# A New Sliding Mode Controller for DC/DC Converters in Photovoltaic Systems

Authors: M. Sarvi; I. Soltani; N. NamazyPour; N. Rabbani
Journal: Journal of Energy
 (2013)

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## Abstract

DC/DC converters are widely used in many industrial and electrical systems. As DC/DC converters are nonlinear and time-variant systems, the application of linear control techniques for the control of these converters is not suitable. In this paper, a new sliding mode controller is proposed as the indirect control method and compared to a simple direct control method in order to control a buck converter in photovoltaic applications. The solar arrays are dependent power sources with nonlinear voltage-current characteristics under different environmental conditions (insolation and temperature). From this point of view, the DC/DC converter is particularly suitable for the application of the sliding mode control in photovoltaic application, because of its controllable states. Simulations are performed in Matlab/Simulink software. The simulation results are presented for a step change in reference voltage and input voltage as well as step load variations. The simulations results of proposed method are compared with the conventional PID controller. The results show the good performance of the proposed sliding mode controller. The proposed method can be used for the other DC/DC converter.


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## Body

## 1. Introduction

Modern electronic systems require high-quality, small, lightweight, reliable, and efficient power supplies. So, the DC/DC converters are widely used in many industrial and electrical systems. The most familiar are switching power supplies, DC drives, and photovoltaic systems. The stability is an important aspect in the design of switch mode power supplies; a feedback control is used to achieve the required performance. Ideally the circuit is in steady state, but actually the circuit is affected by line and load variations (disturbances), as well as variation of the circuit component (robustness). These parameters have a severe effect on the behavior of switch mode power supply and may cause instability. Design of controller for these converters is a major concern in power converters design [1–3].Different control techniques are applied to regulate the DC-DC converters, especially buck converters, in order to obtain a robust output voltage [4, 5].As DC/DC converters are nonlinear and time-variant systems, the application of linear control techniques for the control of these converters is not suitable. In order to design linear control system using classical linear control techniques, the small signal model is derived by the linearization around a precise operating point from the state space average model [4]. The controllers based on these techniques are simple to implement; however, it is difficult to account the variation of system parameters, because of the dependence of small signal model parameters on the converter operating point [6].Sliding mode control is a well-known discontinuous feedback control technique which has been exhaustively explored in many books and journal articles. The technique is naturally suited for the regulation of switched controlled systems, such as power electronics devices, in general, and DC/DC power converters, in particular [2]. Many sliding mode controllers have been proposed and used for DC/DC converters [6–9]. These controllers are direct [6] or indirect control method [6, 7]. The direct method is proposed in [6]. In [7], the output capacitor current of DC/DC converter is used to control the output voltage. The differences of the DC/DC output voltage and the reference voltage enter the proportional-Integrator (PI) type controller, and then the output capacitor current of DC/DC converter is decreased from the output of controller [8]. The output voltage and inductor current are used to control of DC/DC converter in [9]. These references [6–9] have not completely investigated the load and line as well as reference regulations.In this paper, a new sliding mode controller is introduced for DC/DC buck converter as the indirect control method. The proposed controller is compared with a simple direct control method as well as the conventional PID controller. The simulation results are presented for a step change in reference voltage and input voltage as well as for a step load variation. The main contribution of this paper is the presentation of a new indirect sliding mode controller with good accuracy and performance against load and line as well as reference regulations.


## 2. DC/DC Converters

The DC-DC converters can be divided into two main types: (1) hard-switching pulse width modulated (PWM) converters and (2) resonant and soft-switching converters [1].Advantages of PWM converters include low component count, high efficiency, constant frequency operation, relatively simple control and commercial availability of integrated circuit controllers, and ability to achieve high conversion ratios for both step-down and step-up applications. The circuit diagram of the DC/DC buck converter is shown in Figure1. In this figure, the circuit schematic is depicted with the transistor-diode symbols.Figure 1
Semiconductor realization of the DC/DC buck converter.By sensing of the DC output and controlling of the switch duty cycle in a negative-feedback loop, the DC output voltage could be regulated against input line and output load changes [3].


## 3. Solar Array Characteristic

The solar arrays have nonlinearV-I characteristics which depend on the environmental conditions: ambient temperature and insolation.The nonlinear characteristic of a solar sell is obtained as the following equation [10, 11]:
(1)iSA=Iph-Io{exp((qAKT)·(vSA+RSiSA))-1},
where Iph is the generated current under a given insolation condition, Io is the reverse saturation current, q is the charge of an electron, vSA and iSA are the output voltage and current of the solar cell, respectively, A is the ideality factor for a p-n junction, K is Boltzmann’s constant, T is the temperature, and RS is the series resistance of the solar cell. Figures 2 and 3 indicate the power-current (P-I) characteristic of a solar cell for different insolation and temperature, respectively.Figure 2
P-I characteristics of the solar array according to different insolation (for T=25°C).Figure 3
P-I characteristics of the solar array according to different temperature (for S=1000 W/m2).


## 4. The State-Space Model of Buck Converter

To obtain the differential equations describing the buck converter, the ideal topology is used as shown in Figure4. The differential equations describing the DC/DC buck converter dynamics are obtained through the direct application of Kirchoff’s current and Kirchoff’s voltage laws for each one of the possible circuit topologies arising from the assumed particular switch position function value. Thus, when the switch position function exhibits the value u=1, we obtain the topology corresponding to the nonconducting mode for the diode obtained. Alternatively, when the switch position exhibits the value u=0, the second possible circuit topology corresponding to the conducting mode for the diode is obtained.Figure 4
Ideal switch representation of the DC/DC buck converter.The system dynamics is described by the following differential equations.Foru=1,
(2)Ldidt=-v+E,Cdvdt=i-vR.Foru=0,
(3)Ldidt=-v,Cdvdt=i-vR.By comparing the obtained particular dynamic systems descriptions, the following unified dynamic system model can be obtained:(4)Ldidt=-v+uE,Cdvdt=i-vR.


## 5. Sliding Mode Controller Design

In this section, two methods are presented in order to control the output voltage of a DC/DC buck converter. These methods are direct and indirect sliding mode control.Direct control is based on the output voltage feedback, where the output voltage(VC) is directly compared to the reference voltage (Vr).Sliding surface is as follows:(5)S=VC-Vr.Indirect control is based on the inductor current control satisfying desired output voltage(Vr). In this method, because of the possible load variation, at first the output impedance (Ro) is determined by dividing the output voltage by the output current, and then the desired load current (Ild) is calculated by dividing the reference voltage by the output impedance (Ro). By adding a coefficient of capacitor current (k) to Ild, the desired inductor current (Ilr) is determined. So, the sliding surface is defined by comparison between actual indirect current (Il) and Ilr as the following.Sliding surface can be computed as:(6)S=Il-Ilr.


## 6. Simulation of DC/DC Converter with Sliding Mode Control

In order to investigate the proposed controller performance and accuracy, Matlab/Simulink and its facilities are used.Simulations are performed on a typical buck converter with the following parameter values:E=10 V, L=1 mH, C=2200 μF, R=25 W.Figures5 and 6 show the Simulink block diagram of direct and indirect sliding mode control of the proposed buck converter, respectively.Figure 5
Simulink block diagram of direct sliding mode control.Figure 6
Simulink block diagram of indirect sliding mode control.


## 7. Results and Discussions

For the evaluation and validation of the proposed controller accuracy and performance, simulation results are presented in three subsections:(a)
analysis of the proposed sliding mode controller and determining the best sliding mode controller response;(b)
comparison of the proposed sliding mode controller with the conventional PID controller;(c)
investigation of the proposed siding mode controller under environmental (insolation and temperature) conditions.


### 7.1. Simulation Results of the Proposed Sliding Mode Controller

Figures7–9 show the output voltage during a change in reference voltage from V=5 to V=7 at t=0.05 sec in direct and indirect control methods, respectively. Figure 7 shows the DC/DC output voltage for a step change in reference voltage (from V=5 to V=7 at t=0.05 sec) in direct method. Figure 8 is plotted with respect to different values of k, where k is capacitor current coefficient.Figure 7
The DC/DC output voltage for a step change in reference voltage in direct control method.Figure 8
The DC/DC output voltage for a step change in reference voltage with respect to different values ofk in indirect control method.Figure 9
The DC/DC output voltage variation about reference variation time.Figure9 indicates a zoom window about the output voltage variation.In order to evaluate the robustness of sliding mode control, the load is changed from 1 Ω to 0.5 Ω att=0.05 sec. The results are indicated in Figures 10–12 for direct and indirect control methods, respectively. Figure 10 shows the DC/DC output voltage for a step change in load (from 1 Ω to 0.5 Ω at t=0.05 sec) in direct method. Figure 11 is plotted with respect to different values of k. Figure 12 indicates a zoom window about the output voltage variation.Figure 10
The DC/DC output voltage for step load variation in direct control method.Figure 11
The DC/DC output voltage for step load variation with respect to different values ofk in indirect control method.Figure 12
The DC/DC output voltage variation about step load variation time.Figures13–15 show the input voltage regulation, where the line voltage changes from E=10 to E=20 in t=0.05 sec. Figure 13 shows the DC/DC output voltage for a step change in load (from 1 Ω to 0.5 Ω at t=0.05 sec) in direct method. Figure 14 is plotted with respect to different values of k. Figure 15 indicates a zoom window about the output voltage variation.Figure 13
The DC/DC output voltage variation for a step change in line voltage in direct control method.Figure 14
The DC/DC output voltage variation for a step change in Line voltage with respect to different values ofk in indirect control method.Figure 15
The DC/DC output voltage variation about line variation time.These results (Figures7–15) show that the direct control method does not have a good performance. On the other hand, indirect control performance is good but it is related to the k value. By the selection of suitable value of k, the indirect method performance is very good. The results show that the best performance is obtained for k=0.5.


### 7.2. Comparison of Proposed Sliding Mode and PID Controller

For the validation of the proposed sliding mode controller, the best simulated result compares with conventional PID controller.The best result is the one with the wave forms of indirect sliding mode control with capacitor current coefficient(k) equal to 0.5.The noninteracting structure of PID controller is used. This is described by the following equation:(7)GPID=kP(1+1sTi+sTd),
where kP is proportional gain and Ti, Td are integrator and derivative time constants, respectively.The simulation results of comparison between the proposed sliding mode controller and PID controller are indicated in Figures16–19.Figure 16
The DC/DC output voltage variation with a step change in reference voltage.Figure 17
The DC/DC output voltage variation about reference voltage variation time.Figure 18
The DC/DC output voltage variation with a step change in load.Figure 19
The DC/DC output voltage variation about load variation time.Figures16 and 17 show the output voltage of buck converter with sliding mode and PID controller with a step change in reference voltage from 5 V to 7 V at t=0.05 sec. Figure 15 indicates a zoom window about the output voltage variation.Finally, the load resistance is changed from 1 Ω to 0.5 Ω att=0.05 sec. The output voltage variation is indicated in Figure 18. Figure 19 shows a zoom window about the output voltage variation.


### 7.3. Investigation of the Proposed Sliding Mode Controller under Environmental (Insolation and Temperature) Conditions

In order to investigate of the accuracy of the proposed controller, a system includes a solar array, a DC/DC buck converter, and sliding mode controller. For testing the output voltage and current of photovoltaic array and then the output voltage of buck converter versus changing the environmental parameters instabilities, system was examined by changing the insolation and ambient temperature of array. The results are illustrated in following figures.In Figure20, the insolation is changed from 900 (W/m2) to 300 (W/m2) in 30°C ambient temperature at t=0.025 sec. As indicated in these figures, the voltage variation in the output voltage of buck converter is not visible.Figure 20
Output voltage of DC/DC converter for a step change in insolation from 900 (W/m2) to 600 (W/m2) in 30°C.In Figure21, the results of a step change in ambient temperature from 30°C to 20°C in constant insolation (900 (W/m2)) at t=0.025 sec are illustrated.Figure 21
Output voltage of DC/DC converter for a step change in ambient temperature from 30°C to 20°C in constant insolation (900 (W/m2)).


## 7.1. Simulation Results of the Proposed Sliding Mode Controller

Figures7–9 show the output voltage during a change in reference voltage from V=5 to V=7 at t=0.05 sec in direct and indirect control methods, respectively. Figure 7 shows the DC/DC output voltage for a step change in reference voltage (from V=5 to V=7 at t=0.05 sec) in direct method. Figure 8 is plotted with respect to different values of k, where k is capacitor current coefficient.Figure 7
The DC/DC output voltage for a step change in reference voltage in direct control method.Figure 8
The DC/DC output voltage for a step change in reference voltage with respect to different values ofk in indirect control method.Figure 9
The DC/DC output voltage variation about reference variation time.Figure9 indicates a zoom window about the output voltage variation.In order to evaluate the robustness of sliding mode control, the load is changed from 1 Ω to 0.5 Ω att=0.05 sec. The results are indicated in Figures 10–12 for direct and indirect control methods, respectively. Figure 10 shows the DC/DC output voltage for a step change in load (from 1 Ω to 0.5 Ω at t=0.05 sec) in direct method. Figure 11 is plotted with respect to different values of k. Figure 12 indicates a zoom window about the output voltage variation.Figure 10
The DC/DC output voltage for step load variation in direct control method.Figure 11
The DC/DC output voltage for step load variation with respect to different values ofk in indirect control method.Figure 12
The DC/DC output voltage variation about step load variation time.Figures13–15 show the input voltage regulation, where the line voltage changes from E=10 to E=20 in t=0.05 sec. Figure 13 shows the DC/DC output voltage for a step change in load (from 1 Ω to 0.5 Ω at t=0.05 sec) in direct method. Figure 14 is plotted with respect to different values of k. Figure 15 indicates a zoom window about the output voltage variation.Figure 13
The DC/DC output voltage variation for a step change in line voltage in direct control method.Figure 14
The DC/DC output voltage variation for a step change in Line voltage with respect to different values ofk in indirect control method.Figure 15
The DC/DC output voltage variation about line variation time.These results (Figures7–15) show that the direct control method does not have a good performance. On the other hand, indirect control performance is good but it is related to the k value. By the selection of suitable value of k, the indirect method performance is very good. The results show that the best performance is obtained for k=0.5.


## 7.2. Comparison of Proposed Sliding Mode and PID Controller

For the validation of the proposed sliding mode controller, the best simulated result compares with conventional PID controller.The best result is the one with the wave forms of indirect sliding mode control with capacitor current coefficient(k) equal to 0.5.The noninteracting structure of PID controller is used. This is described by the following equation:(7)GPID=kP(1+1sTi+sTd),
where kP is proportional gain and Ti, Td are integrator and derivative time constants, respectively.The simulation results of comparison between the proposed sliding mode controller and PID controller are indicated in Figures16–19.Figure 16
The DC/DC output voltage variation with a step change in reference voltage.Figure 17
The DC/DC output voltage variation about reference voltage variation time.Figure 18
The DC/DC output voltage variation with a step change in load.Figure 19
The DC/DC output voltage variation about load variation time.Figures16 and 17 show the output voltage of buck converter with sliding mode and PID controller with a step change in reference voltage from 5 V to 7 V at t=0.05 sec. Figure 15 indicates a zoom window about the output voltage variation.Finally, the load resistance is changed from 1 Ω to 0.5 Ω att=0.05 sec. The output voltage variation is indicated in Figure 18. Figure 19 shows a zoom window about the output voltage variation.


## 7.3. Investigation of the Proposed Sliding Mode Controller under Environmental (Insolation and Temperature) Conditions

In order to investigate of the accuracy of the proposed controller, a system includes a solar array, a DC/DC buck converter, and sliding mode controller. For testing the output voltage and current of photovoltaic array and then the output voltage of buck converter versus changing the environmental parameters instabilities, system was examined by changing the insolation and ambient temperature of array. The results are illustrated in following figures.In Figure20, the insolation is changed from 900 (W/m2) to 300 (W/m2) in 30°C ambient temperature at t=0.025 sec. As indicated in these figures, the voltage variation in the output voltage of buck converter is not visible.Figure 20
Output voltage of DC/DC converter for a step change in insolation from 900 (W/m2) to 600 (W/m2) in 30°C.In Figure21, the results of a step change in ambient temperature from 30°C to 20°C in constant insolation (900 (W/m2)) at t=0.025 sec are illustrated.Figure 21
Output voltage of DC/DC converter for a step change in ambient temperature from 30°C to 20°C in constant insolation (900 (W/m2)).


## 8. Conclusion

In this paper, a new indirect sliding mode controller for DC/DC converters is presented. The results of analysis and simulation of proposed method are compared by direct sliding mode as well as conventional PID controller. The main conclusions are the following.(i)
The proposed indirect sliding mode controller has more good performance as compared with direct sliding mode as well as PID control.(ii)
The performance of the proposed sliding mode controller is related to suitable selection of its parameter(k).(iii)
The proposed controller is robust to load and line as well as reference variations.(iv)
The proposed controller can be used to all types of DC/DC converters.



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*Source: 871025-2013-04-16.xml*