International Journal of Automation and Computing  2018, Vol. 15 Issue (6): 716-727   PDF    
The Resonance Suppression for Parallel Photovoltaic Grid-connected Inverters in Weak Grid
Qiu-Xia Yang, Kun Li, Cui-Mei Zhao, Hu Wang     
School of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China
Abstract: Obvious resonance peak will be generated when parallel photovoltaic grid-connected inverters are connected to the weak grid with high grid impedance, which seriously affects the stability of grid-connected operation of the photovoltaic system.To overcome the problems mentioned above, the mathematical model of the parallel photovoltaic inverters is established.Several factors including the impact of the reference current of the grid-connected inverter, the grid voltage interference and the current disturbance between the photovoltaic inverters in parallel with the grid-connected inverters are analyzed.The grid impedance and the LCL filter of the photovoltaic inverter system are found to be the key elements which lead to existence of resonance peak.This paper presents the branch voltage and current double feedback suppression method under the premise of not changing the topological structure of the photovoltaic inverter, which effectively handles the resonance peak, weakens the harmonic content of the grid current of the photovoltaic grid-connected inverter and the voltage at the point of common coupling, and improves the stability of the parallel operation of the photovoltaic grid-connected inverters in weak grid.At last, the simulation model is established to verify the reliability of this suppression method.
Key words: Parallel photovoltaic grid-connected inverters     weak grid     grid impedance     resonance peak     branch voltage and current double feedback    
1 Introduction

The development of new energy is becoming increasingly open with the growing shortage of fossil fuels. Among several attractive sources of energy, the solar energy is used more widely as an important renewable energy, besides, the development of photovoltaic power generation is particularly and vigorously pursued[1]. With the increasing maturity of photovoltaic technology and the promulgation of various incentive policies by government, the photovoltaic power generation of China has developed rapidly in recent years[2, 3]. By the end of 2015, the cumulative installed capacity of photovoltaic power generation in China is 43.18 million kilowatts, China has become the country that owns the largest photovoltaic power generation capacity in the world[4]. The construction of photovoltaic photovoltaic (PV) power plants has been listed as key area of national energy development plan in 2010-2020[5].

In recent years, with continuously increasing of the scale of photovoltaic power generation and rapid development of its technology, more and more photovoltaic power stations involve grid-connected parallel inverters[6]. This approach makes full use of solar energy resources, and it also plays a role in meeting the needs of the power grid, such as peak load shaving, supplementing reactive power of grid, etc[7]. However, many photovoltaic power stations are built in remote areas due to the characteristics of distribution of solar energy resources in China[8], far from the power supply area. Consequently, long transmission cables and low power transformers entail high impedance of common coupling point of grid[9]. In such a weak power grid, grid impedance will cause a great impact to stable operation of the grid-connected inverter[10], the situation of parallel inverters connected to the grid is more serious. And, the resonance problem that is generated by the interaction of parallel inverter and weak grid has become a hot research topic both at home and abroad.

Currently, research focuses on controlling the LCL filter of inverter for solving the resonance problem of grid-connected parallel inverters, it can improve the output characteristics of the inverter by controlling the LCL filter, thus avoiding the generation of resonance[11]. In [12], the inverter is shown to be equivalent to an ideal voltage source, it analyzes the resonance characteristics of parallel LCL filter, but the analysis does not consider the impact of control scheme and inverter on the system resonance. He et al.[13] analyze the distribution of resonance frequency in parallel inverter system under the condition of passive damping, however, passive damping consumes active power, which will result in the loss of energy of the grid-connected system. Zhang et al.[14] analyze the small signal equivalent model of multi-inverter parallel system, resonance characteristics are analyzed based on this model, but it does not put forward practical suppression method. Agorreta et al.[15] establish admittance matrix of single-phase parallel inverters system, equivalent circuit model of the system is derived, and the influence of grid impedance on the resonance frequency of LCL filter is analyzed at the same time, but the resonance suppression is not considered.

In this paper, resonance phenomenon and resonance reasons of parallel inverters in the weak grid are analyzed, the analysis is based on the control of the LCL filter of inverter and it combines the characteristics of the inverter itself. In order to solve the resonance problem, this paper proposes the suppression method named branch voltage and current double feedback, its reliability is verified by simulation model at the end of this paper.

2 Resonance analysis of PV grid connected parallel inverter in weak grid 2.1 Mathematical model of parallel inverter in weak grid

Fig. 1 shows the topological structure of the PV grid connected in parallel in the weak grid, where the direct current (DC) voltage $ {U_{{{dc}}}} $ and current $ {i_{{{dc}}}} $ are produced by the photovoltaic cell, and $ {U_{{g}}} $ is grid voltage. LCL filter is constituted by the inverter-side inductance $ {L_1} $, the filter capacitor $ {C_{{f}}} $ and the grid-side inductance $ {L_2} $ after ignoring the parasitic resistances, grid impedance of weak grid is equivalent to inductance $ {L_{{g}}} $, $ {U_{{c}}} $ is the voltage of filter capacitor, $ {i_1} $ is the current of inverter-side, $ {i_{{c}}} $ is the current of filter capacitor, $ {i_2} $ is the current of grid-side, the voltage of common coupling point is $ {U_{{{pcc}}}} $, and $ {i_{{p}}} $ is the injection current of other parallel inverters.

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Fig. 1. Topological structure of the PV grid-connected in parallel in the weak grid

Depending on the topology structure of Fig. 1, and taking the coupling relationship between the major parameters into account, the control structure block diagram can be obtained as shown in Fig. 2. Among them, $ {G_i}(s) $ is the proportional integral (PI) controller, namely $ {G_i}(s) = {K_{{p}}} + \frac{{{K_{{i}}}}}{s} $, $ {G_{{{inv}}}} $ is the amplifying link of the photovoltaic grid-connected inverter, it can be equivalent to $ {G_{{{inv}}}} = {K_{{{pwm}}}} $ (pwm is pulse width modulation), $ {H_1} $ and $ {H_2} $ are the feedback coefficients of the feedback control links respectively.

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Fig. 2. Control block diagram of the PV grid-connected in parallel in the weak grid

The rest of terms are expressed as

$ {G_{{L_1}}}(s) = \frac{1}{{s{L_1}}} \\ {G_{{C_{{f}}}}}(s) = \frac{1}{{s{C_{{f}}}}}\\ {G_{{L_2}}}(s) = \frac{1}{{s{L_2}}}\\ {G_{{L_{{g}}}}}(s) = s{L_{{g}}}. $

As can be seen from Fig. 2, there are three disturbance terms in parallel inverter system in the weak grid, they are respectively: reference current of grid current $ {i_{{{ref}}}} $, grid voltage $ {U_{{g}}} $ and the injection current of other parallel inverters $ {i_{{p}}} $. It is emphasized that, point of common coupling voltage $ {U_{{{pcc}}}} $ and the injection current of other parallel inverters $ {i_{{p}}} $ satisfy the following formula:

$ \begin{equation} {U_{{{pcc}}}}(s) = {U_{{g}}}(s) + s{L_g} [{i_2}(s) + {i_{{p}}}(s)]. \label{eq:1}\end{equation} $ (1)

The transfer functions of these three disturbance terms on the grid-side current $ {i_2} $ are respectively expressed as, $ {G_{{{ref}}}}(s) $, $ {G_{{g}}}(s) $ and $ {G_{{p}}}(s) $:

$ \begin{equation} {G_{{{ref}}}}(s) = \frac{{{i_2}(s)}}{{{i_{{{ref}}}}(s)}} = \frac{{A(s)}}{{1 + C(s)}} \label{eq:2}\end{equation} $ (2)
$ \begin{equation} {G_g}(s) = \frac{{{i_2}(s)}}{{{U_g}(s)}} = \frac{{B(s)}}{{1 + C(s)}} \label{eq:3}\end{equation} $ (3)
$ \begin{equation} {G_{{p}}}(s) = \frac{{{i_2}(s)}}{{{i_{{p}}}(s)}} = \frac{{{G_{{L_{{g}}}}}(s)B(s)}}{{1 + C(s)}} \label{eq:4}\end{equation} $ (4)

where

$ A(s) = {G_{{i}}}(s){G_{{{inv}}}}{G_{{L_1}}}(s){G_{{C_{{f}}}}}(s){G_{{L_2}}}(s) $ (5)
$ \begin{align} B(s) = {G_{{L_2}}}(s) + {G_{{L_2}}}(s){G_{{C_{{f}}}}}(s){G_{{L_1}}}(s) +\notag\\ {G_{{L_2}}}(s){H_1}{G_{{{inv}}}}{G_{{L_1}}}(s) \end{align} $ (6)
$ \begin{align} &C(s) = {H_2}{G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s){G_{{C_{{f}}}}} (s){G_{{L_2}}}(s) +\notag\\ &\qquad\quad~ {H_1}{G_{{{inv}}}}{G_{{L_1}}}(s) + {G_{{C_{{f}}}}}(s){G_{{L_2}}}(s)+ \notag\\ &\qquad\quad~ {G_{{L_1}}}(s){G_{{C_{{f}}}}}(s) + {G_{{L_2}}}(s){G_{{L_{{g}}}}}(s) +\notag\\ &\qquad\quad~ {G_{{L_2}}}(s){G_{{L_{{g}}}}}(s){G_{{L_1}}}(s){G_{{C_{{f}}}}}(s)+ \notag\\ &\qquad\quad~ {G_{{L_2}}}(s){G_{{L_{{g}}}}}(s){H_1}{G_{{{inv}}}}{G_{{L_1}}}(s). \end{align} $ (7)

The bode diagram of these three interference terms can be analyzed as shown in Fig. 3 according to the specific expressions of (2)-(4) and the simulation parameters in Table 1.

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Fig. 3. Bode diagram of the interference terms in parallel-inverter system

Table 1
Simulation parameters

As can be seen from Fig. 3, three resonance points appear in the inverter output current with obvious peak because of the existence of these three interference amounts, and the three resonance points belong to the same resonant frequency.

All the open loop transfer functions (2)-(4) are expressed by $ C(s) $, and the pole-zero distribution of $ C(s) $ also affects the range of resonance frequency, the same open loop transfer function leads to the same resonance, that is why the resonance points of three interference amounts in Fig. 3 are at the same frequency.

2.2 Analysis of causes and influencing factors of resonance

The block diagram as shown in Fig. 2 describes a conventional control method for inverter, namely dual current loop feedback control. As seen in Fig. 3, this conventional inverter control method has not adapted to the case of parallel grid-connected inverter in weak grid. If the control method for inverter has no corresponding improvement, the resonance will be more serious and will affect the stability of the parallel inverter system.

According to (1), grid voltage $ {U_{{g}}} $ and the injection current of other parallel inverters $ {i_{{p}}} $ take resonance interference into inverter system through the grid impedance $ {L_{{g}}} $ in weak grid, and the resonance is also caused by reference current $ {i_{{{ref}}}} $ through LCL filter inside the inverter system. As can be seen from (7), the main coefficients of open loop transfer function are also constituted by the grid impedance $ {L_{{g}}} $ and LCL filter, therefore, theoretically speaking, the parameters of LCL filter and the value of the grid impedance directly affect the resonance distribution of the parallel inverter system.

Fig. 4 shows that how the parameter changing of filter capacitance in LCL filter affects the resonance frequency, the output characteristics of LCL filter remain unchanged during the period. As can be seen, with the increase of the value of filter capacitor, the resonance frequency of the system decreases.

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Fig. 4. Parameter changing of the filter capacitor affects resonance frequency

The values of inverter-side inductance and grid-side inductance in LCL filter also have impact on resonance frequency of the system, the analysis method is the same as the filter capacitor$'$s. Figs. 5 and 6 show that how the parameter changing of inverter-side inductance and grid-side inductance affect the resonance frequency, the output characteristics of LCL filter also remain unchanged during the period. As can be seen, with the increase of the value of inverter-side inductance and grid-side inductance in LCL filter, the resonance frequency of the system also decreases.

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Fig. 5. Parameter changing of the inverter-side inductance affects resonance frequency

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Fig. 6. Parameter changing of the grid-side inductance affects resonance frequency

In the topology structure shown in Fig. 1, when the number of units in parallel inverters increases, the injection current of other parallel inverters $ {i_{{p}}} $ and the grid-side current $ {i_2} $ also increase. When $ N $ identical photovoltaic inverters are connected in parallel, the value of inverter output current $ {i_2} $ will be doubled to $ N{i_2} $, that is equivalent to say that the value of $ {L_{{g}}} $ is increased $ N $-fold if converted to an inverter. Therefore, the number of units in parallel inverters and the value of grid impedance $ {L_{{g}}} $ are equivalent.

Fig. 7 shows that the influence of the number of units in parallel inverter on the resonance frequency and the effect of value changing of the grid impedance on the resonance frequency are also verified as equivalent. As can be seen, with the increase of the number of units in parallel, the resonance frequency of the system decreases.

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Fig. 7. Number of units in parallel affects resonance frequency

Figs. 4-7 confirm that the parameters of LCL filter and the value of the grid impedance directly affect the resonance distribution of the parallel inverter system. In fact, this can be easily explained, the main coefficients of open loop transfer function in (7) are constituted by the grid impedance $ {L_{{g}}} $ and LCL filter, the changing of parameters of the LCL filter and the value of the grid impedance must cause changing of the open-loop transfer function, and the pole-zero distribution of the changing open-loop transfer function changes the range of resonance frequency.

3 Resonance suppression of the PV grid-connected parallel inverter in weak grid

Figs. 4-7 confirm that the grid impedance and the LCL filter of photovoltaic inverter system are the key elements of the cause of the resonance peak, these two factors need to give an important consideration in order to suppress the resonance of the PV grid-connected parallel inverter in weak grid. So, the resonance peak can be suppressed by modifying the relevant parameters of the LCL filter to alter its output characteristic or improving the relevant parameters of the grid impedance.

Improving the impedance parameters of the grid can be realized by installing the resistor and capacitor (RC) damper or adding other passive components, however, this is based on the increased cost of construction and maintenance, which is not conducive to the economical operation of the power system. Theoretically speaking, improving the parameters of grid impedance can not suppress the resonance peak fundamentally, it may produce other factors that affect the stable operation of the inverter system if the result of improving the parameters is not ideal. Therefore, it can not only avoid the resonance that is caused by the interaction with the grid impedance, but also reduce cost of the transformation from the viewpoint of optimizing the control of LCL filter.

In the domestic and foreign research, optimizing the control of LCL filter contains two traditional methods, namely, active damping method and passive damping method. In the following, these two methods will be introduced to discuss their effect on resonance suppression, the topological structure and transfer function of these two methods will be confined to LCL filter for convenience.

3.1 Passive damping method

The most simple and convenient method of optimizing the control of LCL filter is to add resistance in the filter branch to increase the system damping and to keep system stable, namely, passive damping method. In fact, this method changes the output characteristic of the LCL filter to achieve the purpose of optimizing the control of LCL filter. According to the location of the added resistance, the passive damping method can be divided into many kinds of schemes, four of them will be introduced in the next to confine the length of this paper.

Fig. 8 shows one of the four, namely, equivalent circuit of grid-side inductance with a series resistor.

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Fig. 8. Equivalent circuit of grid-side inductance with a series resistor

Its transfer function is expressed as

$ \begin{equation} G_{{L_2}}^{{R_{{{s1}}}}} = \frac{1}{{{s^3}{L_1}{L_2} {C_{{f}}} + {s^2}{L_1}{C_{{f}}}{R_{{{s1}}}} + s({L_1} + {L_2}) + {R_{{{s1}}}}}}. \label{eq:8}\end{equation} $ (8)

The bode diagram of the transfer function when the value of the series resistor is different is shown as Fig. 9. The values of the main parameters are listed in Table 1.

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Fig. 9. Bode diagram of grid-side inductance with a series resistor

The following conclusions can be drawn from Fig. 9.

The system has resonance peak when the damping resistor is not added, and with the value of the grid-side inductance series resistor increasing, the degree of attenuation of the resonance peak becomes greater.

The resistor which is in series with grid-side inductance has no effect on the high frequency attenuation characteristic, but with the value of the series resistor increasing, the low frequency gain of system decreases, which will affect the stability of the system control performance.

When the value of damping resistor is 5 times of the value of the grid-side inductance, the suppression effect of the resonance peak can be relatively ideal, in this case, it will result in a larger loss, therefore, this kind of damping scheme is not generally applicable to engineering practice.

Fig. 10 shows equivalent circuit of grid-side inductance with a parallel resistor.

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Fig. 10. Equivalent circuit of grid-side inductance with a parallel resistor

Its transfer function is expressed as

$ \begin{equation} G_{{L_2}}^{{R_{{{p1}}}}} = \frac{{s{L_2} + {R_{{{p1}}}}}}{{{s^3}{L_1}{L_2}{C_{{f}}}{R_{{{p1}}}} + {s^2}{L_1}{L_2} + s{R_{{{p1}}}}({L_1} + {L_2})}}. \label{eq:9}\end{equation} $ (9)

The bode diagram of the transfer function when the value of the parallel resistor is different is shown as Fig. 11.

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Fig. 11. Bode diagram of grid-side inductance with a parallel resistor

The following conclusions can be drawn from Fig. 11.

The system has resonance peak when the damping resistor is not added, and with the value of the grid-side inductance parallel resistor decreasing, the degree of attenuation of the resonance peak becomes greater.

The resistor which is in parallel with grid-side inductance has no effect on the low frequency attenuation characteristic, but with the value of the parallel resistor decreasing, the high frequency gain of system decreases, which is not conducive to suppress high frequency harmonics.

This kind of damping scheme is not generally applicable to engineering practice because it cannot take into account both the system damping and filtering characteristics.

Fig. 12 shows equivalent circuit of capacitor branch with a parallel resistor.

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Fig. 12. Equivalent circuit of capacitor branch with a parallel resistor

Its transfer function is expressed as

$ \begin{equation} G_{{C_{{f}}}}^{{R_{{{p2}}}}} = \frac{{{R_{{{p2}}}}}}{{{s^3}{L_1}{L_2}{C_{{f}}}{R_{{{p2}}}} + {s^2}{L_1}{L_2} + s{R_{{{p2}}}}({L_1} + {L_2})}}. \label{eq:10}\end{equation} $ (10)

The bode diagram of the transfer function when the value of the parallel resistor is different is shown as Fig. 13.

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Fig. 13. Bode diagram of capacitor branch with a parallel resistor

The following conclusions can be drawn from Fig. 13.

With the value of the damping resistor that is in parallel with the capacitor branch decreasing, the suppression capability of the resonance peak becomes greater.

The resistor which is in parallel with capacitor branch has no effect on the high frequency characteristic or low frequency characteristic.

A smaller damping resistor needs to be used in order to get better damping effect, but at this time the loss will be large, so this kind of damping scheme is also not used generally.

Fig. 14 shows equivalent circuit of capacitor branch with a series resistor.

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Fig. 14. Equivalent circuit of capacitor branch with a series resistor

Its transfer function is expressed as

$ \begin{equation} G_{{C_{{f}}}}^{{R_{{{s2}}}}} = \frac{{s{C_{{f}}}{R_{{{s2}}}} + 1}}{{{s^3}{L_1}{L_2}{C_{{f}}} + {s^2}({L_1} + {L_2}){C_{{f}}}{R_{{{s2}}}} + s({L_1} + {L_2})}}. \label{eq:11}\end{equation} $ (11)

The bode diagram of the transfer function when the value of the series resistor is different is shown as Fig. 15.

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Fig. 15. Bode diagram of capacitor branch with a resistor in series

The following conclusions can be drawn from Fig. 15.

The effect of the resonance suppression can be realized by using the small value of damping resistor that is in series with capacitor branch.

With the value of the series resistor increasing, the attenuation characteristic of the high frequency region of the system will be weakened.

The loss of this kind of passive damping methods is relatively small, and the attenuation effect of the high frequency is not obvious, therefore, the damping resistor that series with capacitor branch is generally applicable to engineering practice to damp the resonance point of the LCL filter.

The passive damping method really can damp the resonance point effectively, but the system loss it brings cannot be ignored. The problem needs to be solved from the angle of control strategy in order to avoid the loss caused by the heating of damping resistor.

3.2 Active damping method

The resonance of the filter can be suppressed and the system can stay stable by using passive damping method, however, the adding of passive damping method may affect the high frequency filtering capabilities, on the other hand, it will increase the loss and lower the efficiency of the system. In order to achieve the same damping effect without increasing the loss, the active damping method can be used to replace the damping resistor by using the additional control algorithm. In the following, several schemes of active damping method will be introduced.

In the active damping method, capacitance current and grid current dual-loop control is commonly used. The control block diagram of it is shown in Fig. 16, it is a conventional control method for inverter and it has many fairly convenient advantages. However, its shortcomings also cannot be ignored: It requires an additional current sensor which increases the cost of the system, the capacitance current has the high frequency component, and the delay which is introduced by the digital control will generate large processing error to high frequency of the system. There are many similar methods such as capacitance current and inverter-side current dual-loop control, they are collectively referred to as multi-state variable combination feedback in active damping method.

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Fig. 16. Control block diagram of capacitance current and grid current dual-loop control

Fig. 17 shows the control block diagram of notch filter based on active damping method. Digital filter is added to the forward path to form a notch filter or a band-stop filter. This method is simple, it needs no modification of the inverter hardware and it does not require additional increase of current sensor.

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Fig. 17. Control block diagram of notch filter based on active damping method

In Fig. 17, open loop transfer function of the system is expressed as

$ G_{{{open}}}^{{{BPF}}} = \frac{{{G_i}(s){G_{{{inv}}}}(1 - {G_{{{BPF}}}}){G_{{L_1}}} (s){G_{{C_{{f}}}}}(s){G_{{L_2}}}(s)}}{{1{\text{ + }} {G_{{C_{{f}}}}}(s)({G_{{L_1}}}(s) + {G_{{L_2}}}(s))}}. $ (12)

where $ (1 - {G_{{{BPF}}}}) $ is equivalent to a notch filter (BPF is band-pass filter).

Transfer function of band-stop filter $ {G_{{{BPF}}}} $ is

$ \begin{equation} {G_{{{BPF}}}} = \frac{{\frac{\omega _0}{Q}}}{{{s^2} + \frac{{\omega _0}s}{Q} + {\omega _0}^2}} \label{eq:12} \end{equation} $ (13)

where $ {\omega _0} $ is center frequency of the band-stop filter, $ Q $ is quality factor of the band-stop filter. The resonance frequency of LCL filter is selected as the center frequency of band-stop filter, which can eliminate the resonance peak. The bode diagram of the addition and removal of the notch filter is shown in Fig. 18, where, $ {G_{{{open}}}} $ is the open loop transfer function of the system before the addition of the notch filter, namely:

$ \begin{equation} {G_{{{open}}}} = \frac{{{G_i}(s){G_{{{inv}}}} {G_{{L_1}}}(s){G_{{C_{{f}}}}}(s){G_{{L_2}}}(s)}}{{1{\text{ + }} {G_{{C_{{f}}}}}(s)({G_{{L_1}}}(s) + {G_{{L_2}}}(s))}}. \label{eq:13}\end{equation} $ (14)
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Fig. 18. Bode diagram of notch filter based on active damping method

As seen from Fig. 18, when the notch filter is not added, the amplitude gain of the amplitude frequency characteristic curve of the system is large at the resonance frequency, which will generate the problem that the system may not be stable. When the notch filter is added, and make center frequency of notch filter the same with the resonance frequency of LCL filter, resonance peak can be almost eliminated. However, most studies show that notch filter based on active damping method is limited in the suppression of high frequency components, and its damping performance is general, these are all restriction factors when considering adopting it.

Fig. 19 shows the block diagram of controlling split capacitor current. The filter capacitor is divided into two parts which are connected in parallel, and then the split capacitor current is sampled to make a feedback term to control the inverter. Thus, the output impedance of the original system is changed, and the control performance has also been improved.

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Fig. 19. Block diagram of controlling split capacitor current

Fig. 20 shows the control block diagram of weighted average current. In this method, the inverter-side current and the grid- side current are added together as feedback quantity according to a certain proportion. This method realizes the reduction of the system order, so it can avoid the problems that are caused by the resonance peak. However, the weighted coefficient is sensitive to the system parameters, and the robustness is poor. The methods in Figs. 19 and 20 are both called model order reduction in active damping method.

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Fig. 20. Control block diagram of weighted average current

3.3 The branch voltage and current double feedback

The analysis results of Section 3.1 have shown that adding damping resistors in series or parallel in filter capacitor branch can reshape the output impedance of the grid connected inverter, which can effectively suppress the resonance peak in the system. However, either in series or in parallel, they all have their own limitations. Similarly, the advantages and disadvantages of each method can be also obtained by analyzing and comparing the passive damping method and the active damping method.

Considering the advantages of all forms above and combining with the grid impedance and the practical engineering application, this paper presents the suppression method named branch voltage and current double feedback.

The topological structure of branch voltage and current double feedback is shown in Fig. 21, for this one branch of filter capacitor, it adds damping resistor $ {R_{{d}}} $ that is connected in series to enhance the capacity of filtering high harmonics of the filter capacitor and adds damping resistor $ {R_{{p}}} $ that is connected in parallel to enhance the capacity of filtering low harmonics of the filter capacitor on the basis of without changing the topology structure of the inverter. In other words, the high order harmonic current in the inverter side flows through the filter capacitor and $ {R_{{d}}} $ branch, and the low order harmonic current flows through the $ {R_{{p}}} $ branch, in this way, the harmonic content of the output current is effectively reduced, thus avoiding the generation of resonance.

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Fig. 21. Topological structure of the branch voltage and current double feedback

With the addition of the damping resistor $ {R_{{d}}} $ and $ {R_{{p}}} $, the current of the filter capacitor branch is no longer just capacitor current, so the capacitor current $ {i_{{c}}} $ is changed to the branch current $ {i_{{{CH}}}} $, similarly, the capacitor voltage $ {U_{{c}}} $ is changed to branch voltage $ {U_{{{CH}}}} $.

Depending on the topology structure of Fig. 21, the control block diagram can be obtained as shown in Fig. 22. The form of $ {G_{{C_{{f}}}}}(s) $ is changed to $ G_{{C_{{f}}}}^*(s) = \frac{1}{{s{C_{{f}}}}} + {R_{{d}}} $ due to adding a damping resistor $ {R_{{d}}} $ that connected in series, meanwhile, a new feedback branch is formed due to adding a damping resistor $ {R_{{p}}} $ that is connected in parallel, in the feedback branch, $ G = \frac{1}{{{R_{{p}}}}} $.

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Fig. 22. Control block diagram of the branch voltage and current double feedback

In the control block diagram as shown in Fig. 22, the feedback of branch current $ {i_{{{CH}}}} $ reflects the effect of damping resistor $ {R_{{d}}} $ that is connected in series, the feedback of branch voltage $ {U_{{{CH}}}} $ reflects the effect of damping resistor $ {R_{{p}}} $ that connected in parallel, thus, resonance suppression method proposed in this paper is called branch voltage and current double feedback.

3.4 The analysis for suppression effect

According to the control block diagram of Fig. 22, the transfer functions of these three disturbance terms (reference current of grid current $ {i_{{{ref}}}} $, grid voltage $ {U_{{g}}} $ and the injection current of other parallel inverters $ {i_{{p}}} $) on the grid-side current $ {i_2} $ can be obtained, they are respectively expressed as, $ {G_{{{Dref}}}}(s) $, $ {G_{{{Dg}}}}(s) $ and $ {G_{{{Dp}}}}(s) $:

$ \begin{equation} {G_{{{Dref}}}}(s) = \frac{{{i_2}(s)}}{{{i_{{{ref}}}}(s)}} = \frac{{{A^*}(s)}}{{1 + {C^*}(s)}} \label{eq:14}\end{equation} $ (15)
$ \begin{equation} {G_{{{Dg}}}}(s) = \frac{{{i_2}(s)}}{{{U_{{g}}}(s)}} = \frac{{{B^*}(s)}}{{1 + {C^*}(s)}} \label{eq:15}\end{equation} $ (16)
$ \begin{equation} {G_{{{Dp}}}}(s) = \frac{{{i_2}(s)}}{{{i_{{p}}}(s)}} = \frac{{{G_{{L_{{g}}}}}(s){B^*}(s)}}{{1 + {C^*}(s)}} \label{eq:16}\end{equation} $ (17)

where

$ \begin{equation} {A^*}(s) = {G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s){G_{{L_2}}}(s) \label{eq:17}\end{equation} $ (18)
$ \begin{align} {B^*}(s) =&{G_{{L_2}}}(s) + {G_{{L_2}}}(s){G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s)+\nonumber \\ & {G_{{L_2}}}(s){H_1}{G_{{{inv}}}}{G_{{L_1}}}(s)+ \nonumber\\ &{G_{{L_2}}}(s)G{G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s) \end{align} $ (19)
$ \begin{align} {C^*}(s) =& {H_2}{G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s) {G_{{L_2}}}(s)+\nonumber \\ &{H_1}{G_{{{inv}}}}{G_{{L_1}}}(s) + G_{{C_{{f}}}}^*(s) {G_{{L_2}}}(s){\text{ + }}D(s) \end{align} $ (20)
$ \begin{align} D(s) =& {G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s) + {G_{{L_2}}}(s){G_{{L_g}}}(s)+\nonumber \\ &G{G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s) +\nonumber\\ &{H_1}{G_{{{inv}}}}{G_{{L_1}}}(s){G_{{L_2}}}(s){G_{{L_g}}}(s)+ \nonumber\\ &{G_{{L_2}}}(s){G_{{L_g}}}(s){G_{{L_1}}}(s)G_{{C_{{f}}}}^*(s)+ \nonumber\\ &{G_{{L_2}}}(s){G_{{L_g}}}(s)G{G_i}(s){G_{{{inv}}}}{G_{{L_1}}}(s) G_{{C_{{f}}}}^*(s). \end{align} $ (21)

What is worth emphasizing is that, it is not difficult to find that a small value of series damping resistor $ {R_{{d}}} $ not only cannot suppress the resonance peak, it will stimulate more serious harmonic resonance. Of course, the suppression effect of the large value of $ {R_{{d}}} $ is obvious, but with the value becoming larger, the attenuation characteristics of the system will become weakend. Similarly, with the value of $ {R_{{p}}} $ becoming smaller, the suppression ability to resonance peak becomes stronger, and when the value of $ {R_{{p}}} $ is too large, it will make the resonance problem becomes even more difficult to control. Therefore, it is particularly important to distribute their values reasonably, after repeated discussions, the values used are shown in Table 1.

The Fig. 23 shows the bode diagram of transfer functions of the interference terms with the suppression method of branch voltage and current double feedback. As can be seen from Fig. 23, the resonance peak is significantly calmed after optimizing the control of the LCL filter, the amplitude frequency characteristics and the phase frequency characteristics of the three curves are greatly improved, and resonance is effectively suppressed, which means, the two forms of passive damping and active damping are combined to achieve a good suppression result.

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Fig. 23. Bode diagram of the interference magnitude under the suppression method

4 Simulation verification

In order to verify the reliability of the resonance suppression method that is presented in this paper, the simulation models are built in Matlab/Simulink according to the control block diagram of Fig. 22. Table 1 shows the main parameters of the simulation models. The simulation analysis focuses on two main parameters of the inverter system: the output current $ {i_2} $ and the point of common coupling voltage $ {U_{{{pcc}}}} $. The harmonic content of these two parameters reflect the effect of the resonance suppression method.

Fig. 24 shows the simulation waveform of $ {i_2} $ and $ {U_{{{pcc}}}} $ before the implementation of the presented suppression method, and Fig. 25 shows the fast fourier transformation (FFT) analysis of the two. It can be seen, the harmonic distortion in waveform of these two parameters is evident, this is consistent with the case in Fig. 3, the total harmonic distortion (THD) of the two are respectively 2.2$\, \%$ and 5.3$\, \%$, among them, the low frequency harmonic content is quite obvious, the high frequency harmonic content is relatively small, but it still cannot be ignored. More significantly, the THD of $ {U_{{{pcc}}}} $ is even more than 5% of the grid connected standard, that is absolutely not allowed.

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Fig. 24. Simulation waveforms before the resonance suppression

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Fig. 25. FFT analysis before the resonance suppression

According to the structure control block diagram in Fig. 22, the implementation of branch voltage and current double feedback can be attained as shown in Fig. 26. First, make the grid voltage $ {U_{{g}}} $ pass through the phase-locked loop and the generator, then the reference current $ {i_{{{ref}}}} $ can be obtained. Next, the branch voltage $ {U_{{{CH}}}} $, the output current of the inverter $ {i_2} $ and the reference current $ {i_{{{ref}}}} $ are involved in the PI control link together, what needs to be explained is that branch voltage $ {U_{{{CH}}}} $ passes through a feedback path which contains the damping resistor $ {R_{{p}}} $. Finally, the output of the PI control link and the branch current $ {i_{{{CH}}}} $ pass through the sinusoidal pulse width modulation (SPWM) generator to generate the gate signal which controls the inverter operation. The ultimate goal of the whole control is to suppress the occurrence of resonance by controlling the output impedance of the inverter.

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Fig. 26. Implementation of branch voltage and current double feedback method

Fig. 27 shows the simulation waveform of $ {i_2} $ and $ {U_{{{pcc}}}} $ after the implementation of the presented suppression method, and Fig. 28 shows the FFT analysis of the two. It can be seen, the harmonic distortion in waveform of these two parameters is almost invisible, the THD of the two are respectively $0.01\, \%$ and $0.1\, \%$, the THD content decreases significantly that compared with Fig. 25, among them, the decline of the low frequency harmonic content is quite obvious, the high frequency harmonic content decreases and maintains a reasonable level, this shows that the proposed control strategy which can enhance the ability to suppress both high frequency harmonics and low frequency harmonics is embodied. Another interesting point to take note of is that the THD of the two is far below the grid connected standards and the waveform is still in good shape. The waveform of Fig. 27 and the FFT analysis of Fig. 28 prove the reliability of the presented resonance peak suppression method.

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Fig. 27. Simulation waveforms after the resonance suppression

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Fig. 28. FFT analysis after the resonance suppression

5 Conclusions

This paper starts from the characteristics of the inverter itself, the resonance suppression of the photovoltaic grid-connected parallel inverter in weak grid is studied, the conclusions drawn are as follows:

1) The resonance peak of the PV grid-connected parallel inverter in weak grid is mainly generated by three interference terms that consist of reference current of grid current, grid voltage and the injection current of other parallel inverters, the resonance frequency that is caused by these three are at the same resonance point.

2) The parameters of LCL filter and the grid impedance are the key factors of changing the resonance frequency. The larger the value of filter capacitor is, the lower the resonance frequency becomes, and with the number of units in parallel increasing, the resonance frequency also becomes lower.

3) This paper presents resonance peak suppression method named branch voltage and current double feedback, under this suppression method, the resonance peak is effectively calmed, the harmonic content of the output current and the point of common coupling voltage drops to a reasonable level, thus avoiding the generation of resonance.

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 61573303) and Natural Science Foundation of Hebei Province (No. E2016203092).

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