Mathematical Model of PWM Grid-Connected Inverter For establishing the mathematical model of three-phase Research on Dual-loop Controlled Grid-connected Inverters on the Basis of LCL Output Filters

Under the premise that the same high-frequency filtering effect is realized, comparing with an L filter, an LCL filter, requiring less total inductance is more suitable for the application in grid-connected devices with low switching frequency and high power, but easier to cause current or voltage resonance and spike, thus further affecting the stability of the system. The design of the parameters not only influences the elimination effect of the ripple at the switching frequency, but also has impact on the operation performance of the devices. This paper establishes the mathematical models of inverter and LCL filter, adopts direct current dual-loop control strategy and analyses the stability, provides design analysis to LCL filter and gives the constraint and restriction conditions for the design of parameters. Moreover, this paper also carries out simulated research by means of Simulink and conducts experiments on the basis of the sample machine established. The result proves the correctness of the control strategy and design analysis herein. DOI: http://dx.doi.org/10.5755/j01.eee.20.1.5589


I. INTRODUCTION
With the extensive application of new-energy distributed power generation, grid connection technology has become an important research subject.
The traditional grid-connected inverter generally uses L filter to eliminate higher harmonics led in by switching device.However, in high-power inverter's applications, to reduce the loss and stress of the switching device, usually lower switching frequency is selected so as to increase the harmonic content in the grid-connected current.A higher inductance is required for the harmonic current to meet the same standard.The growth of the inductance value may cause a series of problems such as over large filter volume, current variation decrease and system performance reduction.With regards to this, substitution of LCL filter for L filter has become a trend in recent years.Comparing with L filter, LCL filter has better effect to eliminate higher harmonics, thus requires less inductance under certain harmonic elimination effect and shows more distinct advantages.In addition, the control technology based on LCL filter has also become one of new search focuses.
This paper establishes the mathematical models of inverter and LCL filter, puts forward direct current dual-loop control Manuscript received May 15, 2013; accepted September 10, 2013.
strategy, provides design analysis to LCL filter and finally proves the correctness of the analysis herein through simulation and experiments.The design boasts high practical value in engineering.

II. ANALYSIS OF SYSTEM STRUCTURE AND MATHEMATICAL MODELS
A. System Structure Figure 1 shows the main circuit topology of three-phase grid-connected inverter.Wherein, iga, igb and igc are the grid currents of the three phases, uga, ugb and ugc are the grid voltages of the three phases, ia, ib and ic are the output currents of inverter, and the forward direction is as shown in the figure; grid inductance and its parasitic resistance are Lt and Rt respectively, inverter inductance and its parasitic resistance are Lf and Rf respectively, Cf is filtering capacitance; ua, ub and uc are neutral-point voltage of inverter bridge, S1~S6 are the power switching transistors of the IGBT, C is filtering capacitance of input DC side, Udc is DC-side voltage, and Lt,Lf and Cf constitute the third-order LCL filter [1].Comparing with L filter, LCL filter has higher filtering capacitance Cf and filtering inductance Lt.The basic principle is as follows: Lt and Cf conducts shunting of the high-frequency ripple contained in Lf current if, and capacitance Cf provides low-impedance path for high-frequency current component generated by switch, which effectively reduces the high-frequency content of current it passing Lt. Therefore, LCL filter can be regarded as capacitance branch Cf connected with Lt branch in parallel, and then connected with Lf in series.it is capacitance branch Cf and Lt branch's shunting to Lf [2]- [6].

B. Mathematical Model of PWM Grid-Connected Inverter
For establishing the mathematical model of three-phase If the voltage and load are three-phase symmetric, namely 0 Based on an overall consideration of (1)-( 5), the below can be obtained:

C. Mathematical Model of LCL Filter
Figure 2 shows the single phase equivalent circuit and its block model of a LCL filter [7].
According to Fig. 2, the transfer function matrix of LCL filter by taking Uk(s) and Ugk(s) as the output quantity, Igk(s), Ik(s) and Uck(s) as state quantity is as follows: where Ugk(s) and Igk(s) are grid voltage and current respectively, Uk(s) and Ik(s) are voltage and current of bridge arm side, and Uck(s) is capacitor voltage.Make (1) be explicit function form, then where:

III. CONTROL STRATEGY AND ITS STABILITY ANALYSIS
Figure 3 shows the block diagram of the principle structure of control strategy.The current feedback is unit feedback.Wherein, K is proportional control link; KPWM is equivalent output gain of relative DC side of the inverter.The inductor current loop of the inverter side in Fig. 3(a) is used to protect the switching transistor and strengthen the system stability.The controller is a proportional controller and the coefficient is K.  .
To get the stability conditions of the system, the close-loop transfer function according to (3) can be as follows The characteristic equation of the close-loop transfer function is: .
According to the Hurwitz stability criterion, the necessary and sufficient conditions of system stability include: all the coefficients in (5) more than zero, and: According to ( 15)- (18), the boundary values of Ki, KP can be obtained.

A. Requirements for parameters design for the filter
Under the premise of saving total inductance materials as far as possible, the LCL filter with optimum performance is designed and the following constraint conditions shall also be satisfied [13]- [15]: To shunt the ripple current component of switch frequency and let the high-frequency component pass through capacitor branch if possible, the condition of Cf Lt X X  shall be met during design, wherein XCf and XLt are impedance values under switch frequency, and take Cf Lt When XCf value is too small, the filtering capacitance value will be high, and the reactive current flowing through the filtering capacitance will be more, as a result the output current of the inverter is increased and the loss is increased accordingly.When XCf value is too big, the filtering capacitance value will be low, and the ripple high-frequency component of capacitor branch will be insufficient, as a result the high-frequency harmonic current component flows into the grid.Moreover, to prevent over low of the power factor of the grid-connected inverter, it is required that the fundamental reactive power absorbed by filtering capacitance is no more than 5 % of the rated active power of the system.Thus where P is the rated output active power of grid-connected inverter;  is the proportion of fundamental reactive power absorbed by filtering capacitance in P; ug is the valid value of grid phase voltage.The impedance drop on LCL filter's inductance is less than 10 % of the rated working voltage of the grid.
The resonance point existent in the LCL filter may lead to oscillation and distortion of grid-connected current, and even affect the inverter's stability.To avoid the influences of the resonance point on the system, it is required to limit the value-taking range of resonance frequency fres Generally, fres is set to be more than 10 times of grid frequency and less than 1/2 of switch frequency where fs is the switch frequency of the inverter, fb is the grid fundamental frequency, and the harmonics in this range is relatively few.

B. Parameters Design for the LCL Filter
The specific parameters are as follows: the rated capacity of the three-phase grid-connected inverter Pn = 40 kVA, grid voltage is ug = 220 V, switch frequency fs = 6 kHz, and voltage of DC side Udc = 800 V [16]- [21].
Design of filtering capacitance.The reactive power absorbed by filtering capacitance Cf is generally less than 5 %.Taking 3 % herein, the below can be obtained Upon calculation, we can get the result Cf = 13.5 μF.
Design of the inductance of inverter side.The design method of the inductance of inverter side Lf can refer to the traditional single L filter.The inductance drop of filter is no more than 10 % of the voltage (r.m.s ) of the grid line, the current ripple is usually limited to 10 %-25 %, take 16 % herein, namely: 16% , 7 where i  is ripple current and n I is rated current.According to the parameters design conditions, the minimum value of Lf is: f L 0.7mH  .
Design of the inductance of grid side.The relationship between the inverter-side inductance and the grid-side inductance is .
The current ripple decay caused by LCL filter is where ( ), ( ) m i h i h is the current harmonic of inverter side and the grid current harmonic of switch frequency respectively.The relationship curve between the ripple current decay of switch frequency and r is as shown in Fig. 4. Take r = 0.6, then the ripple current decay is 5 %, Lt = 1.13 mH.

A. Simulated Analysis of three-phase grid-connected inverter
To demonstrate the correctness and feasibility of the Figure 5 shows the voltage and phaselocked angle waveforms of three-phase power grid.According to the figure, the phaselocked angle ranges from -π to π, between the peak and the valley of the grid voltage and the frequency is 50 Hz.Figure 7 shows the waveforms of three-phase grid voltage and current with full-load three-phase grid-connected LCL filter.Figure 8(a) shows the grid harmonic current spectrum analysis.The current at this time mainly includes fundamental wave and its frequency multiplication component, and switch frequency component.Figure 8(b) shows the inverter-side harmonic current spectrum analysis.The current at this time mainly includes fundamental wave and its frequency multiplication component, switch frequency and its frequency multiplication component.Figure 8(c) shows the harmonic current spectrum analysis of filtering capacitance branch.The current at this time mainly includes switch frequency and its frequency multiplication component.

B. Experimental Research
To further demonstrate the correctness of the analysis in this paper, an LCL filter grid-connected inverter prototype with TMS320 LF2812 as the core processor is established for experimental research.The main experimental parameters are as shown in Table I.
Figure 9 shows the voltage and phase locked angle experimental waveforms of three-phase power grid.Figure 10 shows the experimental waveforms of grid voltage of phase A and three-phase modulation waves.Figure 11 shows the experimental waveforms with full load.Figure 12 shows the sudden load process from no load to full load of the system.According to the figure, there are about 20 power frequency cycles from no load to full load, realizing lazy load.Figure 13 shows the THD curve of grid current with the input power changing.It can be concluded from the figure that, when the grid power is increased, the THD of current is decreased accordingly.Figure 14 shows the system prototype's efficiency curve.From the curve, we can see that the system efficiency is above 93 %, and the efficiency near full load is above 97 %. Figure 15 is a bar chart showing the grid current harmonic analysis and comparison of grid-connected standard, simulated and experimental conditions.It can be concluded from the figure that the system fully satisfies the grid-connected requirements.

VI. CONCLUSIONS
This paper establishes the mathematical models of grid-connected inverter and LCL filter, puts forward direct current dual-loop control strategy, provides design analysis to LCL filter on the basis of the limitation conditions of the design parameters introduced, realizes grid-connected operation of grid current direct control by means of the experimental platform of grid-connected inverter finally and proves the correctness and effectiveness of the parameters design solution based on direct current control strategy.
point of DC side be N, then the switching function from the output points a, b, c to the neutral point of DC side according to (1) and (2) can be obtained as:

Fig. 2 .
Single phase equivalent circuit and its block model of a LCL filter; (a) single phase equivalent circuit, (b) block model.
Figure3shows the block diagram of the principle structure of control strategy.The current feedback is unit feedback.Wherein, K is proportional control link; KPWM is equivalent output gain of relative DC side of the inverter.The inductor current loop of the inverter side in Fig.3(a) is used to protect the switching transistor and strengthen the system stability.The controller is a proportional controller and the coefficient is K. Figure 3(b) is the simplified block diagram of the structure obtained after equivalent conversion.

3 .
Block diagram of direct grid-current controller with dual-loop: (a) Block diagram of the principle structure of control strategy, (b) Block diagram of the equivalent structure of control strategy.In this way, the open-loop transfer function of the system is as follows[8]-[12]

Fig. 4 .Fig. 5 .
Fig. 4. The relationship curve between the ripple current decay of switch frequency and r.

Figure 6 Fig. 7 .
Figure 6 shows the waveforms of grid voltage of phase A and modulation waves CMPR1, CMPR2 and CMPR3.According to the figure, the modulation waveforms are saddle type, which can effectively improve the utilization rate of SC side voltage.The voltage frequency of CMPR1 is the same as that of the neutral point of phase A bridge arm, but the phases are opposite.

Fig. 8 .
Current harmonic analysis; (a) grid current harmonic analysis, (b) inverter current harmonic analysis, (c) current harmonic analysis of filtering capacitance branch.

Fig. 13 .Fig. 14 .
Fig. 13.The THD curve of grid current with the input power changing.

TABLE I .
PARAMETER AND ITS VALUE.