A Novel Stability Improvement Strategy for a Multi-Inverter System in a Weak Grid Utilizing Dual-Mode Control

: Due to the increasing penetration of distributed generations (DGS) and non-negligible grid impedance, the instability problem of the multi-inverter system operating in current source mode (CSM) is becoming serious. In this paper, a closed-loop transfer function model of such a multi-inverter system is established, by which it is concluded that output current resonance will occur with the increase in the grid impedance. In order to address this problem, this paper presents a novel dual-mode control scheme of multiple inverters: one inverter operating in CSM will be alternated into voltage source mode (VSM) if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme. Finally, the validity of the proposed theory is demonstrated by extensive simulations and experiments.


Introduction
Constrained by land resources and light resources, large-scale solar photovoltaic power generation and other distributed generations (DGS) are often installed in remote areas, and due to the impact of long transmission lines and transformers, the point of common coupling (PCC) shows weak grid characteristics with non-negligible grid impedance [1][2][3]. In addition, with the increasing penetration of DGS, it is necessary to connect multiple grid-connected inverters in parallel to the grid in order to increase the power generation efficiency and the scalability of the system [4]. Note that parallel inverters can cause circulating currents and can lead to power semiconductor damage [5][6][7][8][9]. Furthermore, the equivalent grid impedance of a single grid-connected inverter increases, and the grid presents obvious weak grid characteristics [10,11].
At present, there are two main types of grid-connected inverter stability control strategies for weak grids: The first type is the traditional control strategy of grid-connected inverters, which regulates the active and reactive power injected into the grid by adjusting the d-axis and q-axis currents based on grid voltage orientation. The phase-locked loop (PLL) is used to observe the voltage phase of the grid. In this control mode, the inverter is equivalent to a current source, which is called current source mode (CSM) in this paper. Moreover, the existing grid-connected inverters use CSM control to achieve maximum power point tracking (MPPT) so as to ensure the maximum efficiency of new energy generation. So, the CSM scheme is a widely used grid-connected control strategy in multi-inverter in VSM is more stable in an extremely weak grid than CSM [19], a novel stability improvement strategy for a multi-inverter system in a weak grid utilizing dual-mode control is proposed: i.e., one inverter operating in CSM will be alternated into VSM if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme.
This paper is organized as follows. In Section 2, the model of a CSM-only-controlled multiinverter system in a weak grid is established. Section 3 concluded that output current resonance will occur with the increase in the grid impedance. Section 4 explains the proposed dual-mode control strategy in the weak grid and its stability is analyzed. In Section 5, simulations and experimental results of a multi-inverter system consisting of three 100-kW grid-connected inverters are demonstrated. Section 6 is devoted to the conclusion.

Modeling of a Current Source Mode (CSM)-Only-Controlled Multi-Inverter System in a Weak Grid
According to the above introduction, the multi-inverter system often operates under a singlemode control strategy with CSM. Figure 1 shows a typical structure of the three-phase grid-connected inverter operating at CSM, and N such structures are connected in parallel will form a multi-inverter system. In Figure 1, Vdc is the DC voltage, L1 is the inverter side filter inductor, C is the filter capacitor, Rd is the damping resistor, L2 is the grid side filter inductor, iL is the inverter side current, ig represents the grid current, and eg is the grid voltage. The idref and iqref are the reference values of the d axis and the q axis current, respectively. iLd and iLq are the d axis and q axis components, respectively, of the current iL in the synchronous rotating coordinate system. θCSM is the phase obtained by the PLL according to the PCC voltage uoabc. Zg is the grid impedance, as shown in Equation (1).
where Rg and Lg represent the resistive component and inductive component of grid impedance, respectively. Since the stability of the inverter in a weak grid is very susceptible to the inductance component, and the resistive component has the effect of enhancing the stability of the inverter [20], this paper takes Rg = 0. In Figure 1, V dc is the DC voltage, L 1 is the inverter side filter inductor, C is the filter capacitor, R d is the damping resistor, L 2 is the grid side filter inductor, i L is the inverter side current, i g represents the grid current, and e g is the grid voltage. The i dref and i qref are the reference values of the d axis and the q axis current, respectively. i Ld and i Lq are the d axis and q axis components, respectively, of the current i L in the synchronous rotating coordinate system. θ CSM is the phase obtained by the PLL according to the PCC voltage u oabc . Z g is the grid impedance, as shown in Equation (1).
Z g (s) = R g + sL g (1) where R g and L g represent the resistive component and inductive component of grid impedance, respectively. Since the stability of the inverter in a weak grid is very susceptible to the inductance component, and the resistive component has the effect of enhancing the stability of the inverter [20], this paper takes R g = 0. According to Figure 1, the control block diagram of an inverter operating in CSM is shown in Figure 2. Since the d-axis and q-axis controls are the same, the subscripts d and q are omitted in the following text.
Energies 2018, 11, x FOR PEER REVIEW 4 of 19 According to Figure 1, the control block diagram of an inverter operating in CSM is shown in Figure 2. Since the d-axis and q-axis controls are the same, the subscripts d and q are omitted in the following text. In Figure 2, the Gd(s) represents the influence of the sampler and the delays caused by a digital controller in the s-domain. According to [4,21], the transfer function of the delays incurred by the sampler is expressed as 1/Ts, where Ts is the sampling period. The delays incurred by a digitally controlled system contain the computation delay and the pulse width modulation (PWM) control delay.
where Td is the delay time. From Figure 2, the transfer function between the input current command iref(s) and the output current ig(s) can be obtained: According to [19], the inverter operating in CSM is equivalent to the current source, and from Equation (3), the current source equivalent model is shown in Figure 3.  In Figure 2, the G d (s) represents the influence of the sampler and the delays caused by a digital controller in the s-domain. According to [4,21], the transfer function of the delays incurred by the sampler is expressed as 1/T s , where T s is the sampling period. The delays incurred by a digitally controlled system contain the computation delay and the pulse width modulation (PWM) control delay. The expression of G d (s) is: where T d is the delay time. From Figure 2, the transfer function between the input current command i ref (s) and the output current i g (s) can be obtained: According to [19], the inverter operating in CSM is equivalent to the current source, and from Equation (3), the current source equivalent model is shown in Figure 3. According to Figure 1, the control block diagram of an inverter operating in CSM is shown in Figure 2. Since the d-axis and q-axis controls are the same, the subscripts d and q are omitted in the following text. In Figure 2, the Gd(s) represents the influence of the sampler and the delays caused by a digital controller in the s-domain. According to [4,21], the transfer function of the delays incurred by the sampler is expressed as 1/Ts, where Ts is the sampling period. The delays incurred by a digitally controlled system contain the computation delay and the pulse width modulation (PWM) control delay.
where Td is the delay time. From Figure 2, the transfer function between the input current command iref(s) and the output current ig(s) can be obtained: According to [19], the inverter operating in CSM is equivalent to the current source, and from Equation (3), the current source equivalent model is shown in Figure 3.   Z oi_1 (s) and Z oi_2 (s) in Figure 3 can be derived from Equation (3): Generally, photovoltaic and wind power plants employ the same type of the inverter with identical hardware parameters and control algorithms, so the equivalent models of the inverter circuits are consistent. According to Figure 3, an equivalent model of a multi-inverter system with CSM-only control is available, as shown in Figure 4. N represents the number of inverters; for a multi-inverter system, N is larger than 2. i represents any inverter in this system, and 1 ≤ i ≤ N. Zoi_1(s) and Zoi_2(s) in Figure 3 can be derived from Equation (3): Generally, photovoltaic and wind power plants employ the same type of the inverter with identical hardware parameters and control algorithms, so the equivalent models of the inverter circuits are consistent. According to Figure 3, an equivalent model of a multi-inverter system with CSM-only control is available, as shown in Figure 4. N represents the number of inverters; for a multiinverter system, N is larger than 2. i represents any inverter in this system, and 1 ≤ i ≤ N.  From Figure 4, we can get: In Equations (7) and (8), "//" represents the parallel operation, i.e., x//y = xy/(x + y).

Stability Analysis of a CSM-Only-Controlled Multi-Inverter System in a Weak Grid
According to Equation (6), due to the existence of grid impedance, multiple coupling transfer functions exist in multi-inverter systems, which affects the system stability. To illustrate the effect of grid impedance on this multi-inverter system, the main parameters of the inverter operating in CSM are given in Table 1. In this section, the multi-inverter system takes the number of units N = 3 as an example and takes the output current of the first inverter (i = 1) as the research object, and the characteristics of the transfer functions G CSM_P (s), G CSM_N (s), and G CSM_E (s) are studied, respectively. G CSM_P (s) represents the closed-loop transfer function of the reference current to the output current of the CSM-controlled inverters. The Bode diagram and pole-zero map of G CSM_P (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 5a and 5b, respectively. From Figure 5a, when the grid impedance increases, the amplitude of the resonance peak gets larger, and its frequency is gradually moving toward the low-frequency direction. When Z g (s) = 6 p.u., the amplitude of the resonance peak is nearly 20 dB, which means that the current harmonics here will be amplified about 10 times, and the output current will resonate. To further illustrate the stability issue, the distribution of poles near the imaginary axis is given in Figure 5b. With a Z g (s) increase from 1 p.u. to 6 p.u., the poles gradually cross the imaginary axis, so the inverter will operate in an unstable fashion.    Figure 6a, when the grid impedance increases, the amplitude of the resonance peak appears and becomes larger. When Zg(s) = 6 p.u., the amplitude of resonance peak is nearly 20 dB at 1000 rad/s, which means that the current harmonics of one inverter will be amplified in another inverter by about 10 times. Therefore, with the increase of grid impedance, the coupling between the inverters increases, and the harmonics of the other inverters will be amplified and become resonant at the output current of other inverters. To further illustrate the stability issue, the distribution of poles near the imaginary axis is shown in Figure 6b. With the Zg(s) increase from 1 p.u. to 6 p.u., the poles gradually cross the imaginary axis, so the inverter will operate in an unstable fashion.

Magnitude(dB)
With Zg(s) increase, resonance peak becomes larger Zg(s)=6 p.u.  G CSM_N (s) represents the interaction between the CSM-controlled inverters. The Bode diagram and pole-zero map of G CSM_N (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 6a and 6b, respectively. From Figure 6a, when the grid impedance increases, the amplitude of the resonance peak appears and becomes larger. When Z g (s) = 6 p.u., the amplitude of resonance peak is nearly 20 dB at 1000 rad/s, which means that the current harmonics of one inverter will be amplified in another inverter by about 10 times. Therefore, with the increase of grid impedance, the coupling between the inverters increases, and the harmonics of the other inverters will be amplified and become resonant at the output current of other inverters. To further illustrate the stability issue, the distribution of poles near the imaginary axis is shown in Figure 6b. With the Z g (s) increase from 1 p.u. to 6 p.u., the poles gradually cross the imaginary axis, so the inverter will operate in an unstable fashion.   Figure 6a, when the grid impedance increases, the amplitude of the resonance peak appears and becomes larger. When Zg(s) = 6 p.u., the amplitude of resonance peak is nearly 20 dB at 1000 rad/s, which means that the current harmonics of one inverter will be amplified in another inverter by about 10 times. Therefore, with the increase of grid impedance, the coupling between the inverters increases, and the harmonics of the other inverters will be amplified and become resonant at the output current of other inverters. To further illustrate the stability issue, the distribution of poles near the imaginary axis is shown in Figure 6b. With the Zg(s) increase from 1 p.u. to 6 p.u., the poles gradually cross the imaginary axis, so the inverter will operate in an unstable fashion.

Magnitude(dB)
With Zg(s) increase, resonance peak becomes larger Zg(s)=6 p.u.  and the background harmonics of the weak grid will be amplified and become resonant at the output current of the inverter. the inverter increases, and the background harmonics of the weak grid will be amplified and become resonant at the output current of the inverter.

Magnitude(dB)
With Zg(s) increase, resonance peak becomes larger In summary, by constructing the closed-loop transfer function model of the multi-inverter system with CSM-only control, it is found that as the grid impedance increases, a resonance peak appears and the degree of coupling between the inverters and the grid increases, and the inverter is prone to output current resonance.

Proposed Dual-Mode Control Strategy in the Weak Grid
Previous research [19] points out that the grid-connected inverter operating in VSM is more stable in a weak grid than CSM. Due to the object of analysis in this study being only a single inverter, if the inverters in a multi-inverter system are equipped with such a dual-mode adaptive switching operation control strategy which could adaptively switch the operating mode between CSM and VSM, then a novel control strategy for multi-inverter system stability based on dual mode grid impedance adaptation in the weak grid is proposed: One inverter operating in CSM in the system could switch to VSM control based on the grid impedance identification, and then the multi-inverter system will run under dual-mode control and operate stably.
In order to validate this proposed control strategy, similar to the above analysis in Section 2, the model of a multi-inverter system with dual modes is established. Figure 8 shows a typical structure of the three-phase grid-connected inverter operating in VSM. The main circuit of the inverter is the same as that of Figure 1. udref and uqref are the d-axis and q-axis components of the reference voltage loop, respectively. θVSM is the phase of the power angle which can be set according to the desired output power. The ud and uq are the d-axis and q-axis components of the capacitor voltage uCabc, respectively. Differently from CSM control, the inverter operating in VSM is equivalent to a voltage source.
According to Figure 8, the control block diagram of the inverter operating in VSM is shown in Figure 9. In summary, by constructing the closed-loop transfer function model of the multi-inverter system with CSM-only control, it is found that as the grid impedance increases, a resonance peak appears and the degree of coupling between the inverters and the grid increases, and the inverter is prone to output current resonance.

Proposed Dual-Mode Control Strategy in the Weak Grid
Previous research [19] points out that the grid-connected inverter operating in VSM is more stable in a weak grid than CSM. Due to the object of analysis in this study being only a single inverter, if the inverters in a multi-inverter system are equipped with such a dual-mode adaptive switching operation control strategy which could adaptively switch the operating mode between CSM and VSM, then a novel control strategy for multi-inverter system stability based on dual mode grid impedance adaptation in the weak grid is proposed: One inverter operating in CSM in the system could switch to VSM control based on the grid impedance identification, and then the multi-inverter system will run under dual-mode control and operate stably.
In order to validate this proposed control strategy, similar to the above analysis in Section 2, the model of a multi-inverter system with dual modes is established. Figure 8 shows a typical structure of the three-phase grid-connected inverter operating in VSM. The main circuit of the inverter is the same as that of Figure 1. u dref and u qref are the d-axis and q-axis components of the reference voltage loop, respectively. θ VSM is the phase of the power angle which can be set according to the desired output power. The u d and u q are the d-axis and q-axis components of the capacitor voltage u Cabc , respectively. Differently from CSM control, the inverter operating in VSM is equivalent to a voltage source.
According to Figure 8, the control block diagram of the inverter operating in VSM is shown in Figure 9.  In Figure 9, the PIv(s) represents the voltage outer loop proportional-integral (PI) regulator, and PIc(s) is the current inner loop proportional regulator. The expressions of PIv(s) and PIc(s) are shown as follows: where KvP and KvI are proportional and integral coefficients of the PIv(s), respectively. KiP is the proportional coefficient of the PIc(s). According to Figure 9, the transfer function between the input voltage command uref(s) and the output voltage u(s) can be obtained [22]: From Equation (12), the voltage source equivalent model is shown in Figure 10; Gv(s) and Zov(s) in Figure 10 can be derived from Equation (12):  In Figure 9, the PIv(s) represents the voltage outer loop proportional-integral (PI) regulator, and PIc(s) is the current inner loop proportional regulator. The expressions of PIv(s) and PIc(s) are shown as follows: where KvP and KvI are proportional and integral coefficients of the PIv(s), respectively. KiP is the proportional coefficient of the PIc(s). According to Figure 9, the transfer function between the input voltage command uref(s) and the output voltage u(s) can be obtained [22]: From Equation (12), the voltage source equivalent model is shown in Figure 10; Gv(s) and Zov(s) in Figure 10 can be derived from Equation (12): In Figure 9, the PI v (s) represents the voltage outer loop proportional-integral (PI) regulator, and PI c (s) is the current inner loop proportional regulator. The expressions of PI v (s) and PI c (s) are shown as follows: PI v (s) = K vP + (K vI /s) PI c (s) = K iP (11) where K vP and K vI are proportional and integral coefficients of the PI v (s), respectively. K iP is the proportional coefficient of the PI c (s). According to Figure 9, the transfer function between the input voltage command u ref (s) and the output voltage u(s) can be obtained [22]: From Equation (12), the voltage source equivalent model is shown in Figure 10; G v (s) and Z ov (s) in Figure 10 can be derived from Equation (12): Energies 2018, 11, x FOR PEER REVIEW 10 of 19  Figure 10. Voltage source equivalent model of an inverter operating in VSM.

Modeling of the Multi-Inverter System with Dual Modes
According to Figures 3 and 10, when the multi-inverter system contains CSM-and VSMcontrolled inverters at the same time, the equivalent model of a multi-inverter system with dual modes is shown in Figure 11. Figure 11. Equivalent model of a multi-inverter system with dual modes.

Zg(s) eg(s)
From Figure 11, we can get:

Modeling of the Multi-Inverter System with Dual Modes
According to Figures 3 and 10, when the multi-inverter system contains CSM-and VSM-controlled inverters at the same time, the equivalent model of a multi-inverter system with dual modes is shown in Figure 11.  Figure 10. Voltage source equivalent model of an inverter operating in VSM.

Modeling of the Multi-Inverter System with Dual Modes
According to Figures 3 and 10, when the multi-inverter system contains CSM-and VSMcontrolled inverters at the same time, the equivalent model of a multi-inverter system with dual modes is shown in Figure 11. Figure 11. Equivalent model of a multi-inverter system with dual modes.

Zg(s) eg(s)
From Figure 11, we can get: Figure 11. Equivalent model of a multi-inverter system with dual modes.

Stability Analysis of the Multi-Inverter System with Dual Modes
In Section 2, the stability of the multi-inverter system with CSM-only control is analyzed; a similar analysis method will be used in this section to verify that the proposed dual-mode multi-inverter system can effectively improve system stability and reduce coupling between inverters in a weak grid.
The main parameters of the inverter operating in VSM are given in Table 2. It is worth mentioning that the hardware parameters of the VSM-controlled inverter are consistent with the CSM. Therefore, Table 2 only gives the corresponding controller parameters. In this section, the multi-inverter system takes the number of units N = 3 as an example. This system consists of two CSM-controlled inverters and one VSM-controlled inverter to form a dual-mode hybrid system. Moreover, according to reference [19], grid-connected inverters running in VSM can operate stably in a weak grid with high grid impedance, while grid-connected inverters in CSM are prone to output current resonance. Therefore, the stability of this dual-mode multi-inverter system is studied from the point of view of a CSM-controlled inverter in this paper.
In order to correspond to the analysis of Section 2, the output current of the first CSM-controlled inverter (i = 1) in the multi-inverter system is also taken as the research object, and the characteristics of the transfer functions G VSM_P (s), G VSM_N (s), G VSM_E (s), and G VSM_U (s) are studied, respectively.
Similar to the multi-inverter system with CSM-only control, G VSM_P (s) represents the closed-loop transfer function of the reference current to the output current of the CSM-controlled inverters in the dual-mode controlled multi-inverter system. The Bode diagram and pole-zero map of G VSM_P (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 12a and 12b, respectively. Differently from Figure 5, with the increase of grid impedance, the output current of the grid-connected inverter controlled by the current source has no resonance peak, and the pole is far from the virtual axis. It shows that the inverter can operate stably at this time.
transfer function of the reference current to the output current of the CSM-controlled inverters in the dual-mode controlled multi-inverter system. The Bode diagram and pole-zero map of GVSM_P(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 12a and Figure 12b, respectively. Differently from Figure 5, with the increase of grid impedance, the output current of the grid-connected inverter controlled by the current source has no resonance peak, and the pole is far from the virtual axis. It shows that the inverter can operate stably at this time.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears Zg(s)=6 p.u. Similarly, GVSM_N(s) represents the interaction between the CSM-controlled inverters. The Bode diagram and pole-zero map of GVSM_N(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 13a and Figure 13b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, with the grid impedance increases, the degree of coupling between the inverters does not change significantly.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears  Similarly, G VSM_N (s) represents the interaction between the CSM-controlled inverters. The Bode diagram and pole-zero map of G VSM_N (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 13a and 13b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, with the grid impedance increases, the degree of coupling between the inverters does not change significantly.
transfer function of the reference current to the output current of the CSM-controlled inverters in the dual-mode controlled multi-inverter system. The Bode diagram and pole-zero map of GVSM_P(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 12a and Figure 12b, respectively. Differently from Figure 5, with the increase of grid impedance, the output current of the grid-connected inverter controlled by the current source has no resonance peak, and the pole is far from the virtual axis. It shows that the inverter can operate stably at this time.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears Similarly, GVSM_N(s) represents the interaction between the CSM-controlled inverters. The Bode diagram and pole-zero map of GVSM_N(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 13a and Figure 13b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, with the grid impedance increases, the degree of coupling between the inverters does not change significantly.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears  Similarly, G VSM_E (s) represents the coupling degree between the CSM-controlled inverters and grid. The Bode diagram and pole-zero map of G VSM_E (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 14a and 14b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, as the grid impedance increases, the degree of coupling between the inverters and the grid does not have apparent changes.
Similarly, GVSM_E(s) represents the coupling degree between the CSM-controlled inverters and grid. The Bode diagram and pole-zero map of GVSM_E(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 14a and Figure 14b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, as the grid impedance increases, the degree of coupling between the inverters and the grid does not have apparent changes.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears It is worth noting that a new transfer function, GVSM_U(s), appears when the multi-inverter system is operating under the dual-mode control. GVSM_U(s) represents the degree of coupling between the CSM-controlled inverters and the VSM-controlled inverter. The Bode diagram and pole-zero map of GVSM_U(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 15a and Figure 15b, respectively. It can be seen from Figure 15 that there is no resonance peak on the Bode diagram when the grid impedance increases, and the poles are still far from the virtual axis. Therefore, in this kind of dual-mode multi-inverter system, the coupling between CSM-and VSM-controlled inverters is not increased.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears In summary, since the grid-connected inverter operating in voltage source mode (VSM) is more stable in an extremely weak grid than CSM [19], a novel stability improvement strategy of the multiinverter system in a weak grid utilizing dual-mode control is proposed: one inverter operating in CSM will be alternated into VSM if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme. Figure 14. The multi-inverter system with dual-mode control: (a) Bode diagram of G VSM_E (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u.; (b) pole-zero map of G VSM_E (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u.
It is worth noting that a new transfer function, G VSM_U (s), appears when the multi-inverter system is operating under the dual-mode control. G VSM_U (s) represents the degree of coupling between the CSM-controlled inverters and the VSM-controlled inverter. The Bode diagram and pole-zero map of G VSM_U (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u. are shown in Figures 15a and  15b, respectively. It can be seen from Figure 15 that there is no resonance peak on the Bode diagram when the grid impedance increases, and the poles are still far from the virtual axis. Therefore, in this kind of dual-mode multi-inverter system, the coupling between CSM-and VSM-controlled inverters is not increased.
Similarly, GVSM_E(s) represents the coupling degree between the CSM-controlled inverters and grid. The Bode diagram and pole-zero map of GVSM_E(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 14a and Figure 14b, respectively. With the increase of grid impedance, the output current of the grid-connected inverter controlled by CSM has no resonance peak and the poles are still far from the virtual axis. Therefore, as the grid impedance increases, the degree of coupling between the inverters and the grid does not have apparent changes.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears It is worth noting that a new transfer function, GVSM_U(s), appears when the multi-inverter system is operating under the dual-mode control. GVSM_U(s) represents the degree of coupling between the CSM-controlled inverters and the VSM-controlled inverter. The Bode diagram and pole-zero map of GVSM_U(s) with the grid impedance Zg(s) changes from 1 p.u. to 6 p.u. are shown in Figure 15a and Figure 15b, respectively. It can be seen from Figure 15 that there is no resonance peak on the Bode diagram when the grid impedance increases, and the poles are still far from the virtual axis. Therefore, in this kind of dual-mode multi-inverter system, the coupling between CSM-and VSM-controlled inverters is not increased.

Magnitude(dB)
With Zg(s) increase, no resonance peak appears In summary, since the grid-connected inverter operating in voltage source mode (VSM) is more stable in an extremely weak grid than CSM [19], a novel stability improvement strategy of the multiinverter system in a weak grid utilizing dual-mode control is proposed: one inverter operating in CSM will be alternated into VSM if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme. Figure 15. The multi-inverter system with dual-mode control: (a) Bode diagram of G VSM_U (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u.; (b) pole-zero map of G VSM_U (s) with the grid impedance Z g (s) changes from 1 p.u. to 6 p.u.
In summary, since the grid-connected inverter operating in voltage source mode (VSM) is more stable in an extremely weak grid than CSM [19], a novel stability improvement strategy of the multi-inverter system in a weak grid utilizing dual-mode control is proposed: one inverter operating in CSM will be alternated into VSM if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme.

Simulation and Experimental Results
The proposed dual-mode control strategy in a weak grid is confirmed through Matlab/Simulink and an experiment. Figure 16 shows the multi-inverter system with three parallel grid-connected inverters in a weak grid. The system parameters of the simulation as well as the experiments are almost the same and are enumerated in Tables 1 and 2. Energies 2018, 11, x FOR PEER REVIEW 14 of 19

Simulation and Experimental Results
The proposed dual-mode control strategy in a weak grid is confirmed through Matlab/Simulink and an experiment. Figure 16 shows the multi-inverter system with three parallel grid-connected inverters in a weak grid. The system parameters of the simulation as well as the experiments are almost the same and are enumerated in Tables 1 and 2.  Figure 16. The multi-inverter system with three parallel grid-connected inverters in a weak grid.
In Figure 16, the grid impedance is set to Zg(s) = 4 p.u., and the 3# inverter is equipped with the grid impedance dual-mode strategy, so the inverter can adaptively switch between CSM and VSM according to the grid impedance. Moreover, it can be seen from Figure 5 that when the grid impedance is Zg(s) = 3 p.u., the amplitude of the resonance peak is about 6 dB; that is, the harmonic will be amplified about two times. Therefore, for the specific parameters of the inverter in this paper, as shown in Tables 1 and 2, the grid impedance Zg(s) = 3 p.u. is set to the boundary value of CSM and VSM. Based on this value, Figure 17 shows the flow chart of the switching process between CSM and VSM. Notably, there are many kinds of grid impedance identification schemes [23][24][25][26][27], and the method of single harmonic injection [23] is used in this paper. Since the grid impedance identification scheme is not the core work of this paper, it will not be discussed in this paper. In order to avoid chattering between the two modes, a hysteresis switching process as shown in Figure 18 is used in the actual simulation and experiment. As can be seen from Figure 18, the hysteresis center is set to the boundary value Zg(s) = 3 p.u., and the hysteresis width is set to δ, where  Figure 16. The multi-inverter system with three parallel grid-connected inverters in a weak grid.
In Figure 16, the grid impedance is set to Z g (s) = 4 p.u., and the 3# inverter is equipped with the grid impedance dual-mode strategy, so the inverter can adaptively switch between CSM and VSM according to the grid impedance. Moreover, it can be seen from Figure 5 that when the grid impedance is Z g (s) = 3 p.u., the amplitude of the resonance peak is about 6 dB; that is, the harmonic will be amplified about two times. Therefore, for the specific parameters of the inverter in this paper, as shown in Tables 1 and 2, the grid impedance Z g (s) = 3 p.u. is set to the boundary value of CSM and VSM. Based on this value, Figure 17 shows the flow chart of the switching process between CSM and VSM. Notably, there are many kinds of grid impedance identification schemes [23][24][25][26][27], and the method of single harmonic injection [23] is used in this paper. Since the grid impedance identification scheme is not the core work of this paper, it will not be discussed in this paper.

Simulation and Experimental Results
The proposed dual-mode control strategy in a weak grid is confirmed through Matlab/Simulink and an experiment. Figure 16 shows the multi-inverter system with three parallel grid-connected inverters in a weak grid. The system parameters of the simulation as well as the experiments are almost the same and are enumerated in Tables 1 and 2.  Figure 16. The multi-inverter system with three parallel grid-connected inverters in a weak grid.
In Figure 16, the grid impedance is set to Zg(s) = 4 p.u., and the 3# inverter is equipped with the grid impedance dual-mode strategy, so the inverter can adaptively switch between CSM and VSM according to the grid impedance. Moreover, it can be seen from Figure 5 that when the grid impedance is Zg(s) = 3 p.u., the amplitude of the resonance peak is about 6 dB; that is, the harmonic will be amplified about two times. Therefore, for the specific parameters of the inverter in this paper, as shown in Tables 1 and 2, the grid impedance Zg(s) = 3 p.u. is set to the boundary value of CSM and VSM. Based on this value, Figure 17 shows the flow chart of the switching process between CSM and VSM. Notably, there are many kinds of grid impedance identification schemes [23][24][25][26][27], and the method of single harmonic injection [23] is used in this paper. Since the grid impedance identification scheme is not the core work of this paper, it will not be discussed in this paper. In order to avoid chattering between the two modes, a hysteresis switching process as shown in Figure 18 is used in the actual simulation and experiment. As can be seen from Figure 18 In order to avoid chattering between the two modes, a hysteresis switching process as shown in Figure 18 is used in the actual simulation and experiment. As can be seen from Figure 18, the hysteresis center is set to the boundary value Z g (s) = 3 p.u., and the hysteresis width is set to δ, where δ = 5% × 3 p.u. = 0.15 p.u.. At this point, the mode switching process of the grid-connected inverter is described as follows: When the grid becomes stronger to weaker, i.e., the grid impedance Z g (s) > (3 + δ) p.u., the inverter should switch from CSM to VSM; and when the grid becomes weaker to stronger, i.e., the grid impedance Z g (s) < (3 − δ) p.u., the inverter should switch from VSM to CSM.  Figure 18. The hysteresis switching process between CSM and VSM control.

Simulation Verification
The simulation conditions are described as follows: Figure 19a shows the simulation waveform of the A-phase grid current iga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at 0.8 s to the dual-mode control system. Before 0.8 s, all three inverters in the system are running in CSM. It can be found that the current waveform has obvious low-order resonance before 0.8 s, and Figure 19b shows the spectrum of the A-phase grid current iga before 0.8 s. From Figure 19b, the resonance peak is near the 11th harmonics, and this is consistent with the conclusion that when the grid impedance in Figure 5a is Zg(s) = 4 p.u., and the resonant peak frequency is around the 11th harmonics.
After 0.8 s, due to the 3# inverter in the system having a dual-mode adaptive control algorithm based on the grid impedance added, and since Zg(s) = 4 p.u. > 3 p.u., it will adaptively switch from CSM control to VSM at 0.8 s. After 0.8 s, it is found that the output current resonance phenomenon disappears remarkably and the output current quality is high.
In summary, it can be seen that the dual-mode multi-inverter control strategy proposed in this paper can effectively suppress the output current resonance phenomenon caused by the grid impedance in the weak grid and improve the stability of the grid-connected inverter system.

Magnitude(%)
Resonance peak near 11th lower harmonics Figure 18. The hysteresis switching process between CSM and VSM control.

Simulation Verification
The simulation conditions are described as follows: Figure 19a shows the simulation waveform of the A-phase grid current i ga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at 0.8 s to the dual-mode control system.  Figure 18. The hysteresis switching process between CSM and VSM control.

Simulation Verification
The simulation conditions are described as follows: Figure 19a shows the simulation waveform of the A-phase grid current iga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at 0.8 s to the dual-mode control system. Before 0.8 s, all three inverters in the system are running in CSM. It can be found that the current waveform has obvious low-order resonance before 0.8 s, and Figure 19b shows the spectrum of the A-phase grid current iga before 0.8 s. From Figure 19b, the resonance peak is near the 11th harmonics, and this is consistent with the conclusion that when the grid impedance in Figure 5a is Zg(s) = 4 p.u., and the resonant peak frequency is around the 11th harmonics.
After 0.8 s, due to the 3# inverter in the system having a dual-mode adaptive control algorithm based on the grid impedance added, and since Zg(s) = 4 p.u. > 3 p.u., it will adaptively switch from CSM control to VSM at 0.8 s. After 0.8 s, it is found that the output current resonance phenomenon disappears remarkably and the output current quality is high.
In summary, it can be seen that the dual-mode multi-inverter control strategy proposed in this paper can effectively suppress the output current resonance phenomenon caused by the grid impedance in the weak grid and improve the stability of the grid-connected inverter system.

Magnitude(%)
Resonance peak near 11th lower harmonics Figure 19. (a) Simulation waveform of the A-phase grid current i ga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control to the dual-mode control system; (b) spectrum of the A-phase grid current i ga before 0.8 s. THD: total harmonic distortion.
Before 0.8 s, all three inverters in the system are running in CSM. It can be found that the current waveform has obvious low-order resonance before 0.8 s, and Figure 19b shows the spectrum of the A-phase grid current i ga before 0.8 s. From Figure 19b, the resonance peak is near the 11th harmonics, and this is consistent with the conclusion that when the grid impedance in Figure 5a is Z g (s) = 4 p.u., and the resonant peak frequency is around the 11th harmonics.
After 0.8 s, due to the 3# inverter in the system having a dual-mode adaptive control algorithm based on the grid impedance added, and since Z g (s) = 4 p.u. > 3 p.u., it will adaptively switch from CSM control to VSM at 0.8 s. After 0.8 s, it is found that the output current resonance phenomenon disappears remarkably and the output current quality is high.
In summary, it can be seen that the dual-mode multi-inverter control strategy proposed in this paper can effectively suppress the output current resonance phenomenon caused by the grid impedance in the weak grid and improve the stability of the grid-connected inverter system.

Experimental Verification
Experiments were also carried out on an experimental platform consisting of three 100-kW grid-connected inverters to verify the proposed control scheme. The control circuit is implemented on a DSP chip TMS320F28335, which implements the proposed control strategies, as described in the previous sections. In the experiments, a three-phase programmable rectifier is used to represent the DC source. The grid impedance is achieved by a series of practical reactors in a reactor cabinet, and the corresponding grid inductance is 0.25 × 4 mH (that is, Z g (s) = 4 p.u.). The experimental platform is depicted in Figure 20.

Experimental Verification
Experiments were also carried out on an experimental platform consisting of three 100-kW gridconnected inverters to verify the proposed control scheme. The control circuit is implemented on a DSP chip TMS320F28335, which implements the proposed control strategies, as described in the previous sections. In the experiments, a three-phase programmable rectifier is used to represent the DC source. The grid impedance is achieved by a series of practical reactors in a reactor cabinet, and the corresponding grid inductance is 0.25 × 4 mH (that is, Zg(s) = 4 p.u.). The experimental platform is depicted in Figure 20. Corresponding to the simulation, the experimental conditions are described as follows: Figure 21a shows the experimental waveform of the A-phase grid current iga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at time t0 to the dualmode control system.  Corresponding to the simulation, the experimental conditions are described as follows: Figure 21a shows the experimental waveform of the A-phase grid current i ga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at time t 0 to the dual-mode control system.

Experimental Verification
Experiments were also carried out on an experimental platform consisting of three 100-kW gridconnected inverters to verify the proposed control scheme. The control circuit is implemented on a DSP chip TMS320F28335, which implements the proposed control strategies, as described in the previous sections. In the experiments, a three-phase programmable rectifier is used to represent the DC source. The grid impedance is achieved by a series of practical reactors in a reactor cabinet, and the corresponding grid inductance is 0.25 × 4 mH (that is, Zg(s) = 4 p.u.). The experimental platform is depicted in Figure 20. Corresponding to the simulation, the experimental conditions are described as follows: Figure 21a shows the experimental waveform of the A-phase grid current iga of the 1# inverter when the multi-inverter system is converted from the original CSM-only control at time t0 to the dualmode control system. Before t 0 , all three inverters in the system are running in CSM. Similarly to the simulation results, the current waveform has obvious low-order resonance before 0.8 s, and Figure 21b shows the spectrum of the A-phase grid current i ga before t 0 . From Figure 21b, the resonance peak is near the 11th harmonics, and this is consistent with the conclusion that when the grid impedance in Figure 5a is Z g (s) = 4 p.u., the resonant peak frequency is around the 11th harmonics.
After t 0 , due to the 3# inverter in the system having a dual-mode adaptive control algorithm based on the grid impedance added, and since Z g (s) = 4 p.u. > 3 p.u., it will adaptively switch from CSM control to VSM at t 0 . After t 0 , it is found that the output current resonance phenomenon disappears remarkably and the output current quality is high.
It can be concluded from Figure 19 to Figure 21 that the experimental results are in good agreement with the simulation results, and thus the superiority of the proposed control strategy is further verified.

Conclusions
In this paper, a closed-loop transfer function model of the multi-inverter system operating in current source mode (CSM) is established, by which it is concluded that output current resonance will occur with the increase in the grid impedance. In order to address this problem, this paper presents a novel dual-mode control scheme of multiple inverters; i.e., one inverter operating in CSM will be alternated into voltage source mode (VSM) if the grid impedance is high. It is theoretically proved that the coupling between the inverters and the resonance in the output current can be suppressed effectively with the proposed scheme. Finally, the validity of the proposed theory is demonstrated by extensive simulations and experiments.