Hysteresis control for a grid connected dual-buck inverter

1. Introduction

Recently, renewable energy sources as photovoltaic generators, wind turbines, and energy storage systems have been continuously growing, becoming a cost-effective alternative to supply the demand of the market while diversifying the energy matrices and reducing the environmental impact of energy generation [1, 2, 3].

Typically, each renewable generator requires a DC-AC converter (inverter) to inject power into the grid [4, 5, 6]. In the basic approach, the inverter current synchronizes with the grid voltage (i.e. unit power factor), and the magnitude depends on the power generated by the source. Depending on the power of the renewable generator, inverters may be single-phase or three-phase. The latter is used for medium and high power applications, while single-phase inverters are preferred for low power generators.

Single-phase inverters have gained importance in the last years due to their application in microinverters [7, 8] and string inverters for residential and commercial applications [9, 10]. In particular, microinverters offer highpotential in the energy market due to the modularity, simple installation and high efficiency [11]. New technologies in semiconductors like GaN transistors, allow to manage low and medium power with higher switching frequencies, and reduced components sizes as well as better thermal and electrical performances [12]. The conventional full-bridge inverter, is probably the most common topology used in single-phase microinverters. It is formed by two legs connected in parallel to a DC bus, where each leg has two switches connected in series that operate in a complementary fashion. Major challenges with this approach are the avoidance of shoot-through current spikes due to the differences in the turn on and turn off time ratios of the switches. This problem is usually dealt with by introducing a delay in the control, at the cost of the voltage distortion. There are also, significant losses because of the reverse recovery currents passing through the protection diodes [9, 13].

As an alternative, the full-bridge dual-buck microinverter is currently investigated [14, 15, 16, 17]. This topology uses two complementary buck converters, each one operating in a semi-cycle of the grid voltage. Each buck converter has two switches, one commutes at high frequency and the other one remains active while the buck converter is operating. Some authors use two pairs of inductors [18, 19, 20, 21], while others reduce the number of inductors required to two or one [22, 23, 24]. In any case, using nonlinear modulation techniques is possible to deal with the shoot-through, reducing the switching losses while improving reliability [25].

In [22] the authors propose a grid-connected single-phase full-bridge dual-buck microinverter implemented with two inductors. The output of this converter is connected in parallel with a capacitor and the grid, which is represented by a sinusoidal voltage source. The study includes the development of a model to analyze the common mode current with and without a dead band in the transitions between the activation of one buck and the deactivation of the other, analyzing the losses and evaluating the efficiency. The paper does not propose a control system, and instead the authors use a PWM modulation technique to generate the switching signals without providing details on the controller design. In addition, the proposed inverter only used a capacitor as an output filter, which reduces the inverter performance regarding the total harmonic distortion. In [21] the authors present a hysteretic controller for a dual-buck full-bridge microinverter with four inductors. The purpose is to feed a load with a sinusoidal voltage. Therefore, the authors propose an inner loop with a sliding mode controller for the current in the inductor, and an outer loop with a PI controller for the voltage in the output.

Regarding hysteresis control techniques, researchers focus on limiting high variation in the switching frequency by using fix or adaptive hysteretic bands [26, 27, 28, 29]. To minimize harmonics injected into the grid, filters L, LC, LCL are frequently used. Even though LCL filter increases the number of components, it is effective at lower switching frequencies and reduces size and cost. In [30], the authors presents a control technique for gridconnected inverter using a LCL filter, to improve the system stability, control performance, and harmonic suppression. In [31], researcher perform a comparison between a L filter and a LCL filter. The L filter has a marginal greater average life; however the LCL filter offers the best total harmonic distortion supression.

This paper presents an analytical and numerical study of a hysteresis control to regulate the current injected to the grid by a single-phase dual-buck inverter. The implementation uses two inductors and a LC filter. The analysis includes a mathematical model of the inverter and proposes a hysteresis controller of the inductor current. This study also provides guidelines for the selection of the hysteresis band and proposes a methodology to compute the dead band required for the transition between the positive and negative cycles of the grid voltage. A numerical validation through simulation presents the performance of the control and the dead band. The results also show the ability of the controller to operate for different magnitudes of the injected current.

2. Dual-buck DC-AC converter

The open loop operation of the converter can be understood as two individual buck converters working half cycle each in a complementary fashion [22, 32, 33].

Figure 1 presents the schematics for an application where power from a generic DC source vB is used by the converter to inject active power into an AC grid vG defined as:

(1)

The solid state switches determine which converter is active through the control signals U_pe, U_pe ϵ {0, 1} in such a way that only one switch is active at a particular time (i.e. U_pe · U_ne= 0). This guarantees that L_p has positive current (i.e. i(t) > 0) on the positive semicycle of the sinusoidal reference, while L_nhas negative current (i.e. i(t) < 0) on the negative semicycle. Finally, the control signals U_p, U_n ϵ {0, 1} are available to perform the control task at a higher frequency. The dynamic equations for the converter can be written as follows:

(2)

with U_ne= 0, and U_pe = 1, and

(3)

with U_ne= 1, and U_pe = 0. The state variables are the current i in the inductance (L_p for the positive semicycle and L_nfor the negative semicycle), the voltage v_C in the capacitor C and the current i_F in the inductance L_F.

3. Hysteresis Control

ROL Hysteresis control with fixed or adaptive bands have been widely studied in grid connected inverters [34, 35, 36]. Here, we propose a control task that can be stated through the design of an error surface:

(4)

where i_r is a reference current defined in (5), I_rp is the magnitude of the current injected into the grid and its its value is obtained from a higher-level controller, and ωt is obtained from a Phase-Locked Loop (PLL) control system, which tracks the grid frequency and phase:

(5)

(5) The purpose of the control is to force the error in (4) to evolve within a hysteresis band of size H, define

Let us define the control strategy where the control signals U_p(t) and U_n(t) change value whenever e(t) reaches the boundaries of the hysteresis band η. This can be written as:

For simplicity, we will explain the rationale for the control strategy in the positive semi cycle taking advantage of the symmetries in the circuit for the negative semi cycle. Under the assumption that there is always a control action in the boundary, a suitable strategy would be to choose U_p ϵ {0, 1} for two possible scenarios.

• 𝑒(𝑡) = 𝐻. This implies that i(t) < i_r(t), and the control switch must be activated setting U_p=1.

• 𝑒(𝑡) = 𝐻. This implies that i(t) > i_r(t), and the control switch must be activated setting U_p=0.

0. Unfortunately, the strategy described above, does not guarantee that the error will remain in the region η for the whole positive semi-cycle. Indeed, at every zero crossing of the reference current i_r(t) the derivative di(t)/dt vanishes for a particular value of U_p(t) and U_n(t). This conclusion can be drawn by evaluating the first equation of (2) when i_r(t) ≈ 0 A and v_G(t) ≈ 0 V, which occurs at the same time because they are in phase. Therefore, the two possible values of di(t)/dt for U_p(t) = 1 or U_p(t) = 0 are shown in (6) and (6), respectively. Those equations indicate that i(t) is able to increase but not to decrease when i_r(t) ≈ 0 A and v_G(t) ≈ 0 V, as consequence e(t) does not evolve within the hysteresis band H.

(6)

Performing the same analysis for the negative semi-cycle, the values of di(t)/dt for U_n(t) = 1 and Un(t) = 0 are shown in (7) and (7), respectively.

(7)

The next subsection presents a complete mathematical analysis and design of a dead band near the zero-crossing of i_r(t).

3.1. Dead band design

As an analytical tool, we use the average model where the switching control actions U_pand U_ncan be described by an average model [37, 38], as follows:

where T can be defined as a complete cycle of the control action. By doing so, it is possible to use the first dynamic equation from (2) and (3) to obtain an averaged control as:

Where 𝑣̅_𝐶 = v_G(t) + R_F i_r(t). Doing the substitution we have:

Using (5) we obtain an exact expression for the average control action as:

where,

Finally, using trigonometric identities we have:

(8)

(9)

where

Now, it is possible to compute the size of a dead band near the zero crossing of v_G(T) using the angles φ_p and φ_n . Conditions in (8) and (9) show that U_p ϵ [0, 1] if ωt ϵ [0, π−φ_p] and U_n ∈ _[0, 1] if ωt ϵ [π, 2π−φ_p]. Figure 2 presents the computation of the average control actions U¯ p and U¯ n needed to achieve the tracking of a sinusoidal reference i_r(t) = sin(2π f_s t). It can be noticed that U_p < 0 for ωt ϵ [π − φ_p, π]. In the positive semi-cycle, the dead band is calculated as the grid voltage when ωt = π−φ_p as V_dbp = V_p sin(π − φ_p). Similarly, in the negative semicycle the dead band can be defined as the grid voltage when ωt = 2π − φ_n as V_dbn = V_p sin(2π − φ_n). However, |V_dbp| =|V_dbn| considering that v_G(t) is sinusoidal; hence, the proposed dead band is defined at both sides of the point v_G(t) = 0 V, i.e. the hysteresis controller does not operate when condition |V_G|< V_db is met, where V_db=|V_dbp| =|V_dbn| [39].

3.2. Maximum switching frequency

To determine the maximum switching frequency, we compute the shortest period T in the ripple of the error signal. Figure 3 shows conceptually the current. From (2), (4) and (5) it is possible to define the maximum slope of e(t) calculating its derivative term as shown in (10).

(10)

Then, the rising and falling slopes of e(t) are calculated from (10) assuming i(t) ≈ I_rp sin(ωt), v_c ≈ v_G(t) = V_psin(ωt) and the the value of U_p= {0, 1}. i.e. when U_p= 0 the rise slope is:

(11)

when U_p = 1, the falling slope is:

(12)

The variation of the period T with time can be calculated as the time required by the error to increase from −H to +H during the rising slope, plus the time required by the error to reduce from +H to −H during the falling slope, as shown in (13).

(13)

From the definition of F(t), it is clear that the switching frequency vary with the time according to v_G(t) and i_r(t). Figure 4 shows an example of the switching frequency of U_pduring the positive semi-cycle of v_G(t) and i_r(t) for the parameters given in Table 1. It can be observed that the F(t) has two equal maximums (F_max= 112.49 kHz) and the minimum value is near 0 Hz. The behavior of the switching frequency for U_nis symmetric. It is worth noticing that F_maxdepends on the selection of H. Therefore, it is important to analyze the variation of the maximum frequency F_maxfor different values of H. Figure 5 shows an exponential growth of F_maxfor variations of H. This is a useful tool to define H considering the switching characteristics of the semiconductor devices. A similar analysis is carried on in [40] to design the size of the inductor according with the maximum frequency.

3.3. Phase Lock Loop (PLL) implementation

A Phase Lock Loop simulink block to generate a normalized sinusoidal signal v_r(t) in phase with v_G(T), which is then used to generate the reference of the current control is presented in Figure 6. The PLL tracks the frequency and phase by using an internal frequency oscillator and a control system to adjusts the internal oscillator frequency to keep the phases of v_G(t) and v_r(t) close to zero [41].

Figure 7 shows the behavior of the PLL signal v_r(t) under the limits in voltage and frequency allowed for the California Electric Rule No.21 [42]. A 5 % reduction in the magnitude of v_G(t) is introduced in the second period (16 mS > t > 33 mS), in the third period a reduction from 60 Hz to 57 Hz in the grid frequency is performed (33 mS > t > 50 mS), and a 10 V and 1 kHz signal is added to v_G(t) to simulate noise during the all simulation. It can be observed that v_r(t) is in phase with the grid voltage even in the presence of noise and changes in v_G(t) magnitude and frequency.

4. Simulation results

The proposed hysteresis controller is implemented in Matlab/Simulink to evaluate its performance as shown in Figure 8. The circuital elements used to implement the gridconnected power converter are taken from Simscape/Power Systems library; while the hysteresis current controller is implemented by using just two comparators, an S-R flipflop, and a simple Matlab function that implements the dead band from v_G(t) and V_db.

The controller implementation in Simulink is illustrated in Figure 9 and can be divided into five blocks: "positive/negative enable", "U_pe/U_neactivation", "PLL", "error calculation", and "hysteresis control". The first block compares the grid voltage (v_G(t)) with a fixed value (V_db) to determine which Buck is active and which is inactive by defining values of U_pe and U_ne; therefore, if v_G(t) > V_db, then U_pe = 1 and U_ne= 0, which means that the positive Buck is active and the negative one is inactive. Otherwise, if v_G(t) < V_db, then U_pe = 0 and U_ne= 1. These signals (U_pe and U_ne) are used by "U_pe/U_ne activation" block to generate the boolean signals U_pe and U_nerequired to activate the Mosfets S _peand S _ne, respectively. Those boolean signals are generated with the AND blocks.

The PLL block generates a normalized sinusoidal waveform (v_r(t)) in phase with v_G(t), which is multiplied by the scalar Irp to generate i_r(t). The Error calculation block evaluates the instantaneous error e(t) in the current i(t) with respect to the reference (i_r(t)), which is used by the Hysteresis control block. This block compares e(t) with the hysteresis band H to determine if the high frequency switch turns on or off.

On the one hand, for the Buck that operates when v_G(t) > V_db, the hysteresis control operates as follows: if e(t) > H, then flip-flop is set to turn the switch on, and if e(t) < −H, then flipflop is reset to turn the switch off. On the other hand, for the Buck converter that operates when v_G(t) < V_db the logic is the opposite. Therefore, it is enough to use the complement output of the flip-flop. At the output of the hysteresis control block, the AND functions enable the switching signal for the positive or negative Buck converter, by using U_pe and U_ne. Additionally, they also generate the Boolean signals U_pand U_nrequired to activate the high-frequency switches.

4.1. Control validation

The validation of the proposed controller is performed by generating a reference current i_r(t) with different peak values: I_rp = 1.0 A, I_rp = 1.5 A and I_rp = 2.0 A. Moreover, the parameters used in the simulations are introduced in Table 1. The main inverter variables for I_rp = 1.0 A are shown in Figure 10, where the top plot shows that i(t) follows the reference i_r(t) for the positive and negative semi-cycles. Such a plot also illustrates that the average value of the current injected to the grid (i_F(t)) follows i_r(t).

The second plot introduces the capacitor voltage (v_C(t)) and the grid voltage (v_G(t)). It can be observed that the average voltage of v_C(t) follows v_G(t) for the positive and negative semi-cycles. Additionally, the third plot shows the switching signals U_p (green) and U_n(blue) that operate in the positive and negative semi-cycles, respectively. In this plot, it can be clearly identified the dead-band for the transition between the activation of one converter and the other in the middle of the voltage cycle. The last plot in Figure 10 introduces the error in the current e(t) and the hysteresis band defined as H = ±60 mA.

A zoom in of the variables introduced in Figure 10 is shown in Figure 11 to observe the behavior of the hysteresis controller and the transition between the positive and the negative Buck.

The dashed red lines illustrate the dead-band around v_G(t) = 0 V (t = 8.33 ms approximately). At the left of the dead-band the hysteresis controller turns a switch on (i.e. U_p= 1) when the error reaches the upper limit of the hysteresis band (e(t) = 60 _mA), which means that i(t) is below i_r(t) and i(t) increases with U_p = 1. Then, i(t) continues increasing until the error reaches the lower limit (e(t) = −60 mA) and the S _pis turned off to keep e(t) within the hysteresis band. At the right of the dead-band, the controller operation is the opposite to the one described in the previous paragraph. When e(t) = 60 mA the controller turns S _noff to increase i(t); while when e(t) = −60 mA the controller turns S _non.

Moreover, inside the dead band all the switches are off (i.e. u_p= u_n = u_pe = u_ne = 0); therefore, i(t) = 0 A and e(t) remains within the hysteresis band. The proposed controller is also validated for step variations in the reference current as follows: I_rp = 1 for 0 ms < t < 12.5 ms, I_rp = 1.5 for 12.5 ms < t < 29.2 ms, and I_rp = 2 for 29.2 ms < t < 45.8 ms. These particular times are selected to produce the step in I_rp at the minimum current. The main inverter variables for the steps in I_rp are introduced in Figure 12.

It can be observed that i(t) follows i_r(t) even for steps in the amplitude of i_r(t); nevertheless, the ripple in i_F(t) considerably increases after each step in I_rp, which is also evidenced in the ripple of v_C(t). Additionally, it is important to mention that e(t) remains within the hysteresis band most of the time. The error leaves the hysteresis band when there are steps in I_rp and in the zero crossings; Nonetheless, it is worth noting that the controller brings e(t) inside the hysteresis band in a short time.

5. Conclusions

We have presented the hysteresis control of a dual-buck inverter for active power injection in a grid connected application. Using the dynamic equations for the error e(t), we have designed a control strategy aiming to evolve in a set η ∈ [−H, H]. Analytical conditions have been computed for the width of a dead band near the zero crossings of the AC reference i_r(t). We have proposed an expression to compute an approximation of the instantaneous switching frequency and a numerical procedure to design the size of the hysteretic band H. In addition, numerical experiments for validation on the capacity of the control to synchronize with the grid, including perturbations in the voltage magnitude and frequency. Further simulations have shown, the performance of the dead-band and the hysteresis band to obtain a desired maximum switching frequency.

The control law proposed in this paper is easy to implement and does not require the exact model of the entire system, which make the controller suitable for different applications like photovoltaic systems, wind generators, battery chargers, UPSs, among others. The controller considers the nonlinear model of the inverter, which allows the tracking of the current reference for any operating condition. Nevertheless, the control does not warranty a correct current tracking during the dead band, which could produce reference tracking errors if there are disturbances during the dead-band. Another drawback related to the inverter topology is that the dead-band does not allow the injection of reactive power to the grid, since the phase difference between i_r(t) and v_G(t) increases the dead-band size. Additionally, the digital implementation requires the use of specialized microcontrollers with fast comparators, since the switching frequency varies exponentially reaching 500 kHz for H = 10 mA for the converter parameters used in simulations.

As future work, we propose the experimental validation of the controller and the spectral analysis of the injected power. More sophisticated control strategies might be developed using the analytical tools presented in this paper. It would be also interesting to include the proposed inverter and controller into a microgrid to analyze the performance of the proposed system in coordination with the microgrid control and the power quality injected in the connection point. Finally, we are working on the modification of the proposed controller and the inverter topology to inject reactive power.

References

Additional information

How to cite: M.A. Bolaños-Navarrete, J.D. Bastidas-Rodríguez, G. Osorio, “Hysteresis control for a grid connected dual-buck inverter,” Rev. UIS Ing., vol. 20, no. 1, pp. 1-10, 2021, doi: 10.18273/revuin.v20n1-2021001

Revista UIS Ingenierías
ISSN: 1657-4583
Vol. 20
Num. 1
Año. 2021

Hysteresis control for a grid connected dual-buck inverter

Mario AndrésJuan DavidGustavo Bolaños-NavarreteBastidas-RodríguezOsorio
Universidad Nacional de ColombiaUniversidad Nacional de ColombiaUniversidad Nacional de Colombia,ColombiaColombiaColombia