Backstepping Control of a Buck-Boost Converter in an Experimental PV-System

Abstract

This paper presents a nonlinear method to control a DC-DC converter and track the Maximum Power Point (MPP) of a Photovoltaic (PV) system. A backstepping controller is proposed to regulate the voltage at the input of a buck-boost converter by means of Lyapunov functions. To make the control initially faster and avoid local maximum, a regression plane is used to estimate the reference voltages that must be obtained to achieve the MPP and guarantee the maximum power extraction, modifying the conventional Perturb and Observe (P&O) method. An experimental platform has been designed to verify the validity and performance of the proposed control method. In this platform, a buck-boost converter has been built to extract the maximum power of commercial solar modules under different environmental conditions.

I. INTRODUCTION

The PV energy is used to power many autonomous devices and isolated houses, as well as to produce electricity on a large scale through distribution networks. A PV module generates electrical current from the solar irradiance, and the generated power is only maximum for an output voltage under changeable environmental conditions, [1].

In order to reach the Maximum Power Point, the control of this PV output voltage is usually achieved connecting a DC-DC converter at the PV module output, as shown in Fig. 1.

Fig. 1.Structure of an alone PV system.

An adequate control of the digital switch of the DC-DC converter allows the converter voltage input, vPV, to set at desired value to get the MPP Tracking (MPPT). The converter output voltage, vO, will be supplied to the connected load. In grid-connected systems, a DC-AC converter is used to obtain a sinusoidal current to supply to the load or to inject it into the electrical network.

There are different topologies of the DC-DC converter, [2], [3]. In this paper, a buck-boost topology has been designed, which converts the DC power from one voltage level to another higher or lower according to the needs.

The MPPT can be implemented through different control algorithms in order to obtain the maximum power under all conditions, [4]. Many methods have been used. Some of them are based on the well-known principle of Perturb and Observe (P&O) [5]-[7], on Sliding Mode control method [8], Ripple Correlation Control (RCC) [9], Artificial Neuronal Networks or Fuzzy based algorithms [10]-[12], amongst others. The methods have different accuracy and complexity. Some of them can obtain local maximum instead of global maximum, and others have involved structures.

A new control method for MPPT of PV arrays using a buck-boost converter is proposed in this paper. A nonlinear backstepping controller [13], [14] has been designed to track the maximum power point with the help of an off-line calculated regression plane. This plane provides the PV array output reference voltages for different irradiance and temperature values using a modified P&O method. Thus, the MPP tracking is initially faster, a desirable goal of the control [15]. The robustness is increased, global asymptotic stability is guaranteed by means of Lyapunov functions, and the MPP can be ensured even with changeable conditions. A nonlinear control has been chosen due to the nonlinear, time-variant nature and variable structure of the buck-boost. Thus a linear control implies a model linearization that is simple, but cannot control the converter in a wide range.

An experimental platform has been designed to check the real performance of the proposed control. Some papers describe different experimental platform using acquisition & control commercial boards [16]-[18]. In this case, a PV module supplies power to a DC load through a buck-boost converter, and the backstepping controller has been implemented in a low cost microcontroller.

The paper is organized as follows. Section II presents the PV system, including the used PV array model to calculate the regression plane with the reference voltages and buck-boost converter. The proposed design of the control to make the system track the maximum power point is developed in Section III. The experimental platform and the different practical results will be presented in Section IV. Thus, the stationary and transient performance of the designed control will be checked. Finally, some conclusions will be described in Section V.

 

II. PHOTOVOLTAIC SYSTEM

The PV system includes a PV array (the solar generator) and a DC-DC converter, as shown in Fig. 1. Now the solar module is presented as well as its simulation model in order to obtain a regression plane with the desired PV output voltage under different environmental conditions. This plane will be used as initial reference for the control system. Moreover, the power block of the buck-boost converter is explained.

A. PV Array

First, the features of the solar module are detailed. In this work, a commercial solar module is used in the experimental analysis to test the DC-DC converter control. The maximum power of this photovoltaic module is 20 W, and its main electrical parameters are described in Table I for standard conditions, 1000 W/m2 and 25 ℃.

TABLE IELECTRICAL PARAMETERS OF SOLAR MODULE

The equivalent circuit which models the solar cell is presented in Fig. 2, where i is the solar cell output current in A, v is the solar cell output voltage, in V, i1 is the current source in A (it depends on the irradiance and temperature), i0 is the cell reverse saturation current in A, D1 is an anti-parallel diode, Rsh is the shunt electrical resistor, and Rs is a series resistor. The resistors model the module power loss. This circuit is used to model the solar module in Matlab-Simulink, according to Vazquez et al. [10].

Fig. 2.Equivalent circuit of a PV cell.

Fig. 3 shows the I-V and P-V curves of the simulated PV model and the characteristic curves of the used PV module in the experimental platform to prove the accuracy between them under different environmental conditions.

Fig. 3.Comparison between the I-V and P-V curves of the simulated PV model and the used PV module.

In order to make the control initially faster, a regression plane is achieved from I-V and P-V characteristic curves of the solar modules to obtain the theoretical voltage that supplies the maximum power, and a modified P&O is implemented to reach a reference practical voltage around the theoretical one and to get the maximum energy extraction. For that, the PV module was exposed to the sun under changeable irradiance and temperature to obtain the characteristic curves to know the peak of the curve under different environmental conditions. After that, by using the electrical parameters of the PV module, the PV array is modeled to obtain the regression plane. The simulation model was matched with the real module; thus, the characteristic curves obtained from the simulation and the laboratory measured real curves yield the same results, that is to say the same maximum power for each curve. Thus, the regression plane is calculated properly.

The solar module detailed in Table I has been considered to test the proposed control in the next section. The experimental platform has been designed for small-scale, and greater power solar arrays can be analyzed with series and shunt connection of this model.

B. Buck-Boost Converter

This DC-DC converter consists of power electronic components such as capacitors, an inductor, a transistor and a diode connected, as shown in Fig. 4, where its topology is presented. This converter behaves as a non-linear load due to the transistor and the diode.

Fig. 4.Topology of a buck-boost converter.

In Fig. 4, vPV is the PV output voltage in V, iPV is the solar array output current in A, iL is the inductor current in A. and v0 is the buck-boost converter output voltage in V. L is inductor in H, and C1 and C are capacitors in F and are constant parameters.

A Pulse Width Modulation (PWM) method is used to control the commutation of the transistor, allowing the energy to charge and discharge in the storage elements. The output voltage has opposite polarity to the input voltage, and this converter topology can supply a greater or lower voltage than the input voltage.

The main purpose of the proposed backstepping control is to regulate the PV output voltage modifying the buck-boost converter duty cycle, D, so as to obtain the voltage that makes the power maximum. The duty cycle is tON / tC, with tON being the time which the switch is ON, and tC is the switching period (0

The DC-DC converter can work under two working modes, Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM), depending on the inductor current in the operation period. In this work, the buck-boost converter works in CCM. Therefore, the inductor current is never zero.

Using the state averaging method [19], the equations of the converter model are defined in (2), (3) and (4).

 

III. BUCK-BOOST CONVERTER CONTROL

The purpose of the designed control is to regulate the DC-DC converter voltage by means of a backstepping method, adjusting the PV array output reference voltage initially given by an off-line calculated regression plane. Therefore, the maximum power extraction of the solar modules is guaranteed.

A. Reference Voltage

In order to make the control initially faster and avoid local maximum, a regression plane provides the theoretical reference voltage required to achieve the MPP under any conditions of temperature and irradiance that ranges from 0 ℃ to 80 ℃ and from 200 W/m2 to 1200 W/m2, respectively. For that, the PV module used in the experimental platform has to be modeled to obtain the characteristic curves and to calculate the regression plane by linear interpolation for different environmental conditions. Thus, a voltage matrix is achieved which includes the reference voltages that supply the maximum power and, as a consequence, a maximum power matrix can be obtained as well, depending on the temperature and the irradiance.

Some tests have been developed in the lab to check the similarity between the simulated model and a real module. The I-V and P-V curves in both cases are the same under specific values of temperature and irradiance; thus, the solar cell is modeled correctly, as shown in Fig. 3.

Once the theoretical reference voltage is obtained, a practical reference voltage is proposed in this work. For that, by taking into account the PV output voltage and the PV output current, a modified P&O is implemented to obtain an incremental value of reference voltage instead of the duty cycle value. The addition of the theoretical reference voltage and the incremental value gives the practical reference voltage used in the backstepping control as it is described below.

B. Backstepping Controller

The purpose of this control is to regulate the converter input voltage to extract the PV maximum power, taking into account the practical reference voltage mentioned above. A non-linear backstepping controller is designed to control the duty cycle of the buck-boost converter switch to regulate the PV output voltage. Thus, the optimum voltage will be obtained, modifying the voltage around its reference. This type of control is used to design stable controls with a recursive methodology. It must stabilize the origin of a system by means of feedback control laws and using Lyapunov functions to prove the stability of the system. In order to design the controller, the next steps are followed:

The voltage tracking error is defined as it is shown in (5),

where vPVr is the reference output voltage of the PV modules, and it must be reached by the control. This error is defined to enforce the PV output voltage vPV to track the reference voltage vPVr . Thus, the objective is to achieve zero tracking error.

By derivating e1 with respect to time and accounting for (2), (6) is obtained, where iL behaves as a virtual control input.

A Lyapunov function is selected. It must be positive definite and radially unbounded for all t, and the time derivative of Lyapunov function must be negative definite for all t to ensure the solution is locally asymptotically stable. The chosen function and its derivate are defined as below.

will be negative if k1 is constant and positive. This way, the reference current for the control, α1, the so-called stabilization function, can be obtained working out the value of the iL from (8).

Now, the behavior of the current error is studied, z1 = iL - α1, where the inductor current should reach α1 to make the error vanish to achieve the control objective. The time derivative of this error is shown in (10).

The time derivative of α1, (10), replacing iL by z1 + α1, yields (11).

(11) with (2) gives the time derivative of z1, as it is shown in (12).

Similar to what it is done in V1, another Lyapunov function is defined with the same characteristics, being (13).

Its time derivative is (14), accounting for (6) and (12), and replacing iL by z1 + α1.

(14) will be negative when k2 is positive, being a constant, to ensure the stability of the system. Therefore, the term between square brackets must be zero. From (14), the duty cycle derivative must be worked out from the term between square brackets, this term being equal to –k2z1. Thus, Ḋ is worked out, yielding (15), where 0 < D < 1 and α1 ≠ 0.

The parameters design has been achieved empirically, and the used values, k1 and k2 , are presented in table II.

TABLE IIELECTRICAL AND CONTROL PARAMETERS

The proposed control will be tested in Section IV in an experimental platform. (15) will be implemented, and the appropriate performance of the backstepping controller will be checked.

Fig. 5 shows the DC-DC converter control scheme. On the one hand, with the regression plane and T and G values, the theoretical reference voltage is obtained. On the other hand, by taking into account iPV and vPV, a modified P&O method is applied to provide an incremental value of reference voltage instead of the duty cycle value. The final value of the reference voltage is used in the backstepping control. Thus, (15) is implemented to obtain the duty cycle to control the DC-DC converter.

Fig. 5.Control scheme of the DC-DC converter.

Next section will detail the experimental implementation of the proposed control.

 

IV. MEASUREMENT RESULTS

A. Experimental Platform

The experimental platform consists of a commercial solar array (PV module), a buck-boost converter with the dsPIC and the voltage and current sensors in a box, a resistive load, and a PC to supervise the control parameters, as it is shown in Fig. 6. The PV array output is connected to the DC-DC converter input, and the buck-boost converter output is connected to the resistive load. The buck-boost converter input voltage range is 10 V – 40 V, and the maximum DC-DC converter output voltage is 80 V. The maximum power of the buck-boost converter is 50 W, and the output voltage ripple is 0.5% of vo. The MOSFET used in the buck-boost converter is CSD19536KCS, driven by a FOD3180 driver, and the diode is the MBR10200.

Fig. 6.Experimental system supervised by PC.

Fig. 7 shows some details of the DC-DC converter box. The current and voltage values are measured using LEM sensors, LA25-NP (on the top left corner) and LV25-P (on the bottom left corner). The control program runs in a low-cost microcontroller, a dsPIC30F3013 controller (at the bottom centre), to control the DC-DC converter duty cycle of the power block (on the right side of the figure) to regulate the voltage. LM35 temperature sensor and a commercial irradiance meter (a compensated calibrated cell) are used. This cell provides a solar radiation with an accuracy of ±5%.

Fig. 7.Sensor, dsPIC and buck-boost converter.

Moreover, a PC is connected to the PIC to supervise the control performance and all the parameters required. Fig. 8 presents a developed virtual instrument to supervise the tests via Wi-Fi. For that, Visual Basic programming was used.

Fig. 8.Virtual instrument to supervise the control.

Fig. 9 depicts a flow chart in order to summarize the design and implementation of the backstepping controller in the dsPIC. The sampling time for the control loop is 10 ms, and the PWM frequency is 25 kHz with a 10 bits PWM resolution. Besides, the duty cycle change rate is 10 ms, and it is updated with the sampling time.

Fig. 9.Backstepping implementation flow chart.

The integrated development environment (IDE) is MPLAB and the programming language is C. As shown in Fig. 9, the PWM frequency and k1 and k2 constant values are defined. Then, the analogical digital converter reads the value of the PV module output current and voltage, the inductor current, the temperature, and the irradiance. After that, the regression plane and the modified P&O algorithm are applied to obtain the reference voltage that should be reached to achieve the maximum power. Once the reference voltage is known, the backstepping control calculates the time derivative of the buck-boost converter duty cycle, and then this value is integrated. Finally, the PWM is updated in each control loop.

B. Results

The proposed control has been implemented in the experimental platform detailed above to test its performance under changeable environmental conditions, such as a change in irradiance and consequently, in temperature. After some laboratory tests have been performed to obtain the regression plane described in this paper, the practical cases have been developed outdoors, under solar irradiation. Thus, the robustness of the system is evaluated.

The backstepping control parameters and inductor, the capacitors, and the resistive load values are presented in TableII. Constants k1 and k2 are the parameters used in backstepping method, in (15). L is the inductor of the DC-DC converter, and C and C1 are the buck-boost converter output and input capacitors, as shown in Fig. 4.

1) Case 1 - Stationary Analysis: In this case, the proposed control has been tested to check the stationary response under constant irradiance and temperature, although their values fluctuate due to the environmental conditions. As it is shown in Fig. 10, the average irradiance is 835 W/m2, and the average temperature is 38 ℃.

Fig. 10.Irradiance and temperature evolution.

For these values of irradiance and temperature, the theoretical maximum power of the tested module is 18,5 W, about the 90 % of the module peak power, or 20 Wp in this case.

Fig. 11 shows the PV output current, iPV, obtained in this case, and the buck-boost converter input voltage and the reference voltage that must be reached to obtain the MPP.

Fig. 11.DC-DC converter input voltage and reference voltage and PV output current.

The reference voltage tracking efficiency, which is obtained by dividing the obtained voltage that supplies the maximum power by the reference voltage, is greater than 99%. Fig. 12 shows the buck-boost converter input and output power. Thus, the power converter efficiency, defined as the percentage achieved by dividing the DC-DC converter output power by the buck-boost input power, is about 90%.

Fig. 12.DC-DC converter input and output power.

Finally, the control signal is depicted in Fig. 13, where the percentage of the buck-boost converter duty cycle is shown.

Fig. 13.DC-DC converter control signal.

2) Case 2 - Transient Analysis: In this case, the solar module was exposed to the sun during the two minutes in which the irradiance changed. The value of the irradiance is about 840 W/m2 until 38 s, when it changes to 305 W/m2 approximately. Then, the irradiance changes its value at 92 s to 840 W/m2 again, as it is shown in Fig. 14.

Fig. 14.Changeable irradiance.

Fig. 15 shows the PV output current, iPV, obtained when there is a change in the irradiance. Aside from that, it depicts the PV output voltage and the reference voltage that must be tracked to achieve the maximum power point.

Fig. 15.DC-DC converter input and reference voltage and PV output current.

The DC-DC converter input and output power are shown in Fig. 16. Regarding the characteristics curves of the solar module, the maximum power for 840 W/m2 and 38 ℃ is 18.5 W, whereas the maximum power is 5.6 W when the irradiance is 305 W/m2 and the temperature is 39.5 ℃, as in this case.

Fig. 16.DC-DC converter input and output power.

Therefore, the maximum power extraction is always achieved with a performance of about 90%, when the irradiance is 840 W/m2, and of about 99% when the irradiance is 305 W/m2. Moreover, the input and output power are stabilized after a smooth transient response. The duty cycle can be seen in Fig. 17. The buck-boost converter settling time when the duty cycle is changed is 100 ms.

Fig. 17.DC-DC converter control signal.

Finally, a performance comparative between the backstepping control and the well-known P&O is shown to prove the validity of the proposed control. Fig. 18 presents DC-DC converter input and output power obtained with both methods.

Fig. 18.DC-DC converter input and output power obtained with backstepping controller and P&O algorithm.

The P&O algorithm has an oscillatory behavior, and it achieves a tracking efficiency of 96.1%, whereas the backstepping control does not oscillate, and obtains a tracking efficiency of 99%. Regarding the tracking time, the power is stabilized after 3.3 seconds under a change in the irradiance (from 400 W/m2 to 700 W/m2) when the backstepping control is used. When the P&O control is used under the same conditions of irradiance, the power is stabilized after 8.4 seconds.

 

V. CONCLUSION

In this paper, a nonlinear backstepping controller has been designed to control a buck-boost converter in a photovoltaic system. The control aim is to regulate the PV array output voltage in order to track the Maximum Power Point.

The proposed control includes an initial estimation of MPP using a simulation PV model and an off-line calculated regression plane under different temperature and irradiance conditions. The on-line control includes a modifying P&O method and a backstepping controller which calculates the duty cycle of the DC-DC converter switch device.

An experimental platform has been designed to verify the validity and performance of the proposed control method. The results confirm that the control works correctly because the reference voltage is always obtained for any environmental condition, including stationary and transient situations.

The control efficiency ratio or the tracking efficiency is greater than 99%, and the efficiency of power converter block is about 90%.