A novel method for designing the controller of a lcl-Filter-based grid-connected inverter with considering varying system parameters - Nguyen Trung Nhan

Journal of Scıence and Technology 54 (3A) (2016) 39-51 A NOVEL METHOD FOR DESIGNING THE CONTROLLER OF A LCL-FILTER-BASED GRID-CONNECTED INVERTER WITH CONSIDERING VARYING SYSTEM PARAMETERS Nguyen Trung Nhan 1, * , Nguyen Thi Hanh 2 1 Faculty of Electrical Engineering, Industrial University of Ho Chi Minh City, 12 Nguyen Van Bao St, Ward 4, Go Vap Dist, Ho Chi Minh City, Viet Nam 2 Faculty of Information Technology, Industrial University of Ho Chi Minh City, 12 Nguyen Van Bao St, Ward 4, Go Vap Dist, Ho Chi Minh City, Viet Nam * Email: nguyentrungnhan@iuh.edu.vn Received: 15 June 2015; Accepted for publication: 26 July 2016 ABSTRACT Exactly determining the control coefficients for the controller of a three-phase LCL-filter- based inverter is an important and challenging issue in microgrid systems. However, existing LCL-filter-based inverter systems usually assume that all system parameters are determined accurately and remain constant over time, which is not true in real situations. Variations in the system parameters are known to possibly seriously degrade the performance of LCL-filter-based inverter systems. For efficiency and robustness, this paper proposes a novel method for the generalized controller design of a three-phase LCL-filter-based grid-connected inverter system that can address deviations in system parameters. An optimum way to determine the stability bounds under various system parameters cases is introduced. The assessment of the stability bounds is based on the Routh criterion by solving the characteristic equation of the closed-loop control system. Simulations results are presented to validate the correctness and effectiveness of the proposed design method. Keywords: LCL filter, stability criterion, control quality, renewable energy resources. 1. INTRODUCTION In recent years, penetration of renewable energy resources-based distributed generations (DGs) into the power grid is increasing worldwide at a significant rate. This is an inevitable development because of issues of fossil fuels, energy crisis, as well as environmental pollution and warming due to the greenhouse effect is becoming more and more serious, and it is concerned from countries in the world. As a result, renewable energy-based DGs have been rapidly developed. In distributed systems, the presence of DGs has formed a network of distributed generations and they are referred to as the microgid [1]. The high penetration of DGs in traditional power systems is one of the essential factors that could help to solve the energy crisis and improve power supply reliability. However, the involvement of DGs in the main grid Nguyen Trung Nhan, Nguyen Thi Hanh 40 has caused problems, such as harmonic distortion and spinning reserve, especially in the stand- alone mode of the microgrid [2 - 3]. One of the most important approaches to address these problems is the improvement of control quality and the reliability of converters used in DGs systems. To this end, the voltage source inverter (VSI) is the most important module of the converters used in renewable-energy-based DGs [4]. The VSIs acts as a DC/AC converter, which convert a DC voltage to an AC voltage with the grid frequency in order to inject the active and reactive power to the main grid. To improve the injected-current and voltage quality of the VSI, a low-pass filter is required to filter out the harmonic elements. In general, L-, LC-, and LCL-filter are three types of filters that are commonly used to suppress the harmonic elements of the current injected into the main grid. In three types of filters, the LCL-filter has received much attention because it can improve the performance over other types of filters. However, the VSI incorporating LCL-filter is a three-order system. If the controller is inappropriately designed, the system may become unstable due to the resonant peak and the presence of zero impedance to the harmonics at the resonant frequency from the VSI or the main grid [5]. Many control strategies have been proposed to dampen the resonance, and they can be divided into two main strategies: active damping and passive damping. In [6], a passive resistor was added to the LCL-filter to dampen the resonance. This method is a simple to implement and often used for industrial applications. However, a major drawback of this method is the production of extra loss. To decrease the losses in passive damping, a new method for determining the optimized passive damping resistor of LCL and LLCL filters has been proposed in [7]. However, the losses persist here, which constitutes a significant challenge for widely using the VSI with an LCL-filter in practice. Compared with the passive methods, the active damping methods are more commonly used to deal with the resonance due to the LCL-filter. In [8], a control method based on the virtual flux concept has been proposed. In this method, the capacitor currents are estimated from the capacitor voltage by a virtual flux model. Hence, this method can prevent phase lag and the distortion of the remaining high-frequency components. In [9], a resonance compensation method based on adding some compensation terms into the controller of the VSI in the PQR- frame has been proposed. In this method, the AC components of the input current are mitigated to reduce the harmonic components due to the resonance. Usually, a controller of the LCL-filter- based grid-connected inverter requires two feedback signals, the grid-side current and capacitor (or inverter-side) current. To decrease the number of sensors, a control strategy based on the splitting the capacitor of the LCL-filter was proposed in [10]. In this strategy, two capacitors are placed in parallel at the position of the capacitor in the LCL-filter, and the current between the two capacitors is measured as the feedback signal. In [11], the authors analyzed the generalized stability of a grid-connected VSI with an LCL-filter and proposed a control strategy to improve the transient and steady-state performance called composite nonlinear feedback. In [12-13], the authors analyzed and evaluated the applicability of each part of the overall control in a weak grid with the use of a stability criterion. As outlined above, studies that aimed to improve the effectiveness and accuracy of the VSI with an LCL-filter have been implemented and have satisfied our expectations. In general, the controller of a VSI with an LCL-filters issued for fixed system parameters (i.e., the resistances, inductances, capacitors are constants). However, the responses of the control system depend on the system parameters in real control systems. The studies in [14-15] indicated that the stability and injected-current quality of DGs strongly depend on the system parameters. Unfortunately, these system parameters cannot be accurately determined because their values vary with the environment temperature, operating time, and equipment quality. Thus, the quality, stability, and A novel method for designing the controller of a lcl-filter-based grid-connected inverter with 41 target of control systems will not be satisfied during operation. To address this issue, several control methods for DGs have been proposed based on LCL-filter by considering uncertain system parameters. A controller that could reject the uncertain parameters to eliminate the impact of the variation of main grid parameters was proposed in [16-17]. However, its implementation is difficult because its design requires an observation method and a convergence study. To improve the stability of the controller of the VSI with an LCL-filter based on increasing the bandwidth of the proportional-resonant controller, a new control method called weighted average current control was proposed in [18]. In this method, the feedback current is multiplied by an average factor that is determined from system parameters. In references [19 - 21], a method based on full-feed forward functions was proposed to improve the control quality and stability of the control system. In [22 - 23], a new method based on feedback the capacitor current with reduced computation delay for improving robustness of the controller of LCL filter- based grid-connected inverter was proposed. In [24 - 25], to mitigate the impact of the grid voltage disturbance and grid impedance change, a method based on direct grid current control was proposed. The studies in these papers have indicated that the stability and control quality directly relates to the number of feedback signals of the VSI with an LCL-filter controller if the control coefficients are reasonably designed. In these works, the variations of the system parameters are also mentioned in the design of controller. However, these papers do not provide a specific method to determine the precise control parameters. To bridge this gap, a robust controller design method used in VSI with an LCL-filter has been proposed in [26]. In these works, the coefficients of controller are designed based on the root-locus of the transfer function while considering the system parameter variation. However, the changes in the coefficients of controller and system parameters are considered independently of each other. In this paper, a novel method for designing the controller of a three phase LCL-filter-based grid-connected inverter is proposed to bridge the aforementioned gaps. In the proposed method, the coefficients of the controller are generally determined based on the system parameters and the variation of these parameters. The proposed method guarantees the robustness and stability of the controller of the three-phase LCL-filter-based grid-connected inverter for the system performance, irrespective of the change in the system parameters. The rest of the paper is organized as follows. The topology and mathematical model of the three-phase LCL-filter-based grid-connected inverter are presented in Section 2. In Section 3, the multi-loop controller for the three phase LCL filter-based grid-connected inverter is presented. Section 4 will analyze the change of control coefficient bounds versus the change of system parameters. The generalized controller design method and its implementation for improving the control quality of three phase LCL filter-based gird-connected inverter system are presented in Section 5. Section 6 shows the simulation results of the proposed method. Finally, the conclusion is presented in Section 7. 2. MATHEMATICAL MODEL OF THREE PHASE LCL-FILTER-BASED GRID- CONNECTED INVERTER The general topology of the renewable-energy-based distributed generator used in the paper comprises a renewable energy resource, a DC-DC (AC-DC) converter, and a standard three- phase VSI connected to the main grid through an LCL-filter, as shown in Figure 1. Where, L1a, L1b, and L1c are the inverter-side inductors, respectively; R1a, R1b, and R1c are the internal resistances of the inductances L1a, L1b, and L1c, respectively; R2a, R2b, and R2c are the internal resistances of the grid-side inductances L2a, L2b, and L2c, respectively; Ca, Cb, and Cc are the filter Nguyen Trung Nhan, Nguyen Thi Hanh 42 capacitors. Kirchhoff’s laws are applied to the Figure 1 as follows: Figure 1. Topology of the three-phase LCL-filter-based grid-connected inverter. 1 1 inv inv C inv di v v L R i dt (1) 2 2 g C g g di v v L R i dt (2) C inv g dv i C i dt (3) Where, ξ={a,b,c}. The system parameters can be assumed that to be in balance without loss of generality (i.e., L1ξ = L1, R1ξ = R1; L2ξ = L2, R2ξ = R2; and Cξ = C). No zero-sequence injected grid current was also assumed for the three-wire three-phase grid-connected inverter. Therefore, the state-space equations (1)-(3) can be written in the stationary α−β frame as follows: 1 1 1 C ginv inv v vdi R i dt L L (4) 1 1 1 C ginv inv v vdi R i dt L L (5) 2 2 2 g g C g di v vR i dt L L (6) 2 2 2 g g C g di v vR i dt L L (7) Cinv g dv i C i dt (8) C inv g dv i C i dt (9) 3. MULTI-LOOP CONTROLLER FOR LCL-FILTER-BASED GRID-CONNECTED INVERTER As mentioned in Section 1, many methods have been proposed to address the controller design issue of the LCL-filter-based grid-connected inverter. To this end, the active damping based on multi-loop control is most commonly used because of its strong points. The multi-loop A novel method for designing the controller of a lcl-filter-based grid-connected inverter with 43 control method proposed in [21] was adopted in this study. The control block of the three phase LCL-filter-based grid-connected inverter based on the multi-loop controller for the stationary α−β frame is shown in Figure 2. In this control block, the filter-capacitor current is used as the inner loop signal through a control gain (Kc1) and grid current tracked as the outer loop signal through a control gain (Kc2). Note that the control loops in this case act as full-feed forward functions [21]. In Figure 2, Gc(s) is the main controller of the system. This controller might be a proportional-integral (PI) controller, a proportional resonant (PR) controller, harmonic compensation controller, or deadbeat controller. In this paper, a PI controller will be considered. Ginv(s) is the transfer function of VSI. Normally, the switching frequency is much high than the fundamental frequency of the power grid. Therefore, with the three-phase pulse width modulator based on sine-triangle, the transfer function of VSI is a constant and can be approximately determined from the DC input voltage and the amplitude of the triangle carrier. Figure 2. Control block of the three-phase LCL-filter-based grid-connected inverter in the stationary α−β frame. In Figure 2, gi and gi represents the output currents of the multi-loop controller; * gi and * gi represents the reference currents in the stationary α−β frame. From Figure 2, the open-loop transfer function of the system is obtained as follows: 0 ( ). ( ) ( ) ( ) c iG s G sG s H s (10) where H(s) can be expressed in (11) as follows: 3 2 1 2 2 c1 i 1 2 1 2 1 2 2 c1 i 1 2 1 2 H(s) = L L Cs + {L CK G (s) + R L C + L CR }s + + {L + L +R CK G (s) + R R C}s + (R + R ) (11) From (10), the closed-loop transfer function of the system is also obtained as follows: 2 ( ). ( ) ( ) ( ) . ( ). ( ) c i c c i G s G s G s H s K G s G s (12) In this paper, the main controller is a PI controller formed as (13); therefore, the closed-loop transfer function (12) can be expressed in (14) as follows: ( ) ic p K G s K s (13) 4 3 2 4 3 2 1 0 ( ). ( ) ( ) p i iK s K G s G s b s b s b s b s b (14) Nguyen Trung Nhan, Nguyen Thi Hanh 44 where, Kp, Ki are the proportional and integral gains of the PI controller, respectively. The coefficients of the polynomial denominator in (14) were determined as follows: 4 1 2 3 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 0 2 ; ( ) ( ) ; ( ); ( ) ( ) c i c i p i c i i b L L C b L CK G s R L C L CR b L L CR K G s R R C b R R K G s b K K G s (15) Normally, the root locus method is a very common technique used to determine the control coefficients. In this technique, the root locus is plotted as a function of one of the coefficients while all of the other coefficients are fixed. This process is undertaken for all of the control coefficients, and the bounds of coefficients are determined. However, in the multi-loop control system, the integral gain (Ki) and the inner-loop control gain (Kc1) would not strongly affect the system stability (not strongly affect the location of the dominated poles), and they are often determined according to the steady-state error requirement. Contrary to the Ki and Kc1, the changes in the proportional gain (Kp) and outer-loop control gain (Kc2) significantly affect the system stability. Specifically, the proportional gain (Kp) affects not only the system stability but also the steady-state error [27-28]. Therefore, the selection of suitable Kp and Kc2 values is a very important problem in multi-loop control system design. 4. ANALYSIS OF THE CHANGE OF CONTROL COEFFICIENT BOUNDS VERSUS THE CHANGE OF SYSTEM PARAMETERS The quality of the control system depends on control coefficients which can change according to the system parameters. Hence, the changes in the system parameters are well known to be able to seriously degrade the performance and robustness of a VSI with LCL-filter systems. However, system parameters are not constants in real-world situations; they may change over time due to environmental conditions, operating times, the quality of equipment, etc., and may not be accurately determined. This drawback may prevent high quality control and stability in real operations. In this section, we analyze the dependence of the system performance on the system parameters. To validate the change of the system performance on the system parameters, we created a random data set that includes 98 subsets of the system parameters, respectively. In this study, we assumed that the change in system parameters is limited to ± 50% of the initial values (i.e., 0.5{Ri0, Li0, C0} ≤ {Rik, Lik, Ck} ≤ 1.5{Ri0, Li0, C0}), where i = {1, 2}; Ri0, Li0, and C0 are the initial values of the system parameters, as shown in Table 1; Rik, Lik, and Ck are the values of the system parameters at the k th random point (rp). Table 1. Parameters of the system. AC grid line-line voltage Vs = 380 V(rms) AC grid frequency f = 50 Hz Inverter-side impedance L1 = 5.5 mH, R1 = 0.2 Ω Grid-side impedance L2 = 3.5 mH, R2 = 0.2 Ω Filter capacitance C = 20 μF Gi(s) 40 Switching frequency 10 kHz Vdc 400 (V) A novel method for designing the controller of a lcl-filter-based grid-connected inverter with 45 Table 2. The change of dominated poles (DPs) versus the change of system parameters. rp DPs rp DPs rp DPs rp DPs rp DPs rp DPs rp DPs 1 -31.550±5511i 15 56.920±4546i 29 121.62±5350i 43 39.110±3904i 57 43.040±4700i 71 -14.840±4075i 85 -520.62±5964i 2 -443.00±6400i 16 -278.81±6741i 30 32.920±4526i 44 -159.64±4773i 58 25.960±3840i 72 -319.35±5335i 86 -287.25±4697i 3 58.080±5661i 17 -64.040±5856i 31 66.190±7459i 45 -111.50±5991i 59 -43.530±3763i 73 77.900±4458i 87 7.1700±4703i 4 44.690±6142i 18 -16.070±3968i 32 -443.47±4118i 46 -197.60±4026i 60 -143.16±5683i 74 42.010±4128i 88 -217.74±4721i 5 75.780±5049i 19 32.380±4527i 33 -86.450±4998i 47 -330.51±4876i 61 -1.7300±4694i 75 6.8600±5255i 89 -26.930±5156i 6 -151.58±5346i 20 -329.26±7022i 34 118.44±6632i 48 83.520±6258i 62 6.0400±3769i 76 119.57±6661i 90 24.360±5075i 7 -170.17±4960i 21 33.480±3767i 35 -298.96±6224i 49 -136.57±5704i 63 31.240±3968i 77 44.570±4477i 91 -410.52±5267i 8 -210.46±5259i 22 46.070±6170i 36 45.830±6525i 50 -38.920±4265i 64 -247.08±3949i 78 -6.8700±4749i 92 -355.79±5457i 9 -44.010±6166i 23 103.72±4683i 37 -565.20±4232i 51 3.0300±5866i 65 -47.750±4015i 79 -258.44±3804i 93 -51.840±7653i 10 114.69±4998i 24 -594.16±5022i 38 -20.190±3662i 52 -318.27±5077i 66 -132.90±4031i 80 -373.03±4209i 94 -3.5300±5905i 11 -65.970±4742i 25 69.830±5072i 39 -95.470±5159i 53 102.82±5902i 67 -126.30±4287i 81 100.69±5435i 95 -55.570±4547i 12 -77.850±5225i 26 -295.38±6391i 40 -333.51±4487i 54 -318.10±5952i 68 -51.330±4285i 82 -1.0800±3594i 96 -395.12±4963i 13 9.3700±6309i 27 -53.900±4018i 41 -116.45±4569i 55 -194.08±4303i 69 -128.43±5085i 83 -356.74±4815i 97 -246.25±5255i 14 -92.120±4340i 28 -15.420±3923i 42 -26.100±4558i 56 2.0500±4929i 70 -49.770±5292i 84 -256.26±6289i 98 -43.180±5833i The assessment is based on the Routh criterion by solving the characteristic equation of the closed-loop control system. The characteristic equation of the controller system is the closed- loop transfer function denominator polynomial of the system. According to the Routh criterion, a system is stable if and only if all of the roots of the characteristic equation have negative real parts (i.e., all of the roots are in the left-half of the complex-plane). Based on this criterion, we determined the value of the dominated points (DPs) for each value of the system parameters by resolving the characteristic equation of the closed-loop control system. The change of DPs versus the change of system parameters obtained for this case (the value of Kp is determined for the initial values of the system parameters based on the root locus method) is shown in Table 2. This Table shows that have many cases of the DPs values unsatisfied the stability condition (bolded text) based on the Routh criterion. This means that the stability of the control system of the LCL-filter-based grid-connected inverter strongly depend on the system parameters. 5. GENERALIZED DESIGN METHOD FOR THE CONTROLLER OF LCL-FILTER- BASED GRID-CONNECTED INVERTER As outline above, the controller gain obtained by using existing methods may not guarantee stability when system parameters change. As a result, the exact determination of the stability bounds when the system parameters change is extremely important and should be carefully carried out. In this paper, we propose a generalized design method for designing the controller of the three phase LCL-filter-based grid-connected inverter. In proposed method, we determine the optimal stability bound of the coefficient control gains for each value of the system parameters (this value called the local optimal stability bounds) by resolving the characteristic equation of the closed-loop control system. After determining these local optimal stability bounds, the global optimal coefficient control gains of the controller are determined based on the minimum (or maximum) rules. The proposed algorithm can be used to determine the stability bound of all control coefficient gains. However, as mentioned above, we only focused on the global optimal value of the proportional gain (Kp) in this paper. Hence, the global optimal proportional gain of Nguyen Trung Nhan, Nguyen Thi Hanh 46 the controller is determined as follows: 1 2 min{ , ,..., }p opt p opt p opt p opt nK K K K (16) The proposed algorithm was coded in Matlab/M-file and used to obtain the random distribution of the local optimal stability bounds, which are shown in Table 3. This Table show that the stability bounds significantly depend on the system parameters. The global optimal value represents the range for the controller gain Kp that guarantees the stability irrespective of the change in system parameters, whereas the local optimal value represents the range for the controller gain that only guarantee stability for fixed system parameters. Table 3. The change of local optimal stability bounds of the controller versus the change of system parameters. rp Kp_Lopt rp Kp_Lopt rp Kp_Lopt rp Kp_Lopt rp Kp_Lopt rp Kp_Lopt rp Kp_Lopt 1 2.1 15 1.9 29 1.7 43 1.9 57 1.9 71 2.1 85 3.3 2 3.0 16 2.7 30 2.0 44 2.5 58 2.0 72 2.7 86 2.7 3 1.9 17 2.2 31 1.9 45 2.3 59 2.2 73 1.8 87 2.0 4 1.9 18 2.1 32 3.2 46 2.6 60 2.4 74 1.9 88 2.6 5 1.8 19 2.0 33 2.2 47 2.8 61 2.1 75 2.0 89 2.1 6 2.4 20 2.6 34 1.8 48 1.8 62 2.0 76 1.8 90 2.0 7 2.4 21 1.9 35 2.7 49 2.4 63 2.0 77 1.9 91 2.9 8 2.5 22 1.9 36 2.0 50 2.2 64 2.8 78 2.1 92 2.9 9 2.2 23 1.8 37 3.4 51 2.0 65 2.2 79 2.8 93 2.1 10 1.8 24 3.4 38 2.1 52 2.8 66 2.4 80 3.0 94 2.1 11 2.3 25 1.9 39 2.3 53 1.8 67 2.3 81 1.8 95 2.2 12 2.3 26 2.8 40 2.8 54 2.8 68 2.2 82 2.1 96 2.9 13 2.0 27 2.2 41 2.3 55 2.5 69 2.4 83 2.8 97 2.6 14 2.3 28 2.1 42 2.1 56 2.0 70 2.2 84 2.6 98 2.1 Table 4. The change of dominated poles versus the change of system parameters if the value of Kp is accurately determined. rp DPs rp DPs rp DPs rp DPs rp DPs rp DPs rp DPs 1 -221.39±5447i 15 -86.600±4500i 29 -29.840±5306i 43 -87.530±3862i 57 -129.52±4633i 71 -167.92±4017i 85 -717.06±5924i 2 -655.25±6350i 16 -465.44±6702i 30 -126.69±4468i 44 -325.10±4723i 58 -110.17±3790i 72 -556.23±5243i 86 -510.40±4599i 3 -73.360±5632i 17 -249.99±5802i 31 -156.06±7392i 45 -330.17±5917i 59 -171.35±3722i 73 -71.630±4405i 87 -139.50±4658i 4 -94.140±6112i 18 -135.05±3934i 32 -639.42±4048i 46 -361.95±3968i 60 -295.16±5650i 74 -108.89±4071i 88 -390.86±4669i 5 -66.180±5009i 19 -139.58±4458i 33 -304.14±4901i 47 -541.35±4801i 61 -164.68±4639i 75 -200.60±5169i 89 -177.59±5115i 6 -325.13±5298i 20 -591.83±6939i 34 -78.570±6572i 48 -68.360±6221i 62 -127.58±3721i 76 -57.630±6613i 90 -124.51±5033i 7 -374.91±4881i 21 -91.300±3725i 35 -503.92±6170i 49 -328.57±5646i 63 -100.00±3924i 77 -116.37±4418i 91 -641.94±5185i 8 -413.40±5190i 22 -92.110±6141i 36 -163.79±6457i 50 -194.10±4211i 64 -408.39±3896i 78 -125.79±4722i 92 -534.30±5415i 9 -205.55±6127i 23 -52.180±4628i 37 -792.75±4141i 51 -170.85±5817i 65 -170.82±3981i 79 -418.99±3749i 93 -327.65±7556i 10 -75.170±4919i 24 -817.28±4957i 38 -154.36±3614i 52 -505.62±5025i 66 -284.97±3980i 80 -559.49±4145i 94 -134.24±5879i 11 -195.88±4710i 25 -87.920±5022i 39 -240.55±5124i 53 -56.400±5859i 67 -320.73±4198i 81 -62.840±5384i 95 -226.25±4486i 12 -205.89±5198i 26 -462.57±6359i 40 -567.08±4373i 54 -508.79±5906i 68 -192.98±4241i 82 -126.41±3550i 96 -625.27±4874i 13 -154.45±6270i 27 -185.66±3979i 41 -312.80±4485i 55 -371.58±4238i 69 -300.62±5034i 83 -582.99±4724i 97 -441.18±5195i 14 -253.06±4286i 28 -164.04±3867i 42 -197.77±4493i 56 -123.500±4899 70 -219.39±5241i 84 -456.78±6238i 98 -256.40±5757i A novel method for designing the controller of a lcl-filter-based grid-connected inverter with 47 These Tables also show that the global optimal value is relatively near compared with local optimal value and unstable value. Note that the value of Kp can generally be chosen in the range of the global optimal area (namely, Kp ≤ Kp-opt). However, in practice, the steady-state error of the control system will increase if the value of Kp decreases. Therefore, the best value of Kp is the value of Kp-opt determined in (16). To test the correctness of the proposed algorithm, we used the global optimal value of Kp obtained from Table 3 and (16) to obtain the random distribution of the dominated poles when system parameters changes, which are shown in Table 4. This Table shows that the proposed algorithm is correctness. 6. SIMULATION RESULTS The proposed design method for the three-phase LCL filter-based grid-connected inverter control system based on the Routh criterion was verified using a simulation in the Matlab/Simulink environment. The main purpose of the simulation was to test the effectiveness and correctness of the proposed method for designing controller used in the three-phase VSI with an LCL-filter when the system parameters are variable. In this simulation, the main frequency of the power grid is 50 Hz, the power grid voltage in this simulation is 380 V (phase- phase), and the control signal of the IGBTs is generated using a pulse-width modulation generator whose amplitude and frequency of the carrier wave are ± 10V and 10000Hz, respectively. The system parameters used in the simulations are given in Table 1. The value of Kp was assigned equal to the global stability bound (in this study Kp-opt = 1.3). Figure 3 illustrates the grid voltage, the controller output currents ( gi , gi ) in the stationary α−β frame, and the grid-side current in three-phase for a demand reactive power equal to zero (i.e., the power factor equal to unit). Figure 3. Simulation results of the proposed multi-closed-loop control system. (a) Grid voltages. (b) The controller output currents (iα, iβ) in the stationary α−β frame. (c) The grid side current (ig). Nguyen Trung Nhan, Nguyen Thi Hanh 48 Figure 4. Simulation results when the system parameters changed. (a) Kp determined by conventional method. (b) Kp determined by proposed method. To test the dynamic response of the proposed control system, the reference current was changed from 7 ampere to 12 ampere (peak) at t = 0.1 s when the system is operating (note that the change occurs when the a-phase current arrives at the zero value). This Figure shows that the dynamic response of the proposed controller is rapid for this case. Moreover, the steady-state error of the proposed controller is also very small. To verify the system performance is sensitive to the controller coefficients, simulations were performed when the system parameters changed (in this case: C = 0.9C0, L1 = 0.9L10, L2 = 0.85L20, R1 = 0.8R10, and R2 = 0.8R20, as shown in Table 1) for two cases: the value of Kp was assigned equal to the initial stability bound (the value of Kp is determined for the initial values of the system parameters based on the root locus method) and the value of Kp was assigned equal to the global optimal stability bound (as detailed in Section 5), respectively. The simulation result for this case is shown in Figure 4. This Figure shows that the proposed design method guarantees the controller of the LCL filter-based grid- connected inverter robust operation, irrespective of the change in the system parameters while the conventional method is not. This finding demonstrates that the system performance is sensitive to the controller coefficients and can yield a good performance with the conventional PI controller if the controller coefficients are exactly designed. To verify the effectiveness and correctness of the proposed method for designing the controller, simulations were performed in which the value of Kp was assigned to exceed the stability bound. The simulation result for this case is shown in Figure 5, which shows that the system is not stable. A novel method for designing the controller of a lcl-filter-based grid-connected inverter with 49 Figure 5. Simulation results for a value of Kp was assigned to exceed the stability bound. Figure 6. Simulation results for the power factor was 0.8 leading (with Kp = Kp-max-opt). We now consider an REG operated in P-Q mode, namely, simultaneously injecting active and reactive power into the utility grid. The power factor was 0.8 leading. The simulation result for this case is shown in Figure 6. This Figure shows that the control system designed by using the proposed method resulted in efficiency and robust operation modes for all cases. 7. CONCLUSION This paper proposes a generalized method to design the controller of a three phase LCL filter-based grid-connected inverter system. This design results in efficiency and robust operation for both fixed and variable system parameters. The impact of the change in system parameters on the performance of three phase LCL filter-based grid-connected inverter system was analyzed in detail. The proposed design method results in the correct operation of the controller of the three phase VSI with an LCL-filter system for fixed and variable parameters. Implementing the proposed design method using the solution of the equation based on conventional Routh criteria is simple. 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