https://doi.org/10.6113/JPE.2019.19.6.1403

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

Sliding Mode Controller Applied to Coupled Inductor Dual Boost Inverters

Yu Fang^†, Songyin Cao*, and Pat Wheeler^**

^†,*College of Information Engineering, Yangzhou University, Yangzhou, China

^**Department of Electrical and Electronic Engineering, University of Nottingham, Nottingham, ENG, UK

Abstract

A coupled inductor-dual boost-inverter (CIDBI) with a differential structure has been presented for application to a micro-inverter photovoltaic module system due to its turn ratio of a high-voltage level. However, it is difficult to design a CIDBI converter with a conventional PI regulator to be stable and achieve good dynamic performance, given the fact that it is a high order system. In view of this situation, a sliding mode control (SMC) strategy is introduced in this paper, and two different sliding mode controllers (SMCs) are proposed and adopted in the left and right side of two Boost sub-circuits to implement the corresponding regulation of the voltage and current. The schemes of the SMCs have been elaborated in this paper including the establishment of a system variable structure model, selection of the sliding surface, determination of the control law, and presentation of the reaching conditions and sliding domain. Finally, the mathematic analysis and the proposed SMC are verified by experimental results.

Key words: Coupled inductor, Micro inverter, Photovoltaic (PV) power generation, Sliding mode control

Manuscript received Apr. 14, 2019; accepted Jun. 27, 2019

Recommended for publication by Associate Editor Fuxin Liu.

^†Corresponding Author: yfang@yzu.edu.cn Tel: +86-514-87978319, Fax: +86-514-87971865, Yangzhou University

^*College of Information Engineering, Yangzhou University, China

^**Dept. of Electrical & Electron. Eng., The Univ. of Nottingham, UK

Ⅰ. INTRODUCTION

A grid-connected micro-inverter is an effective way to solve the hot spot effects of photovoltaic modules and to improve the efficiency of photovoltaic power generation. There are two main grid-connected power generation methods for low-voltage photovoltaic modules. 1) Single-stage inverters connected to a grid by the line transformer boost mode. 2) Two-stage inverter consisting of a front-stage DC-DC boost circuit and a back-stage inverter. In the first method, when the voltage of a photovoltaic module is low, the turn ratio of the line transformer is large. Hence, the power loss of transformer is large. In the second method, a two-stage circuit is adopted, and the conversion efficiency of the system is not high. It can be seen that these two methods are not conducive to the efficient generation of small and medium power from photovoltaic cells to the grid.

The CIDBI proposed in [1] can meet the voltage boost requirements of photovoltaic modules and AC grid-connected. In other words, it is able to realize one-stage transformerless grid-connected.

The differential CIDBI presented in [1] and shown in Fig. 1 is composed of two coupled inductor boost circuits that generate an ac output voltage that is larger than the dc input. It depends on the instantaneous duty cycle and turn ratio of the coupled inductor, and uses only one power processing stage. Fig. 1 shows that there is one DC source B₁. It can be seen that L₁ and L₂ commonly share the same magnetic core, while L₃ and L₄ use another. Thus, the coupled inductors are constructed. One boost circuit consists of L₁, L₂ and C₂, the switches T₁ and T₂, and their body diodes D₁ and D₂. The other boost circuit consists of L₃, L₄ and C₃, the switches T₃ and T₄, and their body diodes D₃ and D₄. It is clear that the right-side circuit and the left side circuit represent boost converters with a coupled inductor. In [1], a CIDBI can be employed as micro inverter integrated with a much lower voltage PV array and achieve a high energy conversion efficiency due to its one-stage boost circuit architecture. However, it is difficult to achieve a tradeoff between the stability and the fast dynamic response of a CIDBI by using traditional linear controllers. This results in energy loss due to sudden changes in photovoltaic applications.

Fig. 1. Main circuit of CIDBI.

In order to obtain a smoother grid current i_g, an AC output inductor L_ac is added on the basis of Fig. 1, as shown in Fig. 2. Thus, the current i_g through L_ac is shaped by adapting the voltage difference (v_c1-v_c2). In this paper, two different sliding mode controllers are proposed and adopted in the left and right side circuit of the CIDBI to realize grid-connected power generation. In addition, the SMCs for grid-connected power generation are discussed and analyzed. As a result, the energy of low-voltage photovoltaic modules can be transferred into a power grid by a CIDBI with good performance.

Fig. 2. Main circuit of the CIDBI after the insertion of L_ac.

Ⅱ. ANALYSIS AND DESIGN OF SMCS

The main circuit diagram of a CIDBI shown in the Fig. 2 can be divided into two subsystems that are controlled by different sliding modes, as shown in Fig. 3. In order to generate an ac output current i_g from dc sources, the left-side circuit is controlled to shape the grid current i_g, and the right- side circuit is controlled to make the output voltage stable, which is close to v_C2* (reference voltage). These two parts of the controller can realize decoupling control. Next, error state equations of the left circuit and right circuit are deduced.

Fig. 3. Schematic diagram of a CIDBI with two sliding mode controllers.

A. Right-Side Circuit

The CIDBI is based on a boost converter. Hence, the voltage v_C2 must never fall below V_in. That is to say, the output voltage is a dc-biased sinusoidal voltage as presented in equation (1). In addition, equation (1) defines the reference for v_C2.

          (1)

Assume the grid voltage is as follows

          (2)

When in steady operation, the grid current should follow the reference current i_g^*. Here it is assumed that the reference grid current and the grid current are as follows:

             (3)

         (4)

Since the differential output is constructed by the left-side circuit and right-side circuit, the reference ac voltage component is given as follows:

          (5)

In order to keep the voltage v_c2 higher than V_in all the time, the following inequality is met.

          (6)

v_C2^*can be achieved by controlling T₁ and T₂ in the right-side boost circuit, and the pulse control signals v_dr1 and v_dr2 should be complementary. In light of the switching states, the right side of the converter can be equivalent to two sub-circuits as shown in Fig. 4. In addition, the left side of the CIDBI can be considered as the current source. Thus, the calculation formula of the duty cycle ratio d_R(t) for the right side is as follows.

         (7)

Where, N is the turn ratio of the coupled inductor (seen in the Fig. 1: n₂/n₁).

The average power of the output should be equal to that of the input according to the instantaneous power theory. Thus, equation (8) can be obtained as follows.

             (8)

Substituting expressions (1), (2), (4) and (5) into equation (8), the reference value i_dc1* for the dc input current can be expressed as (9).

         (9)

Expression (9) describes the average input current i_dc1 through the coupled inductor L₁/L₂. In fact, the input current contains a high-frequency ripple current that can be neglected since it has no effect on the effective value of the current.

The switching function can be defined in expression (10), where, γ is the state variable.

         (10)

It is clear that the corresponding output energy increases when γ=1. In this state, the inductor L₁ and the output capacitance C₂ are decoupled and the current through L₁ is increased linearly, i.e. the energy storage. In addition, i_g flows to C₂ as shown in the Fig. 4(a). When γ=0, the energy is transferred to C₂ by the coupled inductor L₁/L₂ as shown in the Fig. 4(b).

Fig. 4. Two structures of the right-side converter. (a) T₁ is on and T₂(D₂) off. (b) T₂(D₂) is on and T₁ off.

(a)

(b)

It can be seen from Fig. 4 that there are two error variables that are defined as εv_C2 for the output capacitance voltage, and εi_dc1 for the coupled inductor current in expression (11).

      (11)

As a result, the state error equation of the right-side converter can be written as (12).

           (12)

Where:

           (13)

          (14)

          (15)

          (16)

It is noted that the low-frequency reference functions, when compared with the switching frequency, can be considered as quasistatic, in such a way that the term of the derived function for the reference is generally neglected in equation (12).

B. Left-Side Circuit

The left-side subsystem is seen as the current source and controlled to shape the grid current. Actually, the voltage of L_ac can be regulated only if the voltage v_C1 across C₁ is controlled. As a result, the grid current i_g is obtained.

Additionally, since the inductor L_ac is designed to filer the high-frequency switching ripple, its steady-state low-frequency voltage is small. In addition, the voltage v_C1 should be close to (v_C2+v_g). Thus, v_C1 can be expressed as (17).

          (17)

The ac component of expression (17) is complementary to that of the expression (1). Thus, the phase difference is 180°. In addition, the equations of duty cycle ratio d_L(t) and i_dc2^* are similar to (7) and (9), with rad added to the arguments of the sine or cosine function (or simply the terms in sin(ωt) and cos(ωt) change their signs, while the terms involving 2ωt remain unchanged).

In order to follow i_g^*, the switches T₃ and T₄ in the left-side converter are controlled with a certain duty cycle ratio d_L, and their pulse control signals v_dr3 and v_dr4 should be complementary to each other. In light of the switching state, the left-side converter can be equivalent to two sub circuits as shown in Fig. 5, and the right-side converter is taken as the voltage source in Fig. 5.

Fig. 5. Two structures of the left-side converter: (a) T₃ is on and T₄(D₄) off; (b) T₄(D₄) is on and T₃ off.

(a)

(b)

The two states in Fig. 5 can be described by the switching function as (18).

            (18)

Assuming that the error variable for the filter L_ac is εi_g, the error variable for the filter C₁ is εv_C1, and the error variable for the coupled inductorL₃/L₄ is εi_dc2. This is shown in expression (19).

        (19)

After defining the error variables, the state error equation (20) can be written.

       (20)

Where:

       (21)

          (22)

          (23)

          (24)

The derived function for the reference is neglected in the state error equation (20) as in the previous section. Through the above analysis, the error state equation of the CIDBI is obtained.

Ⅲ. CONTROL STRATEGY

Sliding mode control can change the circuit structure by controlling the switch state. The sliding mode method is implemented in the error state space, and the trajectory through the spatial origin (the desired stable state) is well designed. When the representative point reaches the trajectory, approaches it, and then reaches the origin, zero error is achieved [10]-[16]. Such a trajectory is called a sliding surface (σ), and this motion is called a sliding mode or a sliding mode method. The designed sliding surface space is divided into two subspaces, each of which is associated with a structure. The motion in the vicinity of the sliding surface must point to the sliding surface [17]-[24]. This is a necessary and sufficient condition for the existence of the sliding surface, which is expressed mathematically as follows:

           (25)

Considering the fact that actual switching devices are difficult to operate at infinitely high switching frequencies, the actual sliding mode is not always on the sliding surface, and the quasi-sliding mode is usually used to achieve control [25]-[27].

A. Implementation of the Right-Side SMC

The sliding mode surface σ_R is defined by expression (26), which can be considered as the weight of the error. If S₁ and S₂ are taken to be constant, the sliding mode surface is a straight line on the plane of ×, and the slope is decided by α_R=S₁/S₂. In expression (25), the units: S₂ is ohms; S₁ is dimensionless; and α_R is Siemens.

      (26)

Assuming that S=[S₁, S₂]^T, expression (25) can be expressed by (27).

            (27)

Since the error variables are defined by (12), if the representative point is below the surface σ_R, the energy of the capacitor C₂ and the inductor L₁ is reduced. Then γ should be set to 1, whereas γ should be set to 0 when it is above σ_R. Thus, expression (28) can be obtained.

        (28)

Since the error variables are very small and approach zero by comparison with the reference values, the existing condition (25) for the sliding mode surface on the right side can be written by (29) and (30).

          (29)

        (30)

Substituting matrixes B and D into (29) and (30), the expression (31) and (32) can be obtained.

         (31)

         (32)

It can be concluded from (31) and (32) that S₁ and S₂ choose non-negative values if the sliding mode surface exists as presented in (26).

Fig. 6. Switching function γ.

Fig. 7. Waveform of σ_R.

If the actual system adopts the fixed hysteresis width of 2δ in Fig. 6, the control rules can be rewritten into (33). In addition, the phase plane trajectory oscillates on both sides of the sliding mode surface (σ_R=0), as shown in Fig. 7.

            (33)

It can be seen from Fig. 7 that σ_R must increase from –δ to +δ during ∆t₁ (>0), while σ_R must decrease from +δ to -δ during ∆t₂ (<0). Assuming that the state trajectory remains the same near σ_R=0, formula (34) for the switching frequency can be deduced from Fig. 7.

       (34)

Where, ∆t₁ corresponds to the switching state γ=1, and ∆t₂ corresponds to γ=0. In addition, ∆t₁ and ∆t₂ can be solved by (35) and (36), respectively.

     (35)

         (36)

Substituting (35) and (36) into (34), the maximum switching frequency can be obtained as (37) at the positive peak value of v_C2, assuming i_g^*=0 and the corresponding i_dc1^*=0 in the state of a light load.

          (37)

The duty cycle ratio is defined as follows:

        (38)

Substituting (34) and (35) into (38), the steady switching frequency can be obtained as (39), depending on steady operation points such as the duty cycle, control parameters and value of the grid current.

         (39)

When i_g reaches a negative peak value, the variations of i_dc1 and v_C2 are the largest. At this time, the duty cycle is the largest and the switching frequency the highest. As a result, L₁ and C₂ can be designed as (40) in order to ensure that all of the magnetic components are unsaturated within the scope of the switching frequency.

        (40)

B. Implementation of the Left-Side SMC

Since the left side is a third order system, the sliding mode surface σ_L can be obtained by (41).

        (41)

Let S_L=[S₃, S₄, S₅]^T, and expression (41) can be written as (42).

               (42)

Since the error variables are defined by (19), if the representative point is below the surface σ_L, the energy of the capacitor C₁ and the inductor L₃ is reduced. Then λ should be set to 1, whereas λ should be set to 0 when it is above σ_L. Thus, expression (43) can be obtained.

          (43)

Since the error variable is very small and approaches zero by a comparison with the reference values, the existing condition (25) for the sliding mode surface on the left side can be written by (44) and (45).

           (44)

          (45)

Substituting matrixes B^’ and D^’ into (44) and (45), the expression (46) and (47) can be obtained.

         (46)

            (47)

If expressions (46) and (47) are established, the existing condition for the sliding mode is met. In addition, the sufficient condition for the existence of the sliding mode surface defined by (41) is that S₃, S₄ and S₅ choose nonnegative values.

Just like the right side, the hysteresis comparison control is employed to avoid an excessive switching frequency. Here, assuming that the hysteresis width is 2ζ, the phase plane trajectory oscillates on both sides of the sliding mode surface (σ_L=0).

Defining the on time of switch T₃ for ∆t₁, which corresponds to the switching state λ=1; and the off time for ∆t₂, which corresponds to the switching state λ=0; ∆t₁ and ∆t₂ can be solved by (48) and (49), respectively.

             (48)

               (49)

The same as the right-side converter, the maximum switching frequency occurs at a light load. Assuming i_g*=0 and the corresponding i_dc2*=0, and noticing that the coefficient of the term of S₃ is zero and that the terms of S₄ and S₅ are like (35) and (36), expression (50) can be obtained.

             (50)

It can be seen that the last two terms are the same with the right side in the existing conditions (46) and (47) for the sliding mode surface of the left-side converter, i.e. S₄=S₁ and S₅=S₂. Therefore, only the value of S₃ needs to be confirmed. Since the CIDBI is used for grid connected power generation, in (46) must be established, and this difference value should be equal to v_Lac of L_ac, which result in S₃>0. As a result, the upper limit of S₃ can be solved by (46) under the condition of the maximum of .

The steady state switching frequency can be solved by (51) only if ∆t₁+∆t₂ is calculated during the period of hysteresis, which is like the right side.

             (51)

When i_g reaches the positive peak value, variations of i_dc2 and v_C1 are the largest. Hence, the values of L₃ and C₁ can be chosen by (52).

          (52)

Such a high-pass filter is suitable for cases where the reference of an indirect variable is calculated. The controller tries to offset the sum of the weights of the errors after controlling the power circuit, not just an error. In this way, if one of these references is wrong, it leads to tracing defects. This technology is highly suitable for DC-DC converters. In current applications, the formation of low-frequency AC (twice the grid frequency) is considered to be quasi-static. Therefore, the derivatives of these quasi-static variables can be considered to be zero. In the feedback, the low-frequency components are filtered by a high-pass filter. Thus, it can be considered that the low-frequency components have tracked the reference value in the steady state, and only high- frequency components and the spike need to be regulated.

Thus, it is feasible to collect indirect variables with high- pass filters. Of course, the introduction of high-pass filters increases the order of the system. Generally speaking, the transient response of a system is easier to oscillate than that of the case where the indirect variables have references. The cut-off frequency of a high-pass filter should be set high enough that it has little effect on the dynamic performance of the circuit. On the other hand, high-pass filters should be sensitive to the resonant points of L₁, C₁, or L₂, C₂. It should be noted that the resonant frequency of a switching circuit also depends on the duty cycle ratio. Because the filter system is insensitive to current detection when the switch is on for a long time (especially when starting), the inductance current should be limited for the sake of safety.

In addition, the selection of L_ac should also consider the resonant frequency of the high-pass filter. In particular, the design of the v_C1 high-pass filter should take full account of the resonant frequency of L_ac, C₁ and C₂. In addition, when starting, the DC voltage on v_C2 needs to be established first. Then the grid-connected relay closes, and slowly increases the grid-connected current.

Ⅳ. DESIGN EXAMPLE AND EXPERIMENTAL RESULTS

In order to verify the correctness of the above theoretical analysis, a two-channel experimental prototype has been built as shown in Fig. 8. In the prototype, each channel is connected to a photovoltaic module. The performance parameters of only one channel are given here. The input dc source is a PV array whose open voltage is 44V and the maximum power point (MPP) voltage is 35V. The prototype specifications are listed in Table I.

Fig. 8. Experimental prototype of photovoltaic grid-connected micro-inverters based on the CIDBI.

TABLE I PROTOTYPE SPECIFICATIONS

Output power: Po	480W
Grid maximum voltage: vg	230Vrms
MPP voltage: Vin	35V
Grid frequency: f0	50Hz
Maximum switching frequency: fsmax	130kHz

In order to regulate the grid current i_g linearly with the duty cycle ranging from 0.3 to 0.75, the dc bias voltage V_dc is chosen to be 280V, and the turn ratio of the coupled inductor is 4. Assuming that the grid current is 1.43Arms at the MPP, the reference current and voltage are as follows.

       (53)

It is clear that V_C2max=442.6V.

The DC capacitor C₁ is 6×2200μF; T₁ and T₃ are IPB072N15N3G from Infineon (150V/93A @100°C), Rdson =7.2mΩ; and T₂ and T₄ are SPB17N80C3 from Infineon (800V/17A @100°). Main experimental waveforms are given in Fig. 9. Here, v_g is the utility grid voltage, i_g is the grid connected current, and v_pv is the PV voltage. Fig. 9(a) shows steady operation waveforms with the PI controller in [1], and Fig. 9(b) presents waveforms based on the SMC in this paper. The grid connected current waveform i_g(CH3) in Fig. 9(b) is clearly much better than that in Fig. 9(a). When the output power reaches 221W, the efficiency of the CIDBI is greater than 95.8%, the total harmonic distortion of i_g is equal to 1.52%, and the power factor is not less than 0.99. It is clear that the presented sliding mode controller can be applied to the CIDBI and that it can transfer energy from the dc side to the mains grid with high quality.

Fig. 9. Main steady operation waveforms of grid-connected generation with the: (a) PI controller in reference [1], (b) SMC in this paper.

(a)

(b)

Fig. 10 shows waveforms of the grid voltage and grid- connected current when the prototype starts. As can be seen from this figure, when the prototype starts grid-connected power generation, the current i_g begins to gradually increase. After a period of time, the current reaches its maximum value and constant. Channel 1 of Fig. 11 shows a voltage waveform of capacitor C₂ in the right circuit of the CIDBI, and channel 2 shows a driving waveform of the switch T₁. Channel 3 of Fig. 12 shows a grid-connected current waveform (i_g) when the photovoltaic panel is shielded or not. Channel 2 of Fig. 12 shows the grid voltage. From the above experimental waveforms, it can be seen that using the two sliding mode controllers proposed in this paper can yield both the stability of micro-inverters based on the CIDBI architecture, and a fast dynamic response speed.

Fig. 10. Start-up generation waveform of a micro inverter.

Fig. 11. Driving waveform of T₁ and the voltage on C₂ in the right-side circuit.

Fig. 12. Occlusion experiment of a photovoltaic module.

Ⅴ. CONCLUSIONS

A CIDBI can be used to generate an ac output voltage that is larger than the dc input voltage and to transfer energy from a PV array or dc energy storage device to a grid with only one power processing stage. Aiming at the problem of it being difficult to design the parameters of the traditional regulators for the high-order systems of CIDBI circuits, different sliding- mode controllers are proposed in this paper for the right-side converter and the left-side converter with different switching schemes. In addition, the SMCs can track the time-varying reference functions very well. These functions are only defined for two output variables. The other secondary signals are obtained after high-pass filtering. The use of such SMCs can achieve remarkable overall performance improvements of the system.

ACKNOWLEDGMENT

This paper was supported in part by the National Natural Science Foundation of China under Grant 61873346, in part by the Science and Technology Cooperation Fund of Yangzhou City Hall project under Grant YZ2018136, in part by the Intelligent Energy Internet Research Institute Joint Fund of State Grid Yangzhou Power Supply Company and Yangzhou University under Grant SGTYHT/17-JS-202, and in part by the Open Project Fund of Yangzhou University Jiangdu Institute of High-end Equipment Engineering Technology under Grant YDJD201902.

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Yu Fang was born in Jiangsu Province, China, in 1972. He received his B.S. degree in Electrical Engineering and Automation from the Wuhan University of Technology, Wuhan, China, in 1995; and his M.S. and Ph.D. degrees in Power Electronics and Power Drives from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2004 and 2008, respectively. He worked in the Post-Doctoral Workstation of Southeast University (SU), Nanjing, China; was an Academic Visitor at the University of Nottingham, Nottingham, ENG, UK; and is presently working as a Professor in the College of Information Engineering, Yangzhou University, Yangzhou, China. Dr. Fang has authored more than 80 technical papers published in journals and conference proceedings. His current research interests include modeling, digital control of switching power converters, bidirectional energy converters applied to new energy grid-connected generation and energy internet.

Songyin Cao was born in Jiangsu Province, China, in 1982. He received his M.S. degree from the College of Information Engineering, Yangzhou University (YZU), Yangzhou, China, in 2006; and his Ph.D. degree from the School of Automation, Southeast University, Nanjing, China, in 2011. Since 2010, he has been with the Faculty of Automation, YZU, where he is presently working as an Associate Professor in the College of Information Engineering. From 2012 to 2015, he was a Post-Doctoral Fellow at Beihang University, Beijing, China. From 2014 to 2015, he was an Academic Visitor at Loughborough University, Loughborough, ENG, UK. He has published one book and authored more than 40 technical papers published in journals and conference proceedings. His current research interests include fault tolerant control and inertial navigation systems.

Pat Wheeler received his B.S. [Hons] degree, and his Ph.D. degree in Electrical Engineering for his work on Matrix Converters from the University of Bristol, Bristol, ENG, UK, in 1990 and 1994, respectively. In 1993, he moved to the University of Nottingham, Nottingham, ENG, UK, and worked as a Research Assistant in the Department of Electrical and Electronic Engineering. In 1996, he became a Lecturer in the Power Electronics, Machines and Control Group at the University of Nottingham, where he has been a Full Professor since 2008. He is presently serving as the Head of the Department of Electrical and Electronic Engineering at the University of Nottingham. He is an IEEE PELs ‘Member at Large’ and an IEEE PELs Distinguished Lecturer. He has published over 400 academic papers in leading international journals and conference proceedings.