Analysis and design of photovoltaic three-phase grid-connected inverter using passivity-based control

Received Sep 26, 2021 Revised Jan 19, 2022 Accepted Jan 26, 2022 This paper presents photovoltaic three-phase grid-connected inverter with an inductor-capacitor-inductor (LCL)-filter. For robustness against variation of filter parameters and external disturbance, the passivity-based control (PBC) method has been adopted. In this method, there are two interactively coupled feedforward terms and three damping gains in the control loops which are designed to limit the steady state error of grid current. Boost converter with P&O maximum power point tracker (MPPT) is used for each photovoltaic (PV) string to extract maximum power and to raise the PV voltage to a value suitable for the grid-connected inverter. The outputs of all boost converters are connected in parallel and controlled to fixed reference voltage using proportional-integral (PI) controller to make the direct-current (DC) link voltage robust against variations in sun radiation intensity and system parameters change. The suggested system is analyzed, designed and simulated using PSIM program. 1 kW, 2 kW, and 3 kW PV systems connected to grid of 220 V/50 Hz are tested and the results show the validity of the suggested grid-connected PV systems and robustness against filter parameters variation.


INTRODUCTION
Nowadays, the renewable energy and relative technologies have been significant attention. The more important link between the renewable energy and grid is the grid connected inverter (GCI) with suitable filter to reduce the current harmonics, such as inductor (L), inductor-capacitor (LC), and inductor-capacitorinductor (LCL) filters. The LCL filter has better performance of harmonic attenuation and lower cost compared with other types [1]− [3]. To enhance the performance of GCI with filter, many control strategies are used such as proportional resonance (PR) [4] and proportional integral (PI) controllers with passive damping method [5], [6], active damping method [7]− [10] and hybrid damping method [11], [12]. The damping methods are used to enhance the performance of PR and PI controllers by reduction the resonance effect, in another hand, these methods add extra cost like extra power in passive damping method and extra number of sensors in active damping method. Another non-linear methods are used to enhance the performance of GCI with filter, such as predictive control [13], [14], deadbeat control [15], [16] slide model control [17], [18], adaptive control [19], [20], and passivity based control with wind system [21], railway systems [22], [23], energy storage systems [24], [25], islanded AC microgrid [26] and GCI systems [27]− [34]. The passivity-based control (PBC) strategy has simple modeling process and better performance against parameters variations with different types of filters. The GCI system is coupled with L filter [27], [28] and LC filter [29], [30] where the discrete root locus with unit step response and traditional analysis method like PI parameters design are used, there is one feedforward in control loop and one damping gain of PBC controller which is selected depending on attenuation of delay influence on the inverter, the feedforward term has no effect on stability of the system so it can be neglected. The GCI system is also coupled with the LCL filter [31]− [34] where there are two feedforwards in control loop and three damping gains, the GCI system is used with fixed DC supply voltage. In this paper a PV system (demonstrated in section III-B) is used to supply grid through GCI with LCL filter and PBC strategy is used as shown in Figure 1, where the additional DC voltage loop is used to enhance the performance of GCI system. The rest of this paper is organized as follows: the description of proposed system and mathematical model of LCL filter and PBC controller are introduced in section 2. Design of proposed system is presented in section 3. Many tests of changes in light intensity, filter parameters, and grid voltage are applied to check the robustness of the proposed system in section 4, and the conclusion is summaries in section 5.

MATHEMATICAL MODELING OF LCL FILTERED THREE PHASE GCI WITH PBC 2.1. The mathematical model of LCL filtered system
The mathematical model of LCL filtered system is: Where; L1 is the inverter side inductance, R1 is inverter side parastic resistance, uc is capacitance voltage, u is inverter voltage, C is filter capacitance, L2 is grid side inductance, R2 is grid side parasitic resistance, i1 is inverter current, i2 is grid current, vpcc is grid voltage, and k=a,b,c.
To get better control performance, abc-to-dq transformation is applied to (1), the new model is: According to the theory of passivity, the LCL filtered system is strictly passive, and the controller can be designed by applying PBC [33].

PBC control
The reference state variables are defined as: While the error vector is defined as xe=x * -x, the Euler Lagrange equation can be rewritten as: To accelerate the convergence speed, a damping matrix kd is added to error system, where: Where; k1, k2>0 and k0 can be replaced by PI controller to enhance the steady state response. Substitution (6) into (5) to get: If xe is zero, the left side of (7) is zero too, the control law can be written in details as: Where Vdc is the inverter DC-link voltage, Vref is the reference voltage and kpv, kiv are the proportional integral gains. The diagram of equivalent system using PBC with inverter DC-link voltage control in (8) and (9) is implemented as shown in Figure 2.

Figure 2. Implementation of PBC and inverter DC-link voltage control
The damping gains k1 and k2 can be calculated as explained in [33] by: where; C is the filter capacitance, L1 is the inverter side inductance, Ts is the switching frequency, and ζ is the damping ratio which is 1 √2 [33].

PROPOSED SYSTEM DESIGN 3.1. LCL filter
The power conversion of PV system (boost converter) and inverter injects harmonics into grid which affect the system operation and reduce the overall power factor. LCL filter is used in the proposed system to eliminate the harmonics effect [35]. The LCL filter is shown in Figure 3, and the limits of it is parameters can be calculated by (12).
Where; L1 is the inverter side inductance, VDC is the boosted DC voltage, ∆ 1 is the maximum inverter current ripple which is approximately 20% of grid current, and Fsw is the switching frequency [36]. Where; C is the filter capacitance, I2 is the grid current, Vac is the grid voltage, and Fac is the grid frequency.
Where; L2 is the grid side inductance [37]. A grid inductance Lg is added to L2 and it's value is uncertain, so it is choice practically [38]. According to (12), (13), and (14) Figure 3. LCL filter design

PV system
A PV string of PSIM library consisting of 4 series connected PV panels, it is generated 1000 W of power at 1000 W/m 2 of light intensity. The PV system contains many of PV strings connected in parallel (through boost converter with MPPT for each string) according to the required power, many changes are occurred to the PV system to check the robust of the proposed system of this paper, firstly, the PV system works with different number of PV strings, secondly, the PV system works with different values of light intensity. The low output voltage of PV string is boosted to grid conjunction, a DC-to-DC converter is necessary to use between the PV string and the inverter [35]. in this paper a boost converter with maximum power point tracker (MPPT) is used and the output voltage of boost converter for each string is regulated to 720 V using PI controller. The PV system consisting of PV string and DC-to-DC boost converter is shown in Figure 4. The PV system provides DC power, a DC-to-AC inverter is used to convert the DC power to three phase AC power to supply the grid. The sinusoidal pulse width modulation (SPWM) method is used to control the switches of inverter with carrier frequency of 10000 Hz.

RESULTS
The complete proposed system shown in Figure 1 is simulated, the PSIM program is used for simulation with parameters as explained in Table 1. This system is tested with different PV power (1000 W, 2000 W, and 3000 W) by adding PV systems in parallel. For 1 PV string (4 PV panels connected in series), the PV supplied power is 1000 W, the grid current of 2 A with THD% of 4.07% and power factor of 0.998. The three phase grid current is shown in Figure 5 (a), three phase output voltage is shown in Figure 5 (b), and output voltage and grid current of phase a are shown in Figure 5 (c). The output DC current and voltage of boost converter (input to the inverter) of 720V is shown in Figure 6 (a) and boost voltage is shown in Figure  6 (b). The reference and actual Iq currents and Id current are shown in Figure 7. For 2 PV strings with boost converter for each string (connected in parallel), the PV supplied power is 2000 W, the grid current of 4 A with THD% of 1.85% and power factor of 0.999. The output voltage and grid current of phase a are shown in Figure 8.   For 3 PV strings with boost converter for each string (connected in parallel), the PV supplied power is 3000 W, the grid current of 6A with THD% of 1.38% and power factor of 0.999. The output voltage and current of phase a are shown in Figure 9. The response of the systems above can be summarized as shown in Table 2. To check the robust of the proposed system, there are three tests are applied, the first test is to check the efficiency of the system during weather changes which affect the light intensity, the second test is reduction the inductances of LCL filter, and the last test is to check the system operation with variable grid voltage. A PV system with 2 PV strings is tested in the first test, the light intensity is increased from 173 800 W/m 2 to 1000 W/m 2 which provides 1600 W PV power (3.2 A grid current) then increased to 2000 W PV power (4 A grid current) at time (10 seconds), the Iq reference of this system is shown in Figure 10 with overshoot of 15% and settled after 1.9 seconds. Boost voltage is shown in Figure 11 with overshoot of 0.48%. The grid current of phase a is shown in Figure 12.   Figure 10. The dynamic response of Iq reference current at increasing light intensity Figure 11. The dynamic response of boost voltage at increasing light intensity Figure 12. The dynamic response of grid current of phase a at increasing light intensity The same system is tested again with light intensity of 1000 W/m 2 then decreased to 600 W/m 2 which provides 2000 W PV power (4 A grid current) then decreased to 1200 W PV power (2.4 A grid current) at time (14 seconds), the Iq reference of this system is shown in Figure 13 with downshoot of 47% and settled after 2.5 seconds. Boost voltage is shown in Figure 14, the downshoot is 1.25%. The grid current of phase a is shown in Figure 15. The response of the system after applied first test can be summaries as shown in Table 3.
A PV system with 2 PV strings and light intensity of 1000 W/m 2 is tested in this second test, the grid side inductance is reduced from 15 mH to 10 mH, the THD% of grid current is 3% and power factor of 0.999. The output voltage and current of phase a are shown in Figure 16. The same system is tested again with grid side inductance reduced to 6 mH, the THD% of grid current is 4.7% and power factor of 0.998. The output voltage and current of phase a are shown in Figure 17. The same system is tested with inverter side inductance reduced from 2 mH to 1.5 mH, the THD% of grid current is 2.87% and power factor of 0.998. The output voltage and current of phase a are shown in Figure 18. The response of the system under second test can be summarized as shown in Table 4.    In the third test of a PV system with 2 PV strings, the grid voltage is reduced to 200 V, the PV power is 2000 W, grid current is 4.4 A with THD% of 2.15% and power factor of 0.999. The output voltage and grid current of phase a are shown in Figure 19. The same system is tested again, the grid voltage is increased to 240 V, the PV power is 2000 W, Grid current is 3.4 A with THD% of 2.24% and power factor of 0.999. The output voltage and grid current of phase a are shown in Figure 20. The response of the system under third test can be summarized as shown in Table 5.

CONCLUSION
In this paper, three-phase photovoltaic grid-connected inverter with an LCL-filter is analyzed and designed. PBC method is suggested in order to find systematic strategy to design the damping factors and to make the system robust against system parameters variation. Based on simulation results, the following aspects can be concluded: i) stable DC-link voltage of inverter against variation in sun light intensity and system parameters variation due to the action of DC voltage loop and PBC controller; ii) the designed PBC controller can maintain the system stable even the parameters of the LCL-filter vary; iii) the system stable against the variation of grid voltage, this due to the effectiveness of PBC controller; iv) the total harmonic distortion of grid current is less than 5% for all tests of proposed grid-connected PV system due to the effectiveness of the LCL-filter and PBC controller; v) the system efficiency is approached to 93.24 % and it is calculated by division the three phase output power on the input PV power. ISSN: 2088-8694 