CHAPTER 6 INPUT VOLATGE REGULATION AND EXPERIMENTAL INVESTIGATION OF NON-LINEAR DYNAMICS IN PV SYSTEM 6. INTRODUCTION The DC-DC Cuk converter is used as an interface between the PV array and the load, but other types of converters can be used for the same purpose. The input voltage of the converter is controlled in order to regulate the operating point of the array. Besides reducing losses and stress because of the bandwidth-limited regulation of the converter duty cycle, controlling the converter input voltage reduces the settling time and avoids oscillation and overshoot, making easier the functioning of maximum power point tracking methods. The voltage regulation problem is addressed that starts with the linear modelling of the PV module and the design of controller. 6. NEED FOR INPUT VOLTAGE REGULATION In photovoltaic power systems, both photovoltaic modules and switching-mode converters present non-linear and time-variant characteristics, which result in a difficult control problem. Figure. illustrates that the changing radiation varies the photovoltaic current dramatically. The fast dynamics of insolation is usually caused by a cover of mixed rapid moving clouds. The PV array operating
point can be adjusted by regulating the voltage or current at the terminals of the array. If the photovoltaic current is used as the set point, the MPP tracking requires fast dynamics to follow a wide operating range from A to the shortcircuit current, depending heavily on weather conditions. Nevertheless, the changing insolation slightly affects the voltage of MPP (V m ). Figure. shows the effect of temperature on the I-V characteristics. Unlike the current of the MPP, the photovoltaic voltage of the MPP is usually bounded by 7%-8% of the open circuit voltage. This gives a lower bound and upper limit of the tracking range. When regulation of photovoltaic voltage is implemented, the MPP tracker can quickly decide the initial point according to the percentage of the open-circuit voltage. The value of VMPP is continuously tracked and updated by the MPP tracker. Therefore, the regulation performance of the photovoltaic voltage is important for MPP tracking. The voltage control is preferred because the voltage at the MPP is approximately constant. The PV current, on the other hand, changes greatly when the solar irradiation varies. This research work discussed the voltage control problem as shown in Figure 6. in solar PV-powered Cuk converter MPPT system. The PV array feeds the DC-DC Cuk converter. The Cuk converter is used as an interface between solar PV module and load, since the Cuk converter is the good choice for the maximum power point tracking circuits.
3 Figure 6. Input voltage regulation for Cuk converter-based PV system The input voltage of the converter is controlled in order to regulate the operating point of the solar PV module. However, both photovoltaic modules and switching-mode converters demonstrate non-linear and time-variant characteristics, which make a controller design difficult. This research work proposes to design a voltage controller to regulate the input voltage of the converter for the change in irradiation. The voltage controller improves the transient response to the input voltage of the converter, avoids oscillation, overshoot, making easier the functioning of MPPT methods and ensures period - operation. 6.3 LINEARIZATION OF SOLAR PV MODULE AT MPP PV module of L35-37Wp has non-linear I-V charecteristic which is shown in Figure 6.. The operating characteristic of a solar cell consists of two regions: the current source region, and the voltage source region. In the current source region, the internal impedance of the solar cell is high and this region is located on the left side of the current-voltage curve. The voltage source region, where the internal impedance is low, is located on the right side of the current-voltage curve. As can be observed from the characteristic curve, in the current source region, the output current remains almost constant as the terminal voltage changes and in the voltage source
4 region, the terminal voltage varies only minimally over a wide range of output current. The terminal current I pv remains at a somewhat constant level in the constant-current region up to MPP voltage. The current decreases if the voltage is further increased in the constant voltage region eventually diminishing to zero when the open circuit condition is reached. Figure 6. Non-linear I-V characteristics of L35-37Wp solar module The Thevenin s equivalent circuit at MPP is shown in Figure 6.3. Table 6. shows the values of the Thevenin s equivalent circuit of the L35-37Wp module at MPP. Figure 6.3 Linear equivalent circuit valid at the linearization point
5 Table 6. Thevenin s equivalent circuit values for solar PV at MPP Maximum power at MPP (P max ) 37W Short circuit current (I sc ).5A Voltage at MPP (V pv ) 6.4 Current at MPP (I pv ).5 Thevenin s equivalent voltage (V eq ) 34.6V Equivalent resistance (R eq ) 7.89 Open circuit voltage (V oc ) V The linear equivalent circuit of Figure 6.3 is valid at the linearization point (V, I) and is a good approximation of the solar PV module for the computer simulation. The dynamic behavior of the solar PV powered MPPT system depends strongly on the point of operation of the module. 6.4 SMALL SIGNAL MODELLING FOR INPUT VOLTAGE CONTROL In Figure 6.4, the small signal model of the Cuk converter-based solar PV system is analyzed to obtain small-signal converter transfer function. The small signal model describes the behavior of V pv with respect to the duty cycle of the Cuk converter Figure 6.4 Cuk converter with Solar PV module linear model
6 The bar over a variable name (e.g. v, ) means the discrete-time average value of the variable within one switching period of the converter. By writing the circuit state equations with average variables, the high-frequency components are eliminated and natural system behavior is analyzed. The average capacitor state equation is c v - = (6.) The average output inductor state equation is V - V o - L - R L = (6.) The circuit constituting L, C, MOSFET, and diode may be replaced by the average equivalent quatripole with terminals --3-4, which is equated with following equations, where d is the duty cycle of the transistor V = v (-d / -d) = v K (6.3) = (-d / -d) = k (6.4) where k= -d/ -d The relation between the input voltage of the converter and the capacitor voltage is given by v = R v +v (6.5) To obtain the input voltage control for Cuk converter-based solar PV system, the small signal variables are introduced into the state equations.
7 The steady state DC values are capitalized and the small signals are marked with a hat. v = V c + v v = V pv + v = I L + (6.6) k = K - k By substituting 6.4, 6.5 in 6. and eliminating non linear product terms, the small signal equation is derived using Laplace transformation I L k(s) - v (s) s C R c v (s)- (s)k s C v (s) = (6.7) Similarly, solving 6., 6.3, 6.5, 6.8, -Vc k(s) - R L (s)-sl (s)- s C R c K v (s)+ v (s)k- s C R c Vc k(s)= (6.8) Solving 6.5, 6.7, and 6.8, the small signal input voltage to duty cycle is obtained which is given below: G vd (s) = = ( )( ( ) ( ) ( ) ( ) (6.9) ( ) For analyzing the input voltage regulation of Cuk converter-based solar PV system, the converter parameters are chosen as follows: L =L =5e-6H, C =C =e-6f, V pv = 6.4V, d=.44, K= -.79, I L =.84A, V o = 3V load resistance R=, switching frequency f s =5kHz, diode (BY9). The switch S in the power stage is realized using a MOSFET (IRF 84). The component values used in linear model of solar PV module
8 circuit are C = F, R c =.5, R L =., R eq =7.9, d=.44, K= -.79, V eq = 34.6V 6.5 DESIGN OF SINGLE FEEDBACK LOOP VOLTAGE CONTROLLER The voltage controller (PI controller) actuates on the converter duty cycle and directly regulates the input voltage (V pv ) of the Cuk converter. Figure 6.5 shows how the controller is constituted. Figure 6.6 shows the single feedback loop voltage controller to regulate the input voltage of the Cuk converter. Figure 6.5 Input Voltage-controlled Cuk converter-based solar PV system Figure 6.6 Voltage controller with single feedback loop
9 The Cuk converter-based solar PV system is compensated with voltage controller of (4s+4)/s and the feedback gain (H) is /6. The crossover frequency of the compensated system is. 3 rad/sec with phase margin of 4. The voltage controller makes the Cuk converter-based solar PV system operate at points other than the point at which the I-V curve was linearised. The bode plots of open loop system (G vd ) and closed loop system are illustrated in Figures 6.7 and 6.8. To verify the validity of small signal modelling in time domain the closed loop system is tested with unit step input. Figure 6.7 Bode plots of the open loop system G vd (s) and the closed loop compensated system
3 Figure 6.8 Bode plot of compensated system At low frequency, the compensated system gain is 7dB which is high enough to minimize the steady state error. The unit step response of the transfer function is shown in Figure 6.9. Figure 6.9 Unit step response of the compensated system
3 6.6 SIMULATION RESULTS The diagram of the input regulated closed loop system designed in MATLAB/Simulink is presented in Figure 6. that includes linearised model of solar PV array which consists of voltage source in a series with the equivalent panel resistance, Cuk converter, voltage controller and a load. Figure 6. Input voltage regulation of solar PV powered Cuk converter in MATLAB /Simulink
3 To test the system operation under change in irradiation condition, the system is modeled in which indent to change the solar panel resistance. Initially, the solar panel resistance is kept as 7.89 which corresponds to irradiation of W/m. At t=.5 sec, the panel resistance is reduced by 5% which corresponds to 5 W/m. The voltage controller is designed in such a way that converter input voltage regulation of 6.4V is achieved for both the conditions. The simulated input voltage regulation for the change in irradiation is shown in Figure 6.. Figure 6. Regulated Input voltage waveform due to change in irradiation levels 6.7 IMPLEMENTATION OF INPUT VOLTAGE REGULATION FOR CUK CONVERTER-BASED SOLAR PV SYSTEM The experimental setup is shown in Figure 6.. It consists of a power circuit and a control circuit. The power circuit consists of inductors L and L made of ferrite core, and capacitors C and C are of plain polyester. Power MOSFET (IRF84) is used as active switch S. The converter is assumed to operate in continuous conduction mode. The control circuit consists of the following blocks: voltage divider, V ref generation, difference
33 amplifier, inverting amplifier, and a Schmitt trigger. A reference voltage is generated and fed to non-inverting input of the difference amplifier. Voltage from the divider circuit is given to the inverting input of the difference amplifier LM358. This input voltage is regulated irrespective of the temperature and irradiation change. The experimental setup and the Piece spice PCB layout for PID controller are shown in Figures 6. and 6.3. Figure 6. Photography of an experimental setup Figure 6.3 PCB layout for PID controller
34 Voltage measurement is required at the point where the solar PV module output is connected to the input of Cuk converter. The voltage at this point is the operating voltage of the PV module. The unregulated input voltage of the converter, i.e., solar PV panel output voltage is shown in Figure 6.4. Figure 6.4 Unregulated input voltage without voltage controller (Horizontal scale: 5* -6 sec/div, Vertical scale= 5V/div) The input voltage is regulated using PI voltage controller. The regulated input voltage is shown in Figures 6.5 and 6.6. Figure 6.5 Regulated input voltage with PID controller (Horizontal scale: 5* -6 sec/div, Vertical scale= 5V/div)
35 Figure 6.6 Regulated input voltage with voltage controller (Horizontal scale: 5* -6 sec/div, Vertical scale= 5V/div) The unregulated input voltage takes a time of 3ms to reach steady state voltage of 7V without voltage controller. Using PI voltage controller, the input voltage of the converter takes a time of 5ms to reach the steady state voltage of 6.4V. The input voltage is regulated as constant for the change in irradiations using single feedback loop voltage controller. 6.8. EXPERIMENTAL INVESTIGATION OF NON-LINEAR DYNAMICS IN CUK CONVERTER-BASED SOLAR PV SYSTEM Power electronics is a field spawned by many real-life applications in industrial, commercial and aerospace environments. At the same time, it is also a field rich in nonlinear dynamics. As one of the most popular members in power electronics circuits, the DC-DC converter has found in widespread application for many decades. In non-linear circuits and systems a great variety of strange phenomena have been observed, including sub-harmonics, quasi-periodic oscillations, and chaotic behaviors.
36 The non-linear behaviors have been intensively studied in the crossdisciplinary science of chaos. In particular, it has recently been observed that a large number of power electronic circuits can exhibit deterministic chaos. Power converters can work under linear control or non-linear control. Most research works focus on linear feedback controlled converters, which may exhibit interesting bifurcation and chaos when some parameters are varied. Period-doubling bifurcation, Hopf bifurcation, border-collision bifurcation, and chaos have been reported in these converters. On the other hand, nonlinear controlled converters can also exhibit bifurcation and chaos, although little is known about these nonlinear phenomena. From the experimental point of view, the chaos may be defined as bounded steady-state behavior which is not an equilibrium point, not periodic, and not quasi-periodic. In time domain, a chaotic trajectory is neither periodic nor quasi-periodic but looks random. Also DC-DC converters exhibit different non-linear phenomena including bifurcations, quasi-periodicity and chaos under both voltage mode and current mode control schemes. Due to the non-linear dynamics in the power electronic circuits, their operation is characterized by the cyclic switching of circuit topologies, which gives rise to a variety of non-linear behaviors like bifurcation and chaos. The behavior of a chaotic system is a collection of many orderly behaviors, none of which dominates under ordinary circumstances. Since current and voltage mode controlled converters have wide industrial application, control of chaos has an important significance. Non-linear phenomena jeopardize the performance of converters, and suppression of bifurcation and chaos has been an important subject in
37 designing converters. Non-linear phenomena in DC-DC converter used for solar PV system have drawn attention only recently. Studying the non-linear behavior in DC-DC converter used for solar PV system is not only interesting, but also very useful. To track the maximum power from the solar PV module, the output voltage of the solar PV module (input voltage of the DC-DC converter) has to be chaos- free and the solar PV voltage to ensure period- operation so that the oscillation near to maximum power point is nil. An attempt to control chaos in the Cuk converter-based solar PV system is made in this research by adopting a conventional PID controller. Hence, Cuk converter used for solar PV system is designed to operate period - operation. Different methods are proposed for controlling chaos in non-linear systems which can be classified into two general categories namely, feedback control methods and non-feedback control methods. Feedback methods include the Ott-Grebogi-Yorke (OGY) method, Variable Ramp Compensation (VRC), Time-Delayed Feedback Control (TDFC) method, etc. Examples of non-feedback methods include adaptive control and Resonant Parametric Perturbation (RPP). In adaptive control, conventional controllers such as PID controller and sliding mode controllers are used to control chaos in non-linear system. Even though most of the approaches proposed until now are very interesting, they mainly present theoretical or simulated results. As a consequence, there is a lack of experimental analysis on the parameter domains for which chaotic behavior may occur. Therefore, this research work aims to bridge this gap by presenting an experimental study of some dynamic phenomena that can occur in solar PV powered voltage mode-controlled Cuk converter system.
38 In particular, this research illustrates a novel hardware implementation able to show some pathways through which the solar PV powered Cuk converter may enter into chaos. The analysis of chaos in a voltage mode-controlled Cuk converter-based solar PV system has been performed and the use of conventional control method to suppress chaos has been discussed. The chaotic behavior of voltage mode controller Cuk converterbased solar PV system in continuous conduction mode is analyzed using the block diagram shown in Figure. 6.7. The non-linear dynamics are analyzed by varying the V ref. 6.8. System Description Figure 6.7 describes the block diagram of experimental setup of the proposed voltage-controlled Cuk converter-based solar PV system, which is constituted by a power stage and a control circuit. The power stage includes an inductor L, L, capacitor C, C, a switch S, a load resistance, a solar PV module ( L35-37Wp). For analyzing the chaotic behavior in Cuk converter-based solar PV system, the converter parameters are chosen as follows: L =L =5e-6H, C =C =e-6f, V in = 6.4V, load resistance R=, switching frequency f s =5kHz, diode (BY9). The switch S in the power stage is realized using a MOSFET (IRF84). The control circuit consists of a voltage divider, a comparator LM3, PID controller, Schmitt trigger-gate drive circuit (555 timer).
39 Figure 6.7 Block diagram of the experimental setup The LM3 compares the reference voltage V ref with the voltage across solar PV module (input voltage of Cuk converter) using a voltage divider which is proportional to the input voltage of the Cuk converter. Therefore, the output of the comparator is high when the input voltage reaches the value V ref, whereas it is low when the input voltage is less than V ref. In order to generate a square wave with amplitude of 5 V, frequency f s = /T = 5 khz and duty cycle d =.4, the integrated device 555 Timer (NE555N) IC (along with proper resistors and capacitors) is used. The measurements have been recorded by using a RIGOL digital storage oscilloscope. The Cuk converter has two modes of operation. The converter is assumed to operate in continuous conduction mode, that is, the input inductor current of Cuk converter never falls to zero. Hence, there are two possible constituent linear circuit configurations.
4 When deriving the state equations for the Cuk converter, all capacitor voltages and inductor currents are chosen as state variables. The converter can be modeled by the following equation : = A x + B V in the switch S is on = A x + B V in the switch S is off (6.) where x= v v i i and V in is the input voltage of the Cuk converter. In mode, the switch is on and the diode is off. During this mode, the system matrices A and B are given by A = C R L L C C, B = L and B are In mode, the switch S is off and the diode is on. In this case, A A = C L R L C C, B = L
4 Also the Cuk converter can work in discontinuous mode which can be either discontinuous-inductor-current or discontinuous-capacitor-voltage mode. For discontinuous-inductor-current operation, it is characterized by the presence of a duration in which both the switch S and diode are open, i.e., i + i =. This happens when the inductances are relatively small. During this mode of operation, the system matrices are given by A 3 = C R L L L L L L L L C C, B 3 = L L L L The discontinuous capacitor voltage mode is characterized by the presence of a duration in which the capacitor voltage, V, is zero. This happens when the capacitance of the capacitor C is relatively small, such that the value of V drops to zero within the switch on-time, introducing a duration in which both the switch and diodes are conducting. During this duration, the system matrices are given by A 4 = C L R C C, B 4 = L
4 6.8. Route to Chaos For analyzing the dynamic behavior in solar PV-powered Cuk converter system, the reference voltage V ref is varied. The input voltage of the Cuk converter is kept as 6.4 V. 6.9 INVESTIGATION OF DYNAMIC BEHAVIOR FOR PARAMETER VARIATION The dynamic behavior of input voltage of the solar PV-powered Cuk converter system is experimentally analyzed by varying V ref. The fundamental period - waveform has been found with V ref =5.68V. The solar PV powered Cuk converter system has stable periodic behavior. The input voltage of the period- operation is shown in Figure 6.8. Figure 6.8 Experimental period- waveform in the input voltage when V ref =5.68V (Horizontal scale: 5* 6 sec/div, Vertical scale= 5*mV/div) When the reference voltage (V ref ) is 5.V, the system has unstable periodic behavior. The -periodic orbit loses its stability via flip bifurcation
43 and gives a -periodic waveform. The input voltage of the period - operation is shown in Figure 6.9. Figure 6.9 Experimental period- waveform when V ref =5.V (Horizontal scale: 5* -6 sec/div, Vertical scale= 5mV*/div) Varying V ref further, it is observed that the converter changes from a stable operation to an unstable operation. The input voltage of Cuk converter-based solar PV system has unstable aperiodic behavior. Chaotic waveform is observed for the V ref = 4.8V as shown in Figure 6.. Figure 6. Experimental chaos waveform when V ref =4.8V (Horizontal scale: * -6 sec/div, Vertical scale= 5mV/div)
44 6. ANALYSIS OF INPUT VOLATGE OF PV-POWERED CUK CONVERTER WITH PI CONTROLLER The control of non-linear dynamics in the input voltage of the solar PV module fed Cuk converter is implemented using PI controller. The feedback resistance R f =k potentiometer, input resistance R i =k potentiometer are selected for P (Proportional) controller and feedback capacitor =. F, R i =k are selected for I (Integral) controller. The input voltage of the Cuk converter is regulated and its ripple is experimentally analyzed. The PI (Proportional plus integral) controller improves the transient response of the input voltage. The time taken to reach the regulated input voltage is ms. The fundamental period- waveform shown in Figure 6. has been found with V ref =5.68V. Figure 6. Experimental stable Period- waveform when V ref = 5.68V using PI controller (Horizontal scale: * -6 sec/div, Vertical scale= mv*/div) But the converter reference voltage is decreased as V ref = 4.8V, and chaotic unstable behavior is observed as shown in Figure 6..
45 Figure 6. Experimental unstable chaotic waveform when V ref =4.8V (Horizontal scale: 5* -6 sec/div, Vertical scale= 5mV/div) The input gate pulse to switch (S) corresponds to an unstable chaotic mode as shown in Figure 6.3. Figure 6.3 Gate pulse to switch during the chaos mode of operation with PI controller (Horizontal scale: * -6 sec/div, Vertical scale:5v/div)
46 6. ANALYSIS OF INPUT VOLATGE OF PV POWERED CUK CONVERTER WITH PID CONTROLLER The control of non-linear dynamics is investigated experimentally in the input voltage of Cuk converter-based solar PV system with PID (Proportional plus Integral plus Derivative) controller which is shown in Figure 6.4. The feedback resistance R P =k potentiometer, input resistance R P =k potentiometer are selected for P controller and feedback capacitor C i =. F, input resistance R i =k potentiometer are selected for I controller. Input series resistance R c =, input capacitor C d =. F, feedback resistance R D =k potentiometer are selected for D controller. The input voltage is regulated and the non-linear dynamics are experimentally analyzed for the supply disturbances using PID controller. Figure 6.4 Diagram of the PID controller
47 The V ref is varied from 4.V to 5.76 V and it is observed that the converter is always operated in period- stable region for all the parameter variations which is given in Figure 6.5. Figure 6.5 Experimental period- waveform for 3V< V ref < 5.76V (Horizontal scale:* -6 sec/div, Vertical scale:mv*/div) The gate pulse corresponds to input voltage regulation of Cuk converter with PID controller as shown in Figure 6.6. Figure 6.6 Gate pulse to switch(s)-during the period - mode of operation with PID controller (Horizontal scale: * - 6 sec/div, Vertical scale= 5V/div)
48 6. CONCLUSION The input voltage of the Cuk converter is regulated using a voltage controller (PI compensator) in order to control the solar PV module output voltage for the change in irradiation. The small signal modelling is analyzed for input regulated Cuk converter-based solar PV system. The non-linear dynamics such as chaos is investigated experimentally in Cuk converterbased solar PV system. The PID controller is designed to regulate the input voltage of Cuk converter and to operate the solar PV- powered Cuk converter system is chaos-free with period- operation in which all the waveforms repeat at the same rate as the driving clock for the parameter variations.