and ω0 is the fundamental frequency. The damping factor ξ = 0.001 is set from a practical point of view [39]. According to Fig. 2a, the open loop transfer function G (s) is derived as4G(s)=Gc(s)Gd(s)KPWMs3L1L2C1+s2Gd(s)KPWMkdL2C1+s(L1+L2)2Fig.Current control loop of the inverter(a) Without the grid impedance, (b) With the grid impedanceThe grid injected current is derived as (5), and Yo (s) is the equivalent output admittance of the inverter.5ig(s)=G(s)1+G(s)iref(s)−Yo(s)upcc(s)=I∗(s)−Yo(s)upcc(s)The terminal behavioural model of the inverter seen from the PCC is drawn with a Norton equivalent circuit. The equivalent impedance model of the grid‐connected inverter as shown in Fig. 3, through which it can be determined whether there is resonance in the inverter‐grid system.3Fig.Equivalent impedance model of the inverterRelationship analysis of system stability and resonanceResonanceGenerally, the grid impedance includes a resistor Rg and an inductor Lg, then Zg = Rg + sLg. In Fig. 3, the series resonance will occur when the admittance Yo (s) is matched with the grid impedance Zg at certain frequencies. Assuming that Yo (s) takes the following form:61Yo(jω)=R(ω)+jX(ω)where R (ω) and X (ω) are the resistance and reactance of the equivalent admittance, respectively (they are expressed in Appendix). Since the grid impedance is uncertain, the resonance has three cases:If both R (ω) and X (ω) are totally cancelled with the grid impedance, the (7) is established, which is called complete resonance.7R(ω)+Rg=0X(ω)+ωLg=0If only X (ω) is cancelled with the grid impedance and R (ω) + Rg > 0, the (8) is established, which is called positive incomplete resonance.8R(ω)+Rg>0X(ω)+ωLg=0If only X (ω) is cancelled with the grid impedance and R (ω) + Rg < 0, the (9) is established, which is called negative incomplete resonance.9R(ω)+Rg<0X(ω)+ωLg=0StabilityThe stability of the inverter‐grid system can be determined using the Nyquist impedance stability criterion which has a significant advantage especially for the case of multiple inverters in parallel. However, the Bode diagram is better if we want to investigate the system stability margin.The current control loop that accounts for the grid impedance as shown in Fig. 2b, where L2eq =L2 + Lg. According to Fig. 2b, the open loop transfer function G ′(s) can be derived as10G′(s)=Gc(s)Gd(s)KPWM s3L1L2eqC1+s2Gd(s)KPWMkdL2eqC1+sL1+⋯⋯+sL2eq+Rg1+sGd(s)KPWMkdC1+s2L1C1.By means of (10), the stability of the grid‐connected inverter can be analysed using the system phase margin.Relationship of system stability and resonanceThrough derivation, the deduction (11) is established, which means that if the grid‐connected inverter is in the complete resonance at a certain frequency, the phase margin of the system is zero at that frequency. In this case, the system is critical stable.If the grid‐connected inverter is in the positive or negative incomplete resonance at a certain frequency, we cannot determine the value of G ′(jω), since R (ω) + Rg is uncertain. Also, the phase margin of the system cannot be determined. In this case, the system can be stable or unstable11R(ω)+Rg=0X(ω)+ωLg=0⇒G′(jω)=−1According to Liserre et al. [3], the system parameters are designed, as shown in Table 1. A pure inductor is usually used to draw the worst case of inductive–resistive grid impedance [14]. Fig. 4 shows the Bode diagrams of the transfer function G′ (s) for different proportional gain kP, also R (ω) + Rg and X (ω) + ωLg are illustrated. There are three cutoff frequencies in each Bode diagram, since G ′(s) has a pair of poles on the right half plane. In order to stabilise the grid‐connected inverter, the requirement is that the phase margin at ωc1 is positive and the phase margin at ωc2 and ωc3 is negative.1TableSystem parameters of the grid‐connected inverterParametersValueParametersValueDC link voltage Udc400 Vinverter side inductance L13 mHgrid voltage (RMS) ug220 Vfilter capacitance C15 μFoutput power P5 kWgrid side inductance L21.5 mHfundamental frequency50 Hzdamping factor kd0.04sampling frequency fs20 kHzresonant gain kR1amplitude of the triangular carrier1grid impedance Rg, Lg0 Ω,0.5 mH4Fig.Bode diagrams of the transfer function G′(s) and plots of R(ω) + Rg and X(ω) + ωLg(a) kP = 0.046, (b) kP = 0.066, (c) kP = 0.076Fig. 4a is the case that kP =0.046. ωc2 is 2330 Hz, and the phase margin at ωc2 =−31.3°. Therefore, the inverter is stable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 1578 Hz and ω2 = 2120 Hz. Both R (ω1) + Rg and R (ω2) + Rg are greater than zero. Therefore, the inverter is in the positive incomplete resonance at ω1 and ω2.Fig. 4b is the case that kP =0.066. ωc2 =2060 Hz, and the phase margin at ωc2 =−0.8°. Therefore, the inverter is critically stable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 1192 Hz and ω2 = 2058 Hz. R (ω1) + Rg > 0, and R (ω2) + Rg =0. Therefore, the inverter is in the positive incomplete resonance at ω1 and in the complete resonance at ω2.Fig. 4c is the case that kP =0.076. ωc2 =1880 Hz, and the phase margin at ωc2 =11.6°. Therefore, the inverter is unstable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 740 Hz and ω2 = 2036Hz. R (ω1) + Rg > 0, and R (ω2) + Rg < 0. Therefore, the inverter is in the positive incomplete resonance at ω1 and in the negative incomplete resonance at ω2.The above three cases are summarised as shown in Table 2. We can find that system stability and resonance are two different things. Resonance is a local concept, only at a certain frequency. While the system stability is a global concept. There can be several roots for the equation X (ω) + ωLg = 0, since the grid impedance is uncertain. It means the inverter can be in resonance state at several frequencies, but the stability of the system depends on the value of R (ω) + Rg. If all the resonances of the inverter are positive incomplete resonance, the inverter is stable. If there is no less than one negative incomplete resonance, the inverter is unstable. If the complete resonance occurs, the resonance frequency is the system cutoff frequency and the phase margin of the system is zero at this frequency.2TableSystem stability and resonance of the inverterkPStabilityResonance0.046stablepositive incomplete resonance at ω1positive incomplete resonance at ω20.066critical stablepositive incomplete resonance at ω1complete resonance at ω20.076unstablepositive incomplete resonance at ω1negative incomplete resonance at ω2Harmonic amplification of grid injected currentGenerally the most concerned thing in the grid‐connected inverter is the harmonic amount of the grid injected current. The grid injected current can be polluted by the distorted grid voltage and the inverter output voltage harmonics.Harmonic amplification caused by the distorted grid voltageAccording to Fig. 3, the grid injected current can be polluted by the distorted grid voltage, as shown in (12). igh and ugh are the h harmonic of the grid injected current and the grid voltage, respectively. According to (12), in order to reduce the value of igh, the denominator should be as large as possible12igh=ugh(R+Rg)2+(X+ωLg)2As mentioned above, the system phase margin is zero and the system is in complete resonance occurring simultaneously. In addition, R (ω) and X (ω) are continuous functions. Therefore, both of R (ω) + Rg and X (ω) + ωLg are approaching zero along with the system phase margin approaching zero. On the contrary, R (ω) + Rg and X (ω) + ωLg will not close to zero simultaneously if the grid‐connected inverter has sufficient phase margin. Thus, the grid injected current will not be polluted.Fig. 5 shows the magnitude of the denominator of (12) for kP =0.046 and 0.066, respectively. The other parameters of the inverter are listed in Table 1. In Fig. 5a, the minimum value=5.78 at 2206 Hz, so the minimum attenuation of the grid voltage harmonics=5.78. Therefore, the grid injected current will not be polluted in this case. In Fig. 5b, the magnitude is very close to zero around 2058 Hz, and the tip=0.24. Therefore, the harmonics of the distorted grid voltage will be amplified badly around 2058 Hz.5Fig.Magnitude of the denominator of (12)(a) kP = 0.046, (b) kP = 0.066Harmonic amplification caused by the inverter output voltage harmonicsIn Fig. 2, the inverter output voltage ur contains a large number of harmonics. The transfer function with ur as input and ig as output is derived as (13). Similarly, the denominator of (13) should be as large as possible to attenuate the inverter output voltage harmonics.13Φ(s)=ig(s)ur(s)=1((1+G′(s))/G′(s))Gc(s)Gd(s)KPWMAs mentioned above, G ′(s)=−1 and the system phase margin equals to zero are identical. Obviously, Ф(s) is approaching infinite when G ′(s) approaches −1. Therefore, in order to reduce the harmonic amplification caused by the inverter output voltage harmonics, the system should have sufficient phase margin.Fig. 6 shows the magnitude of the denominator of (13) for kP =0.046 and 0.066 respectively. In Fig. 6a, the minimum value is 6.17 at 2178 Hz, so the minimum attenuation of the inverter output voltage is 6.17. In Fig. 6b, the magnitude is very close to zero around 2058 Hz, and the tip is 0.22. Therefore, the harmonics of the inverter output voltage will be amplified badly around 2058 Hz.6Fig.Magnitude of the denominator of (13)(a) kP = 0.046, (b) kP = 0.066To sum up, the minimum values of the denominators of (12) and (13) depend on the minimum phase margin of the system, i.e. the system stability. Therefore, the grid‐connected inverter is expected to have higher stability to obtain high‐quality grid injected current.Simulation verificationTo verify the analysis above, the time‐domain simulations are performed based on a grid‐connected inverter like in Fig. 1. The system parameters are shown in Table 1. In addition, the grid voltage is arranged to contain 2000, 2050, and 2100 Hz harmonics that respectively have 1% amplitude of the fundamental grid voltage. For kP =0.046, the simulated PCC voltage and grid injected current are shown in Fig. 7a, and the voltage is shrunk by a factor of five in order to clearly observe the two curves in one figure. The spectrum of the grid injected current is shown in Fig. 7b. Corresponding to Fig. 4a, the grid‐connected inverter works stably and the grid injected current is good. The total harmonic distortion (THD) of the grid injected current is 1.95%. The harmonic amplification around 2050 Hz is inconspicuous.7Fig.Simulation results with the distorted grid voltage for kP = 0.046(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentFor kP =0.066, the simulated PCC voltage and grid injected current are shown in Fig. 8a. The grid voltage is arranged not to contain harmonics. The spectrum of the grid injected current is shown in Fig. 8b. The PCC voltage and the grid injected current are both distorted, and the current THD is 7.53%. The inverter output voltage contains a large number of harmonics around the switching frequency, however, the grid injected current is essentially free of high‐frequency harmonics thanks to the high performance of LCL filter. The current harmonics are mainly around 2050, and 2050 Hz harmonic in the grid injected current is 2.15 A. Meanwhile 2050 Hz harmonic of the inverter output voltage is only 0.28 V. According to Fig. 6b, the harmonics in the inverter output voltage are amplified.8Fig.Simulation results with the ideal grid voltage for kP = 0.066(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentFor the case, that kP =0.066, and the grid voltage is arranged to contain harmonics. The waveforms of the PCC voltage and the grid injected current are shown in Fig. 9a. The spectrum of the grid injected current is shown in Fig. 9b. The PCC voltage and the grid injected current are both highly distorted, and the current THD is 28.84%. The current harmonics are mainly around 2050, and 2050 Hz harmonic is 8.74 A. According to Fig. 5b, the harmonics in the grid voltage are amplified.9Fig.Simulation results with the distorted grid voltage for kP = 0.066(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentExperimental verificationFig. 10 depicts the experimental prototype system. The inverter is connected to the actual grid through an isolation transformer and an autotransformer. The inverter bridge adopts the integrated module CCS050M12CM2, and its drive circuit adopts the integrated module CGD15FB45P. The dc link voltage is kept constant by the programmable dc power supply AGP1010. The control algorithm is implemented using the DSP controller TMS320F28335 Experimenter Kit. The ac voltage sensor is DVDI‐001 and it measures the voltage between the isolation transformer and the autotransformer. The leakage inductance of the autotransformer is small and can be neglected so that the measured voltage is regarded as the grid voltage. The ac current sensor is TA1419. Since the input of the A/D module in the DSP is required to be non‐negative, the dc bias circuit is employed to shift the sampling signals. The system parameters are listed in Table 1, and the 1.28 mH inductance provided by the isolation transformer must be taken into account. In addition, the switching frequency is equal to the sampling frequency. In order to keep the unified parameters, the current controller is discretised by the impulse response invariance.10Fig.Photograph of the experimental prototype systemFor the case that kP =0.046, the experimental waveforms of the PCC voltage and grid injected current are shown in Fig. 11. It can be seen that the inverter was running stably and the grid injected current has little harmonics. By using the saved data of the grid injected current waveform acquired by a Tektronix DPO4104 digital phosphor oscilloscope, the current harmonics can be analysed by FFT function. The current THD is 1.95%.11Fig.Experimental waveforms of the PCC voltage and grid injected current for kP = 0.046For the case that kP =0.066, the experimental waveforms of the PCC voltage and grid injected current are shown in Fig. 12a. Both the PCC voltage and the grid injected current contain a large number of harmonics. As the inverter is very close to the critical stable state. The spectrum of the grid injected current is shown in Fig. 12b, and the current THD is 10.54%. The current harmonics mainly appear around 2050 Hz. The current THD is less than that in the simulation shown in Fig. 9, which is due to the fact that harmonics of the actual grid voltage at 2000, 2050, and 2100 Hz are less than that of the simulated grid voltage.12Fig.Experimental results for kP = 0.066(a) Waveforms of the PCC voltage and grid injected current; (b) Spectrum of the grid injected currentConclusionsIn this paper, the comprehensive analysis of system resonance and harmonic amplification of grid‐connected inverters are presented based on the equivalent impedance model and the current control model. It is pointed out that the resonance is a local concept and the system stability is a global concept. The resonances can arise due to the uncertain grid impedance. The complete resonance will gradually be induced when the stability of the inverter is reduced. Also the negative incomplete resonance will arise when the inverter is unstable. In addition, it is also revealed that the cause of harmonic amplification is the low stability of the system, and the centre frequency of harmonic distribution is where the system has the lowest stability. In order to reduce the harmonic content of the grid injected current and to avoid the complete resonance and negative incomplete resonance, the system requires sufficient stability margin. Finally, simulation and experimental results have verified the comprehensive analysis in this paper.AcknowledgmentsThis work was supported by the National Natural Science Foundation of China (grant nos. 51477021 and 51707026), the China Postdoctoral Science Foundation (grant no. 2018M643410), Chongqing Special Postdoctoral Science Foundation (grant no. XmT2018033), and the National ‘111’ Project of China under Grant B808036.9 References1Ehsan, A., Yang, Q., Cheng, M.: ‘A scenario‐based robust investment planning model for multi‐type distributed generation under uncertainties’, IET Gener. Transm. 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Electron., 2009, 56, (3), pp. 706–71739Castilla, M., Miret, J., Matas, J., et al.: ‘Control design guidelines for single‐phase grid‐connected photovoltaic inverters with damped resonant harmonic compensators’, IEEE Trans. Ind. Electron., 2009, 56, (11), pp. 4492–450110AppendixR (ω) and X (ω) are expressed as below:A=1−L1C1ω2,B=12−2.25Ts2ω2,C=kdKPWMC1ωD=L1+L2−L1L2C1ω2,E=kPKPWM−L2Cω.R(ω)=−81Ts2CDω3−81Ts2AEω2+B2CDω+AB2E(AB+9TsCω)2+(BC+9TsAω)2.X(ω)=81Ts2ADω3+9TsC(2BD−9TsE)ω2+B(ABD−18TsAE)ω−B2CE(AB+9TsCω)2+(BC+9TsAω)2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "IET Generation, Transmission & Distribution" Wiley

Analysis of resonance and harmonic amplification for grid‐connected inverters

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Abstract

IntroductionThe penetration level of distributed generation system (DGS) based on renewable energy is gradually increasing because of the mature power generation technology and urgent demand for clean and sustainable energy [1, 2]. The grid‐connected inverter is widely used in DGS due to its advantages of potential for full control of both dc link voltage and power factor [3]. To reduce the high‐frequency harmonics caused by grid‐connected inverters, the LCL filter is widely used with the advantage of smaller values of inductance and capacitance [4]. Currently, the active damping methods are usually employed to solve the resonance problem of LCL filter [5, 6]. However, the uncertain grid impedance threatens the stability of LCL‐filtered grid‐connected inverters. The grid impedance which depends on the different operations of the power system can vary over a wide range [7]. When the LCL‐filtered grid‐connected inverter is connected to a weak grid with large grid impedance, the system stability can be affected and harmonic resonance may occur. Consequently, the grid injected current can be rich in harmonics which will significantly damage the system and equipments [8, 9].In the previous works, the impedance‐based stability criterion has been extensively implemented to analyse and discuss the stability of grid‐connected converters. In [10–16], the impedance‐based stability criterion without coupling of the grid impedance and inverters is introduced. It has been concluded that an inverter is stable if the ratio of the grid impedance and the inverter output impedance satisfies the Nyquist stability criterion. In [14], the Nyquist impedance‐based stability criterion is applied for illustrating how the individual responses of interconnected complex sources and loads affect the total closed‐loop stability. The presented Nyquist application indicates how the individual impedance responses contribute to the closed‐loop system poles. Wang et al. [17] analysed a power‐electronics‐based ac system where two active rectifiers and a power factor correction capacitor are connected to the point of common coupling (PCC). It has been pointed out that large grid impedance can induce system instability, and the interactions of paralleled converters may result in unstable oscillations when multiple harmonic compensators are employed. For n‐parallel grid‐connected inverters, the inverters are coupled due to the grid impedance and the equivalent grid impedance for each inverter will become n times bigger. Thus, the impedance characteristics of the system are changed [17–19]. Based on the above stability criteria, some stability enhancement methods have been proposed, such as the impedance shaping method [14], the adaptive active damper method [20], and the passivity‐based method [21] and so on.The stability of grid‐connected inverters can also be analysed based on the inverter current control models that account for the coupling of the grid impedance and inverters. In [22–24], the modelling and control analysis of grid‐connected inverters are presented, and it is shown that the inverters do not behave as expected due to the grid impedance. Generally, the current control bandwidth will reduce remarkably when the inverter is connected to a weak grid with large grid impedance [7]. Therefore, the possible wide range of the grid impedance can lower the stability of the inverters with multiple selective harmonic compensators, when the frequencies of harmonics to be suppressed are close to or above the current control bandwidth [7, 14, 25, 26]. In addition, the resonance damping methods of the LCL filter may fail in grid‐connected inverters. Especially for considering the digital control delay, the phase characteristics of the system become worse and the system stability is threatened. In [27], a digitally controlled LCL‐filtered grid‐connected inverter with the capacitor‐current‐feedback active damping is investigated. It turns out that the LCL filter resonance frequency is changed by a virtual impedance which is caused by the proportional feedback of the capacitor current. If the actual resonance frequency is higher than one‐sixth of the sampling frequency, the virtual impedance will contain a negative resistor component. As a result, the inverter becomes much easier to be unstable if the resonance frequency is moved closer to fs/6. Lu et al. [28] comprehensively analysed the stability and robustness of a grid‐connected inverter with LCL‐filter in the discrete domain, where the grid voltage is fed forward. The critical resonance frequency of LCL filter is identified and two more boundary frequencies are revealed in a weak grid. It has been demonstrated that the feedforward of grid‐voltage would significantly alter the inverter stability in a weak grid.The current‐controlled grid‐connected inverter is usually modelled with a Norton circuit which consists of a current source in parallel connection with an equivalent admittance. The actual grid can be modelled with a Thevenin circuit that consists of a voltage source in series connection with a grid impedance. When the equivalent admittance is matched with the grid impedance, the series resonance can arise taking the grid voltage background harmonics as the voltage source. Similarly, the parallel resonance can arise taking the inverter output current harmonics as the current source [10–12]. In a word, series and parallel resonance phenomena can arise due to the varied grid impedance [29–31]. The harmonics in the grid injected current can be excessive due to the resonances, and the harmonic problem will be more serious in an inverter intensive microgrid [14, 32, 33]. In [34], the frequency‐varying resonances caused by the interaction between the inverters and grid impedance are analysed. By modelling the output admittance of the peak‐current controlled flyback micro‐inverter, Wang et al. [35] introduce a way to analyse the quasi‐resonance issue of the impedance network. It is pointed out that the flyback micro‐inverter has less risk of harmonic resonance compared with the conventional string inverter because the resonant frequencies of the flyback micro‐inverter are normally out of the dominant frequency range. In [36], the resonance issues caused by the submarine transmission cable for an offshore wind farm is investigated. The high shunt capacitance of the submarine cable can cause severe high‐frequency resonance, and a series of considerable resonant peaks are found because of the high‐order LC configuration.Nevertheless, the system stability and resonance of grid‐connected inverters were just investigated separately. Although the relationship between system stability and resonance is analysed in [37], the process is not clear and the argument is inadequate. Thus, their relationship needs to be identified further. In this paper, in order to expediently describe the relationship of system stability and resonance, the resonance is divided into three cases: positive incomplete resonance, complete resonance, and negative incomplete resonance, which exactly correspond to the three stability states: stable, critical stable, and unstable. In the grid‐connected inverter, if all the resonances are positive incomplete resonance, the inverter is stable. If there is no less than one negative incomplete resonance, the inverter is unstable. If the complete resonance occurs, the resonance frequency is the system cutoff frequency and the phase margin is zero at this frequency. In summary, when the stability of the inverter is reduced, the complete resonance will gradually be induced. Moreover, the negative incomplete resonance will arise when the inverter is unstable. In addition, the principle of harmonic amplification of the grid injected current is studied by using the grid voltage background harmonics and the inverter output voltage harmonics as the harmonic sources. It is found that the harmonic amplification is always accompanied by low stability of the system. The lower the stability of the system, the more serious the harmonic amplification. The centre frequency of harmonic distribution is where the system has the lowest stability. Finally, simulation and experimental tests are done to verify the effectiveness of the proposed comprehensive analysis.The remaining part of this paper is organised as follows: in Section 2, the modelling of a single phase grid‐connected inverter is given. Next, the system stability and resonance of grid‐connected inverters are analysed in Section 3 based on the equivalent impedance model and the current control model. Then, the harmonic amplification of the grid injected current is discussed in Section 4. In Sections 5 and 6, the simulation and experimental results, respectively are presented to verify the analysis in this paper. Finally, Section 7 concludes this paper.Description and modelling of single phase grid‐connected inverterFig. 1 shows the specific structure of a single phase LCL‐filtered grid‐connected inverter. The LCL filter consists of an inverter side inductor L1, a filter capacitor C1, and a grid side inductor L2. The grid is considered as an Thevenin circuit that consists of a voltage source ug in series connection with a grid impedance Zg. ur is the inverter output voltage, ig is the grid injected current, and upcc is the voltage at the PCC. The current reference iref is generated using the synchronisation signal sin θ and the current amplitude command Iref. kd is the active damp factor. For the grid‐connected inverter, the dc link voltage can be considered as a constant, namely Udc [24].1Fig.Single phase LCL‐filtered grid‐connected inverterThe digitally controlled inverter has an inherent computation delay and a pulse‐width modulation delay caused by the zero‐order hold effect, also the sampler has to be taken into account [22]. The product of these three elements in the s‐domain is derived as1Gd0(s)=e−Ts⋅s1−e−Ts⋅sTs⋅sIn order to obtain rational transfer functions, the second‐order Padé approximation is applied. Then, Gd0 (s) is approximated as (2). That has been demonstrated the approximation Gd (s) maintains the s‐domain analysis with a fair agreement between simplicity and accuracy [38]2Gd(s)=(1.5Ts⋅s)2−9Ts⋅s+12(1.5Ts⋅s)2+9Ts⋅s+12The current control loop of the inverter is modelled as shown in Fig. 2a, where KPWM is the transfer function of the inverter bridge. The current controller Gc (s) adopts a damped proportional resonant controller which is expressed as3Gc(s)=kP+kR2ξω0ss2+2ξω0s+ω02,where kP is the proportional gain, kR is the resonant gain, and ω0 is the fundamental frequency. The damping factor ξ = 0.001 is set from a practical point of view [39]. According to Fig. 2a, the open loop transfer function G (s) is derived as4G(s)=Gc(s)Gd(s)KPWMs3L1L2C1+s2Gd(s)KPWMkdL2C1+s(L1+L2)2Fig.Current control loop of the inverter(a) Without the grid impedance, (b) With the grid impedanceThe grid injected current is derived as (5), and Yo (s) is the equivalent output admittance of the inverter.5ig(s)=G(s)1+G(s)iref(s)−Yo(s)upcc(s)=I∗(s)−Yo(s)upcc(s)The terminal behavioural model of the inverter seen from the PCC is drawn with a Norton equivalent circuit. The equivalent impedance model of the grid‐connected inverter as shown in Fig. 3, through which it can be determined whether there is resonance in the inverter‐grid system.3Fig.Equivalent impedance model of the inverterRelationship analysis of system stability and resonanceResonanceGenerally, the grid impedance includes a resistor Rg and an inductor Lg, then Zg = Rg + sLg. In Fig. 3, the series resonance will occur when the admittance Yo (s) is matched with the grid impedance Zg at certain frequencies. Assuming that Yo (s) takes the following form:61Yo(jω)=R(ω)+jX(ω)where R (ω) and X (ω) are the resistance and reactance of the equivalent admittance, respectively (they are expressed in Appendix). Since the grid impedance is uncertain, the resonance has three cases:If both R (ω) and X (ω) are totally cancelled with the grid impedance, the (7) is established, which is called complete resonance.7R(ω)+Rg=0X(ω)+ωLg=0If only X (ω) is cancelled with the grid impedance and R (ω) + Rg > 0, the (8) is established, which is called positive incomplete resonance.8R(ω)+Rg>0X(ω)+ωLg=0If only X (ω) is cancelled with the grid impedance and R (ω) + Rg < 0, the (9) is established, which is called negative incomplete resonance.9R(ω)+Rg<0X(ω)+ωLg=0StabilityThe stability of the inverter‐grid system can be determined using the Nyquist impedance stability criterion which has a significant advantage especially for the case of multiple inverters in parallel. However, the Bode diagram is better if we want to investigate the system stability margin.The current control loop that accounts for the grid impedance as shown in Fig. 2b, where L2eq =L2 + Lg. According to Fig. 2b, the open loop transfer function G ′(s) can be derived as10G′(s)=Gc(s)Gd(s)KPWM s3L1L2eqC1+s2Gd(s)KPWMkdL2eqC1+sL1+⋯⋯+sL2eq+Rg1+sGd(s)KPWMkdC1+s2L1C1.By means of (10), the stability of the grid‐connected inverter can be analysed using the system phase margin.Relationship of system stability and resonanceThrough derivation, the deduction (11) is established, which means that if the grid‐connected inverter is in the complete resonance at a certain frequency, the phase margin of the system is zero at that frequency. In this case, the system is critical stable.If the grid‐connected inverter is in the positive or negative incomplete resonance at a certain frequency, we cannot determine the value of G ′(jω), since R (ω) + Rg is uncertain. Also, the phase margin of the system cannot be determined. In this case, the system can be stable or unstable11R(ω)+Rg=0X(ω)+ωLg=0⇒G′(jω)=−1According to Liserre et al. [3], the system parameters are designed, as shown in Table 1. A pure inductor is usually used to draw the worst case of inductive–resistive grid impedance [14]. Fig. 4 shows the Bode diagrams of the transfer function G′ (s) for different proportional gain kP, also R (ω) + Rg and X (ω) + ωLg are illustrated. There are three cutoff frequencies in each Bode diagram, since G ′(s) has a pair of poles on the right half plane. In order to stabilise the grid‐connected inverter, the requirement is that the phase margin at ωc1 is positive and the phase margin at ωc2 and ωc3 is negative.1TableSystem parameters of the grid‐connected inverterParametersValueParametersValueDC link voltage Udc400 Vinverter side inductance L13 mHgrid voltage (RMS) ug220 Vfilter capacitance C15 μFoutput power P5 kWgrid side inductance L21.5 mHfundamental frequency50 Hzdamping factor kd0.04sampling frequency fs20 kHzresonant gain kR1amplitude of the triangular carrier1grid impedance Rg, Lg0 Ω,0.5 mH4Fig.Bode diagrams of the transfer function G′(s) and plots of R(ω) + Rg and X(ω) + ωLg(a) kP = 0.046, (b) kP = 0.066, (c) kP = 0.076Fig. 4a is the case that kP =0.046. ωc2 is 2330 Hz, and the phase margin at ωc2 =−31.3°. Therefore, the inverter is stable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 1578 Hz and ω2 = 2120 Hz. Both R (ω1) + Rg and R (ω2) + Rg are greater than zero. Therefore, the inverter is in the positive incomplete resonance at ω1 and ω2.Fig. 4b is the case that kP =0.066. ωc2 =2060 Hz, and the phase margin at ωc2 =−0.8°. Therefore, the inverter is critically stable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 1192 Hz and ω2 = 2058 Hz. R (ω1) + Rg > 0, and R (ω2) + Rg =0. Therefore, the inverter is in the positive incomplete resonance at ω1 and in the complete resonance at ω2.Fig. 4c is the case that kP =0.076. ωc2 =1880 Hz, and the phase margin at ωc2 =11.6°. Therefore, the inverter is unstable. In this case, the equation X (ω) + ωLg = 0 has two roots ω1 = 740 Hz and ω2 = 2036Hz. R (ω1) + Rg > 0, and R (ω2) + Rg < 0. Therefore, the inverter is in the positive incomplete resonance at ω1 and in the negative incomplete resonance at ω2.The above three cases are summarised as shown in Table 2. We can find that system stability and resonance are two different things. Resonance is a local concept, only at a certain frequency. While the system stability is a global concept. There can be several roots for the equation X (ω) + ωLg = 0, since the grid impedance is uncertain. It means the inverter can be in resonance state at several frequencies, but the stability of the system depends on the value of R (ω) + Rg. If all the resonances of the inverter are positive incomplete resonance, the inverter is stable. If there is no less than one negative incomplete resonance, the inverter is unstable. If the complete resonance occurs, the resonance frequency is the system cutoff frequency and the phase margin of the system is zero at this frequency.2TableSystem stability and resonance of the inverterkPStabilityResonance0.046stablepositive incomplete resonance at ω1positive incomplete resonance at ω20.066critical stablepositive incomplete resonance at ω1complete resonance at ω20.076unstablepositive incomplete resonance at ω1negative incomplete resonance at ω2Harmonic amplification of grid injected currentGenerally the most concerned thing in the grid‐connected inverter is the harmonic amount of the grid injected current. The grid injected current can be polluted by the distorted grid voltage and the inverter output voltage harmonics.Harmonic amplification caused by the distorted grid voltageAccording to Fig. 3, the grid injected current can be polluted by the distorted grid voltage, as shown in (12). igh and ugh are the h harmonic of the grid injected current and the grid voltage, respectively. According to (12), in order to reduce the value of igh, the denominator should be as large as possible12igh=ugh(R+Rg)2+(X+ωLg)2As mentioned above, the system phase margin is zero and the system is in complete resonance occurring simultaneously. In addition, R (ω) and X (ω) are continuous functions. Therefore, both of R (ω) + Rg and X (ω) + ωLg are approaching zero along with the system phase margin approaching zero. On the contrary, R (ω) + Rg and X (ω) + ωLg will not close to zero simultaneously if the grid‐connected inverter has sufficient phase margin. Thus, the grid injected current will not be polluted.Fig. 5 shows the magnitude of the denominator of (12) for kP =0.046 and 0.066, respectively. The other parameters of the inverter are listed in Table 1. In Fig. 5a, the minimum value=5.78 at 2206 Hz, so the minimum attenuation of the grid voltage harmonics=5.78. Therefore, the grid injected current will not be polluted in this case. In Fig. 5b, the magnitude is very close to zero around 2058 Hz, and the tip=0.24. Therefore, the harmonics of the distorted grid voltage will be amplified badly around 2058 Hz.5Fig.Magnitude of the denominator of (12)(a) kP = 0.046, (b) kP = 0.066Harmonic amplification caused by the inverter output voltage harmonicsIn Fig. 2, the inverter output voltage ur contains a large number of harmonics. The transfer function with ur as input and ig as output is derived as (13). Similarly, the denominator of (13) should be as large as possible to attenuate the inverter output voltage harmonics.13Φ(s)=ig(s)ur(s)=1((1+G′(s))/G′(s))Gc(s)Gd(s)KPWMAs mentioned above, G ′(s)=−1 and the system phase margin equals to zero are identical. Obviously, Ф(s) is approaching infinite when G ′(s) approaches −1. Therefore, in order to reduce the harmonic amplification caused by the inverter output voltage harmonics, the system should have sufficient phase margin.Fig. 6 shows the magnitude of the denominator of (13) for kP =0.046 and 0.066 respectively. In Fig. 6a, the minimum value is 6.17 at 2178 Hz, so the minimum attenuation of the inverter output voltage is 6.17. In Fig. 6b, the magnitude is very close to zero around 2058 Hz, and the tip is 0.22. Therefore, the harmonics of the inverter output voltage will be amplified badly around 2058 Hz.6Fig.Magnitude of the denominator of (13)(a) kP = 0.046, (b) kP = 0.066To sum up, the minimum values of the denominators of (12) and (13) depend on the minimum phase margin of the system, i.e. the system stability. Therefore, the grid‐connected inverter is expected to have higher stability to obtain high‐quality grid injected current.Simulation verificationTo verify the analysis above, the time‐domain simulations are performed based on a grid‐connected inverter like in Fig. 1. The system parameters are shown in Table 1. In addition, the grid voltage is arranged to contain 2000, 2050, and 2100 Hz harmonics that respectively have 1% amplitude of the fundamental grid voltage. For kP =0.046, the simulated PCC voltage and grid injected current are shown in Fig. 7a, and the voltage is shrunk by a factor of five in order to clearly observe the two curves in one figure. The spectrum of the grid injected current is shown in Fig. 7b. Corresponding to Fig. 4a, the grid‐connected inverter works stably and the grid injected current is good. The total harmonic distortion (THD) of the grid injected current is 1.95%. The harmonic amplification around 2050 Hz is inconspicuous.7Fig.Simulation results with the distorted grid voltage for kP = 0.046(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentFor kP =0.066, the simulated PCC voltage and grid injected current are shown in Fig. 8a. The grid voltage is arranged not to contain harmonics. The spectrum of the grid injected current is shown in Fig. 8b. The PCC voltage and the grid injected current are both distorted, and the current THD is 7.53%. The inverter output voltage contains a large number of harmonics around the switching frequency, however, the grid injected current is essentially free of high‐frequency harmonics thanks to the high performance of LCL filter. The current harmonics are mainly around 2050, and 2050 Hz harmonic in the grid injected current is 2.15 A. Meanwhile 2050 Hz harmonic of the inverter output voltage is only 0.28 V. According to Fig. 6b, the harmonics in the inverter output voltage are amplified.8Fig.Simulation results with the ideal grid voltage for kP = 0.066(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentFor the case, that kP =0.066, and the grid voltage is arranged to contain harmonics. The waveforms of the PCC voltage and the grid injected current are shown in Fig. 9a. The spectrum of the grid injected current is shown in Fig. 9b. The PCC voltage and the grid injected current are both highly distorted, and the current THD is 28.84%. The current harmonics are mainly around 2050, and 2050 Hz harmonic is 8.74 A. According to Fig. 5b, the harmonics in the grid voltage are amplified.9Fig.Simulation results with the distorted grid voltage for kP = 0.066(a) waveforms of the PCC voltage and grid injected current; (b) spectrum of the grid injected currentExperimental verificationFig. 10 depicts the experimental prototype system. The inverter is connected to the actual grid through an isolation transformer and an autotransformer. The inverter bridge adopts the integrated module CCS050M12CM2, and its drive circuit adopts the integrated module CGD15FB45P. The dc link voltage is kept constant by the programmable dc power supply AGP1010. The control algorithm is implemented using the DSP controller TMS320F28335 Experimenter Kit. The ac voltage sensor is DVDI‐001 and it measures the voltage between the isolation transformer and the autotransformer. The leakage inductance of the autotransformer is small and can be neglected so that the measured voltage is regarded as the grid voltage. The ac current sensor is TA1419. Since the input of the A/D module in the DSP is required to be non‐negative, the dc bias circuit is employed to shift the sampling signals. The system parameters are listed in Table 1, and the 1.28 mH inductance provided by the isolation transformer must be taken into account. In addition, the switching frequency is equal to the sampling frequency. In order to keep the unified parameters, the current controller is discretised by the impulse response invariance.10Fig.Photograph of the experimental prototype systemFor the case that kP =0.046, the experimental waveforms of the PCC voltage and grid injected current are shown in Fig. 11. It can be seen that the inverter was running stably and the grid injected current has little harmonics. By using the saved data of the grid injected current waveform acquired by a Tektronix DPO4104 digital phosphor oscilloscope, the current harmonics can be analysed by FFT function. The current THD is 1.95%.11Fig.Experimental waveforms of the PCC voltage and grid injected current for kP = 0.046For the case that kP =0.066, the experimental waveforms of the PCC voltage and grid injected current are shown in Fig. 12a. Both the PCC voltage and the grid injected current contain a large number of harmonics. As the inverter is very close to the critical stable state. The spectrum of the grid injected current is shown in Fig. 12b, and the current THD is 10.54%. The current harmonics mainly appear around 2050 Hz. The current THD is less than that in the simulation shown in Fig. 9, which is due to the fact that harmonics of the actual grid voltage at 2000, 2050, and 2100 Hz are less than that of the simulated grid voltage.12Fig.Experimental results for kP = 0.066(a) Waveforms of the PCC voltage and grid injected current; (b) Spectrum of the grid injected currentConclusionsIn this paper, the comprehensive analysis of system resonance and harmonic amplification of grid‐connected inverters are presented based on the equivalent impedance model and the current control model. It is pointed out that the resonance is a local concept and the system stability is a global concept. The resonances can arise due to the uncertain grid impedance. The complete resonance will gradually be induced when the stability of the inverter is reduced. Also the negative incomplete resonance will arise when the inverter is unstable. 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Journal

"IET Generation, Transmission & Distribution" – Wiley

Published: May 1, 2019

Keywords: invertors; electric current control; power grids; power system stability; harmonic amplification; uncertain grid impedance; grid voltage background harmonics; system stability; positive incomplete resonance; complete resonance; negative incomplete resonance; inverter output voltage harmonics; single phase grid‐connected inverter; MATLAB/SIMULINK software

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