An enhanced control strategy of three - phase four - wire inverters under nonlinear load conditions

Journal of Science Technology and Food 21 (1) (2021) 3-14 
 AN ENHANCED CONTROL STRATEGY OF THREE-PHASE 
 FOUR-WIRE INVERTERS UNDER NONLINEAR LOAD 
 CONDITIONS 
 Van Tan Luong1*, Pham Dinh Tiep1, Le Nguyen Hoa Binh2 
 1Ho Chi Minh City University of Food Industry 
 2Van Lang University 
 *Email: luongvt@hufi.edu.vn 
 Received: 19 January 2021; Accepted: 05 March 2021 
 ABSTRACT 
 An enhanced nonlinear control technique based on a coordination between feedback 
linearization (FBL) approach and sliding mode control (SMC) is proposed for a three-phase 
split-capacitor inverter under the nonlinear load conditions. A nonlinear model of system with 
pulse-width modulation (PWM) voltage-source inverter (VSI) including the output inductor-
capacitor (LC) filters is derived in the d-q-0 synchronous reference frame, not by small signal 
analysis. The controllers for d-q-0 components of three-phase line-to-neutral load voltages are 
designed by linear control theory. With the proposed coordination scheme, three-phase 
split-capacitor inverter provides an excellent control performance for regulating the load 
voltages with nearly zero steady-state errors in both the transient and steady states. The proposed 
scheme is verified by the simulation results which show that three-phase split-capacitor inverter 
gives a low total harmonic distortion (THD) for the load voltages under the balanced or 
unbalanced nonlinear load conditions. 
Keywords: Nonlinear load, three-phase inverter, feedback linearization, sliding mode control, 
unbalanced load. 
 1. INTRODUCTION 
 Recently, three-phase inverter has been widely applied for standalone applications. These 
applications considered as loads could be the vehicles, trucks, or the photovoltaic power 
systems, and so on [1, 2]. These loads could be the three-phase loads and/or single-phase loads 
which can cause a three-phase unbalanced load, an irregularly distributed single-phase load or 
a balanced three-phase load operating at a fault condition. If the imbalanced loads appear in 
the system, the components of the unwanted negative-and zero-sequence currents are 
produced. The negative-sequence component of the currents can cause the excessive heating 
in machines, saturation of transformers and ripple in rectifiers. Meanwhile the zero-sequence 
currents cause excessive power losses in neutral lines and affect protection. 
 The three-phase inverter is connected to a load by a four-wire system, in which the neutral 
point of both source and load sides is also grounded. Several different methods have been 
applied to provide the neutral point of the source side. In the one way, the / (delta/winding) 
transformer has been used, in which the and Y windings are connected to the inverter and 
the load, respectively [3]. For this, the zero-sequence current is trapped in the windings. 
However, the use of the transformer can make its topology bulky, heavy and costly. In the 
other ways, the three-phase split-capacitor inverters and the four-leg inverters equipped with 
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Van Tan Luong, Pham Dinh Tiep, Le Nguyen Hoa Binh 
the eight switches have been employed. Nevertheless, the two switches must be required to 
add to the four-leg inverters and the three-dimension space vector modulation is so 
complicated [4]. Fortunately, a three-phase three-leg inverter with split direct current (DC) bus 
is one topology which can implement the three-phase four-wire system with a neutral point, 
as seen in the connection point of the load in Figure 1. Compared to a three-phase three-wire 
system, this topology can cope with the zero-sequence to regulate the output voltages to be 
balanced and the zero-sequence current can flow in the connection between the neutral point 
and the mid-point of the capacitive divider. 
 Several researches focusing on improving the quality of the output voltages for the 
inverters and uninterruptable power supply (UPS) have been suggested. A repetitive control is 
used to regulate the inverters for UPS applications, but this controller shows slow response 
and lack of systematical method to stabilize the dynamic error of the system [5, 6]. Although 
this method can obtain high performance of the output voltage, the techniques for the control 
design is relatively complicated. In [7], a control strategy applying the technique of the 
symmetrical sequence decomposition to extract the positive-, negative- and zero-sequence 
components from the unbalanced three-phase signals have been developed. The proportional-
integral (PI) controllers for the current and voltage are used to regulate the output voltages of 
the inverter. However, the using of twelve PI controllers and the processes of the sequence 
decomposition and composition could increase the computation time. Also, this control 
strategy is only suitable for the case of unbalanced linear loads. Sliding-mode control 
techniques are applied for regulating ... , i lq i l0
Thus, a controller based on sliding mode is suggested, so that an input–output of the system 
controller is linearized and can be implemented with a digital method for convenience. 
3.2. Sliding mode input-output feedback linearization control 
 The sliding surfaces with the errors of the indirect component voltages are expressed as [15]: 
 s= e + e + e dt
 1 1 11 1 12 1
 s= e + e + e dt (13) 
 2 2 21 2 22 2
 s= e + e + e dt
 3 3 31 3 32 3 
 If the system states operate on the sliding surface, then s1= s 2 = s 3 = 0 and 
 s= s = s = 0
 1 2 3 . Substituting (13) into yields 
 e1= − 111 e − 121 e;; e 2 = − 212 e − 222 e e 3 = − 313 e − 323 e (14) 
 It is guaranteed in (14) that the system states ( ) will exponentially converge 
towards the reference values when they are kept the sliding surface to zero. The equivalent 
control concept of a sliding surface is the continuous control that allows the maintenance of 
the state trajectory on the sliding surface ss==0 . The equivalent control is achieved from 
(13) as 
 2 1 1 1
 s= z − i + + 2 v + i +  i
 11 q ld ld q
 CLCCCf f f f f
 2 1 1 1
 s= z + i + + 2 v + i −  i (15) 
 22 d lq lq d
 CLCCCf f f f f
 11
 s3= z 3 + vll 0 + i 0
 (LLCf+ 3 n) f C f
 where z1 , z2 and z3 coincide with the new inputs of the system, whose expressions are 
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Van Tan Luong, Pham Dinh Tiep, Le Nguyen Hoa Binh 
expressed as 
 *
 z1= vld + 11 e 1 + 12 e 1
 *
 z2= vlq + 21 e 2 + 22 e 2 (16) 
 z= v* + e + e
 3l 0 31 3 32 3 
 The equivalent control is obtained by making s1= s 2 = s 3 = 0 as: 
 2
 u11eq= zLC f f −21 Li f q +( +  LCvLi f f) ld + f ld +  Li f q
 2
 u2eq= zLC 2 f f +21 Li f d +( +  LCvLi f f) l q + f lq −  Li f d (17) 
 1 
 u3eq= z 3 L f C f + L f v l 0 + L f i l 0
 (LLfn+ 3 )
 The equivalent obtained control is similar to the one achieved in (12). In order to drive 
the state variables to the sliding surface s1= s 2 = s 3 = 0 , in the case of s1, s 2 , s 3 0 , the 
control laws are defined as 
 u1=+ u 1eq u 1 st
 u2=+ u 2eq u 2 st (18) 
 u=+ u u
 3 3eq 3 st
 where u1st =  1s i gn( s 1 ) , u2st =  2s i gn( s 2 ) , u3st =  3s i gn( s 3 ) , 1 > 0, 2 > 0, 3 > 0. 
 Lf
 i
 ia la
 ib ilb s
 ic ilc
 vlabc
 Cf
 in
 Ln
 *
 va
 *
 vb SVPWM 
 3D 
 v*c abc abc abc
 dq0 dq0 dq0
 v v
 ld lq i
 v l0
 id iq i0 l0 ild ilq
 s1
 s2
 vlq
 vld
 s3 * vl0
 - vld
 u X
 1 + *
 v - vlq
 ilq ild il0 vlq ld vl0 (15) X
 u2 (18) +
 *
 - vl0
 u3 X
 u +
 1eq iq
 u2eq
 (17) id
 u3eq
 i0
 Figure 2. Block diagram of the proposed inverter control scheme. 
The reaching law can be derived by substituting (17) and (18) into (15), which gives 
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An enhanced control strategy of three-phase four-wire inverters under nonlinear load 
 s= −sgn i s ; s = −  sgn i s ; s = −  sgn i s (19) 
 1 1( 1) 2 2( 2) 3 3( 3 ) 
 The stability and robustness can be tested, using Lyapunov’s function which is presented 
in [15]. 
 Figure 2 shows the block diagram of the proposed controller, in which the dq0-axis load 
voltages use the sliding mode input-output feedback linearization control. The outputs of 
controller ( ***) are applied for SVPWM-3D (space vector pulse-width modulation -
 va,, v b v c
three dimensions). 
 4. SIMULATION RESULTS 
 To verify the feasibility of the proposed method, PSIM simulations have been carried out 
for the unbalanced and nonlinear loads. The DC-link voltage at the input of inverter from a 
three-phase ac source is 500 [V], the switching frequency of inverter is 10 [kHz]. The filter 
inductor Lf is 3 [mH] and the filter capacitor Cf is 100 [µF] which correspond to a cut-off 
frequency at 450 [Hz]. The parameters of loads and controllers are shown in the Table 1 and 
Table 2, respectively. 
 Table 1. Parameters of loads 
 Type of load Parameters 
 Ls = 1 [mH], C = 4.7 [mF], 
 Balanced nonlinear load R = R = R = 50 [Ω] 
 dca dcb dcc
 Ls = 1 [mH], C = 4.7 [mF], 
 Unbalanced nonlinear load R = 50 [Ω], R = R = 1 [kΩ] 
 dca dcb dcc
 Table 2. Parameters of controllers 
 Gain of controller 
 Controller type 
 Balanced nonlinear load Unbalanced nonlinear load 
 k = 5.4 k = 17.5 
 Current controller p p
 ki = 4000 ki = 13100 
 PI 
 k = 0.21 k = 0.32 
 Voltage controller pv pv
 kiv = 682 kiv = 896 
 3 6
 Proposed controller (FBL and SMC) k11 = k21 = k31= 5 × 10 , k12 = k22 = k32 = 8.4 × 10 
 Table 3. THD of load voltages 
 THD [%] 
 Load type Controller type 
 THD (phase A) THD (phase B) THD (phase C) 
 PI 2.14 0.58 0.45 
 Balanced 
 Proposed controller 
 nonlinear load 0.94 0.45 0.35 
 (FBL and SMC) 
 PI 2.5 0.6 0.5 
 Unbalanced 
 Proposed controller 
 nonlinear load 1.05 0.38 0.39 
 (FBL and SMC) 
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Van Tan Luong, Pham Dinh Tiep, Le Nguyen Hoa Binh 
 (a) Load voltages [V] (a) Load voltages [V]
 VLa VLb VLc VLa VLb VLc
 (b) Load currents [A] (b) Load currents [A]
 ILa ILb ILc
 ILa ILb ILc
 (c) Zero current [A] (c) Zero current [A]
 I
 L0 IL0
 (d) DC-link voltages [V] (d) DC-link voltages [V]
 VDC1 V 
 DC2 VDC1 VDC2 
 Figure 3. Dynamic response of PI controller Figure 4. Dynamic response of proposed 
 under the conditions of balanced nonlinear loads: controller under the conditions of balanced 
 (a) Load voltages, (b) Load currents, (c) Zero nonlinear loads: (a) Load voltages, (b) Load 
 current, (d) DC-link voltages. currents, (c) Zero current, (d) DC-link voltages. 
 The simulation results for the system using the proposed controller and PI controller in 
the case of the balanced nonlinear loads are shown in Figures 3 and 4, respectively. Each 
illustration shows the load voltages (VLA, VLB, VLC), load currents (ILA, ILB, ILC), neutral current 
(IL0) and voltages across two DC capacitors (VDC1, VDC2). 
 As can be clearly seen in Figures 3 and 4, the phase load voltages (Figures 3 (a) and 4(a)) 
become sinusoidal and are maintained at rated values under the balanced and nonlinear load 
conditions. However, the phase-A load voltage in the proposed controller is more almost 
sinusoidal, in the comparison with the traditional PI one. Also, the total harmonic distortion 
(THD) of the load voltage given in Table 3 shows that a THD of phase-A voltage in the case 
of using a PI controller is 2.14%, which is greater than that of using the proposed controller 
(0.94%). 
 The simulation results for the standalone inverter system using the proposed controller 
and the PI controller in case of unbalanced nonlinear loads are illustrated in Figures 5 and 6, 
respectively. By using the PI controller, the load voltages, load currents, neutral current and 
voltages across two DC capacitors are shown from Figure 5(a) to (d), respectively. Similarly, 
the load voltages, load currents, neutral current and voltages across two DC capacitors in the 
proposed controller are illustrated from Figure 6(a) to (d), respectively. Compared with the 
traditional PI controller, the load voltages in the proposed one as shown in Figure 6(a) give 
better performance. On the other hand, the load voltages in the proposed method are regulated 
to be rated and are almost sinusoidal. 
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An enhanced control strategy of three-phase four-wire inverters under nonlinear load 
 In the case of an unbalanced nonlinear load, the THD (Table 3) of the phase-A load 
voltage using the PI controller is 2.5%, which is still greater than the proposed controller 
(1.05%). 
 (a) Load voltages [V] (a) Load voltages [V]
 V V 
 VLa VLb VLc La Lb VLc
 (b) Load currents [A] (b) Load currents [A]
 ILa ILc ILa ILc
 ILb ILb
 (c) Zero current [A] (c) Zero current [A]
 IL0 IL0
 (d) DC-link voltages [V] (d) DC-link voltages [V]
 VDC1 VDC2 VDC1 VDC2 
 Figure 5. Dynamic response of PI controller Figure 6. Dynamic response of proposed 
 under the conditions of unbalanced nonlinear controller under the conditions of unbalanced 
 loads: (a) Load voltages, (b) Load currents, nonlinear loads: (a) Load voltages, (b) Load 
 (c) Zero current, (d) DC-link voltages. currents, (c) Zero current, (d) DC-link voltages. 
 (a) (b)
 1st 1st
 3rd 5th 7th 5th 7th
 Figure 7. FFT spectra of the phase-A load voltage under the conditions of balanced nonlinear loads: 
 (a) Using the PI controller, (b) Using the proposed controller. 
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Van Tan Luong, Pham Dinh Tiep, Le Nguyen Hoa Binh 
 (a) (b)
 1st 1st
 3rd 5th 7th 9th 5th 7th
Figure 8. FFT spectra of the phase-A load voltage under the conditions of unbalanced nonlinear loads: 
 (a) Using the PI controller, (b) Using the proposed controller. 
 To clarify the output voltage quality of the inverter, a fast Fourier transform (FFT) spectra 
analysis of the phase-A load voltage is performed in the two cases (balanced and unbalanced 
nonlinear loads), which are shown in Figures 7 and 8. In the case of using a PI controller, the 
load voltage in phase-A contains the high order frequency components such as 3rd, 5th, 7th (both 
cases) and 9th (just in case of unbalanced nonlinear loads) since the PI controller bandwidth 
does not respond to high frequencies well. In the case of using the proposed controller, the 
THD of the load voltage has been greatly reduced, compared to the case of using a PI 
controller. Specifically, the phase-A voltage no longer contains 3rd-order frequency 
components in the balanced nonlinear load conditions (Figure 7(b)) and 3rd and 9th-order 
frequency components in the unbalanced nonlinear load conditions (Figure 8(b)). Thus, it can 
be seen that the proposed control method achieves better performance than the PI controller in 
the cases of balanced nonlinear loads and unbalanced nonlinear loads. 
 5. CONCLUSION 
 The paper proposed a novel output voltage control of three-phase split-capacitor inverter 
based on the feedback-linearization technique and sliding mode control. This control method 
can regulate the load voltages in the case of unbalanced or unbalanced nonlinear loads. With 
this method, the load voltages are kept mostly balanced and sinusoidal with a low THD value 
for the simulation. The response of the three-phase split-capacitor inverter with the proposed 
strategy is better than the existing PI method. In the future, the proposed method can be used 
for unbalanced and distorted distribution grid voltage conditions. 
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Van Tan Luong, Pham Dinh Tiep, Le Nguyen Hoa Binh 
 TÓM TẮT 
 CHIẾN LƯỢC ĐIỀU KHIỂN NÂNG CAO CỦA BỘ NGHỊCH LƯU BA PHA BỐN DÂY 
 TRONG TRƯỜNG HỢP TẢI PHI TUYẾN 
 Văn Tấn Lượng1*, Phạm Đình Tiệp1, Lê Nguyễn Hòa Bình2 
 1Trường Đại học Công nghiệp Thực phẩm TP.HCM 
 2Trường Đại học Văn Lang 
 *Email: luongvt@hufi.edu.vn 
 Kỹ thuật điều khiển phi tuyến nâng cao dựa trên sự phối hợp giữa kỹ thuật tuyến tính hóa 
hồi tiếp (FBL) và điều khiển trượt (SMC) được đề xuất cho bộ nghịch lưu chia tụ ba pha trong 
trường hợp tải phi tuyến. Mô hình phi tuyến của hệ thống với bộ nghịch lưu nguồn điện áp 
điều chế độ rộng xung (PWM) bao gồm các bộ lọc LC ở ngõ ra được hình thành trong hệ tọa 
độ quay (dq0), mà không cần dùng phương pháp phân tích tín hiệu nhỏ. Bộ điều khiển đối với 
các thành phần d-q-0 của điện áp pha của tải được thiết kế theo kỹ thuật điều khiển tuyến tính. 
Với chiến lược kết hợp đề xuất, bộ nghịch lưu chia tụ ba pha tạo ra vận hành điều khiển tốt để 
điều khiển điện áp tải với sai số gần như bằng không cả trạng thái quá độ và xác lập. Chiến 
lược đề xuất được kiểm chứng bởi các kết quả mô phỏng, cho thấy rằng bộ nghịch lưu chia tụ 
ba pha cho độ méo hài tổng (THD) thấp đối với điện áp tải trong trường hợp tải phi tuyến cân 
bằng và không cân bằng. 
Từ khóa: Tải phi tuyến, bộ nghịch lưu ba pha, tuyến tính hóa hồi tiếp, điều khiển trượt, tải 
không cân bằng. 
 14