This paper presents a nonlinear method to control a DCDC 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 buckboost 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 buckboost 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 DCDC converter at the PV module output, as shown in
Fig. 1
.
Structure of an alone PV system.
An adequate control of the digital switch of the DCDC converter allows the converter voltage input,
v_{PV}
, to set at desired value to get the MPP Tracking (MPPT). The converter output voltage,
v_{O}
, will be supplied to the connected load. In gridconnected systems, a DCAC 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 DCDC converter,
[2]
,
[3]
. In this paper, a buckboost 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 wellknown 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 buckboost 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 offline 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, timevariant nature and variable structure of the buckboost. 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 buckboost 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 buckboost 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 DCDC 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 buckboost 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 DCDC 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/m
^{2}
and 25 ℃.
ELECTRICAL PARAMETERS OF SOLAR MODULE
ELECTRICAL 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,
i_{1}
is the current source in A (it depends on the irradiance and temperature),
i_{0}
is the cell reverse saturation current in A,
D_{1}
is an antiparallel diode,
R_{sh}
is the shunt electrical resistor, and
R_{s}
is a series resistor. The resistors model the module power loss. This circuit is used to model the solar module in MatlabSimulink, according to Vazquez et al.
[10]
.
Equivalent circuit of a PV cell.
Fig. 3
shows the IV and PV 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.
Comparison between the IV and PV 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 IV and PV 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 smallscale, and greater power solar arrays can be analyzed with series and shunt connection of this model.
 B. BuckBoost Converter
This DCDC 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 nonlinear load due to the transistor and the diode.
Topology of a buckboost converter.
In
Fig. 4
,
v_{PV}
is the PV output voltage in V,
i_{PV}
is the solar array output current in A,
i_{L}
is the inductor current in A. and
v_{0}
is the buckboost converter output voltage in V.
L
is inductor in H, and
C_{1}
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 buckboost converter duty cycle,
D
, so as to obtain the voltage that makes the power maximum. The duty cycle is t
_{ON}
/ t
_{C}
, with t
_{ON}
being the time which the switch is ON, and t
_{C}
is the switching period (0<
D
<1). The output/input voltage ratio is calculated as it is shown in (1).
The DCDC 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 buckboost 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. BUCKBOOST CONVERTER CONTROL
The purpose of the designed control is to regulate the DCDC converter voltage by means of a backstepping method, adjusting the PV array output reference voltage initially given by an offline 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/m
^{2}
to 1200 W/m
^{2}
, 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 IV and PV 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 nonlinear backstepping controller is designed to control the duty cycle of the buckboost 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
v_{PV}^{r}
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
v_{PV}
to track the reference voltage
v_{PV}^{r}
. Thus, the objective is to achieve zero tracking error.
By derivating
e
_{1}
with respect to time and accounting for (2), (6) is obtained, where
i_{L}
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 k
_{1}
is constant and positive. This way, the reference current for the control,
α_{1}
, the socalled stabilization function, can be obtained working out the value of the
i_{L}
from (8).
Now, the behavior of the current error is studied,
z_{1}
=
i_{L}

α_{1}
, where the inductor current should reach
α_{1}
to make the error vanish to achieve the control objective. The time derivative of this error is shown in (10).
The time derivative of
α_{1}
, (10), replacing
i_{L}
by
z_{1}
+
α_{1}
, yields (11).
(11) with (2) gives the time derivative of
z_{1}
, as it is shown in (12).
Similar to what it is done in
V_{1}
, another Lyapunov function is defined with the same characteristics, being (13).
Its time derivative is (14), accounting for (6) and (12), and replacing
i_{L}
by
z_{1}
+
α_{1}
.
(14) will be negative when k
_{2}
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 –
k_{2}z_{1}
. Thus,
Ḋ
is worked out, yielding (15), where 0 <
D
< 1 and
α_{1}
≠ 0.
The parameters design has been achieved empirically, and the used values, k
_{1}
and k
_{2}
, are presented in
table II
.
ELECTRICAL AND CONTROL PARAMETERS
ELECTRICAL 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 DCDC 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
i_{PV}
and
v_{PV}
, 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 DCDC converter.
Control scheme of the DCDC 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 buckboost 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 DCDC converter input, and the buckboost converter output is connected to the resistive load. The buckboost converter input voltage range is 10 V – 40 V, and the maximum DCDC converter output voltage is 80 V. The maximum power of the buckboost converter is 50 W, and the output voltage ripple is 0.5% of
v_{o}
. The MOSFET used in the buckboost converter is CS
D_{1}
9536KCS, driven by a FOD3180 driver, and the diode is the MBR10200.
Experimental system supervised by PC.
Fig. 7
shows some details of the DCDC converter box. The current and voltage values are measured using LEM sensors, LA25NP (on the top left corner) and LV25P (on the bottom left corner). The control program runs in a lowcost microcontroller, a dsPIC30F3013 controller (at the bottom centre), to control the DCDC 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%.
Sensor, dsPIC and buckboost 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 WiFi. For that, Visual Basic programming was used.
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.
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
k_{1}
and
k_{2}
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 buckboost 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
k_{1}
and
k_{2}
are the parameters used in backstepping method, in (15).
L
is the inductor of the DCDC converter, and
C
and
C_{1}
are the buckboost 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/m
^{2}
, and the average temperature is 38 ℃.
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 W
_{p}
in this case.
Fig. 11
shows the PV output current,
i_{PV}
, obtained in this case, and the buckboost converter input voltage and the reference voltage that must be reached to obtain the MPP.
DCDC 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 buckboost converter input and output power. Thus, the power converter efficiency, defined as the percentage achieved by dividing the DCDC converter output power by the buckboost input power, is about 90%.
DCDC converter input and output power.
Finally, the control signal is depicted in
Fig. 13
, where the percentage of the buckboost converter duty cycle is shown.
DCDC 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/m
^{2}
until 38 s, when it changes to 305 W/m
^{2}
approximately. Then, the irradiance changes its value at 92 s to 840 W/m
^{2}
again, as it is shown in
Fig. 14
.
Changeable irradiance.
Fig. 15
shows the PV output current,
i_{PV}
, 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.
DCDC converter input and reference voltage and PV output current.
The DCDC converter input and output power are shown in
Fig. 16
. Regarding the characteristics curves of the solar module, the maximum power for 840 W/m
^{2}
and 38 ℃ is 18.5 W, whereas the maximum power is 5.6 W when the irradiance is 305 W/m
^{2}
and the temperature is 39.5 ℃, as in this case.
DCDC 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/m
^{2}
, and of about 99% when the irradiance is 305 W/m
^{2}
. Moreover, the input and output power are stabilized after a smooth transient response. The duty cycle can be seen in
Fig. 17
. The buckboost converter settling time when the duty cycle is changed is 100 ms.
DCDC converter control signal.
Finally, a performance comparative between the backstepping control and the wellknown P&O is shown to prove the validity of the proposed control.
Fig. 18
presents DCDC converter input and output power obtained with both methods.
DCDC 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/m
^{2}
to 700 W/m
^{2}
) 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 buckboost 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 offline calculated regression plane under different temperature and irradiance conditions. The online control includes a modifying P&O method and a backstepping controller which calculates the duty cycle of the DCDC 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%.
Acknowledgements
This work was supported by the Ministerio de Educación, Cultura y Deporte of Spain under FPU grant (Formación de Profesorado Universitario).
BIO
Jesus R. Vazquez was born in Huelva, Spain, on December 24. He received the degree in electrical engineering from the University of Seville, Spain in 1995. He obtained the Ph. D. degree in 2004. For one year, he was with the electrical department of Nissan Motor Ibérica S.A., Barcelona, Spain. Since 1996, he has been with the Electrical Engineering Department at the Escuela Técnica Superior de Ingeniería of the University of Huelva. He teaches Electric Circuits, Power Quality and Photovoltaic Systems, and his research interests include power quality, active power filters, renewable energy, distributed generation, and artificial network application.
Aranzazu D. Martin was born in Bollullos del Condado, Huelva, Spain. She received the degree in Industrial Engineering from the University of Huelva, Spain in 2008. She obtained her master’s degree in Control Engineering, Electronic Systems and Industrial Computer Science in 2011. Since 2008, she has been with the Electrical Engineering Department at University of Huelva, working toward her Ph. D. She is currently working as a researcher with the FPU (Formación de Profesorado Universitario) grant. Currently, her research interests include renewable energy, control systems, distributed generation and power quality.
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