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Novel Control for a Buck Converter used in a DMPPT System
2nd IFAC Workshop on Convergence of Information Technologies and Control Methods with Power Systems May 22-24, 2013. Cluj-Napoca, Romania
Novel Control for a Buck Converter used in a DMPPT System Stefan Daraban*, Dorin Petreus*, Daniel Moga ** *Technical University of Cluj-Napoca, Applied Electronics Department, Romania (e-mail: stefan.daraban@ ael.utcluj.ro, [email protected]). **Technical University of Cluj-Napoca, Automation Department, Romania (e-mail: [email protected]) Abstract: This paper is focused on the digital implementation of a voltage and a current loop for a buck converter used in distributed maximum power point systems (DMPPT). It presents the importance of an additional control to help the maximum power point tracking (MPPT) algorithm. The paper gives a detailed small signal analysis and describes the method for compensating the current and voltage loop, making the stability of the system independent of the operating point on the power voltage (P-V) characteristic. The paper also proposes an original Perturb and Observe (P&O) algorithm with variable step size to search a local maximum. The calculation of the step size is independent from the slope of the solar panel’s P-V characteristic, making the tracking performances independent of the PV module. Simulation and experimental results are finally provided to verify the performance of the system Keywords: DMPPT System, Power Supplies, Digital Control, Solar Energy, Battery Charging. 1. INTRODUCTION Solar power is one of the renewable energy sources with the highest potential. A lot of research work and significant investment funds are necessary before renewable energy can replace fossil fuels dominance. The average efficiency of commercial PV arrays is close to 20%, (Parida et al. 2011). The maximization of the power extracted from the solar panel is achieved by placing a high efficiency DC-DC converter between the PV array and the load. The converter matches the impedance of the load to the impedance of the solar panel for maximum power transfer (Pastor et al. 2011). Digital control offers many advantages, in particular in power management and renewable energy systems, where there are multiple power sources, loads, power buses and converter modules. Classical topologies of MPPT systems have at most one closed loop for the input voltage and the maximum power point tracking algorithm implemented (Safari and Mekhilef 2011). An analog version of a MPPT system for a boost converter is presented in (Enrique et al. 2009). This type of implementation is suitable for low cost systems but the performance decreases with the sudden change of irradiance and with aging. By far the most popular algorithm is the P&O, due to its ease of implementation and good performance. Adaptive algorithms can modify their perturbation step so that when the operating point is far from the MPP, the step size is big, and when it is close to the maximum power point, the step becomes small. Intelligent MPPT systems based on fuzzy logic and neuronal networks are discussed in (Zhang and Bai 2008, Ansari et al. 2010).
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The major drawback of the above systems is that these algorithms fail to find the global maximum power point (GMPP) under partial shading conditions (Shaiek et al. 2013). Bypass diodes are mounted on each panel to eliminate the hot spot effect (Silvestre et al. 2009). These diodes introduce local maximum power points (LMPP) in the power voltage (P-V) characteristic. A hardware solution is to replace the bypass diodes with power converters, thus eliminating LMPP, (Femia et al. 2008, Petrone et al. 2012, Shenoy, et al. 2013). The performance of the algorithm can be enhanced by adding control loops. Practical implementations employing a control loop for the input voltage are presented in (Petreus et al. 2011,). The addition of a control loop on the input voltage helps the system to respond faster to luminosity changes, (Villalva et al. 2010), but it still has limitations in terms of response time of the system and over current protection. A loop can then be employed on the input current, thus providing increased immunity to output voltage disturbances. This also leads to a more accurate control of the output current, making it suitable for battery and supercapacitor charging. Furthermore, it introduces current limiting when a short circuit is present at the output. Additionally, the added loop simplifies the compensation network of the control loops using simple proportional-integrative (PI) controllers, (Etz et al. 2012, Tsang and Chan 2011). The present paper improves the solution proposed in (Petreus and Daraban 2011), by solving the problem of partial shading with a DMPPT system. This paper presents a control method and an algorithm that are independent of the characteristics of the solar panel. The operating point has a great influence on the stability of the
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system as will be pointed out later in the paper. An algorithm with variable step is proposed which helps in finding the MPP, for a solar panel with single maxima, in a short time. The step size is calculated independently of the characteristics of the solar panel, thus obtaining the same performance for different solar panels. 2. DIGITAL AVERAGE CURRENT MODE CONTROL The efficiency of the system is influenced by the power topology and the control structure. For the last twenty years engineers have been concentrating on harvesting solar energy just by developing complex MPPT algorithms. Additional hardware can improve energy harvesting.
The response of the system for a luminosity step change is presented in Fig. 4. The first I-V characteristic has its MPP in ‘A’ and the second I-V characteristic has its MPP in ‘C’. When step luminosity occurs from the first characteristic to the second characteristic, the operating point moves from ‘A’ to ‘B’. From this moment, the MPPT algorithm searches the MPP on the second characteristic and finds it in ‘C’. After a second luminosity step, the system is back on the first I-V characteristic and has the operating point in ‘D’. Another tracking begins and gets the operating point to the MPP located in ‘A’. This closed perimeter (‘ABCD’) shows the response of the power loop to an irradiance step change.
The P-V characteristics for two strings of PV panels are presented in Fig. 1. In order to avoid the hot spot effect for a string bypass diodes are connected to each panel. For the other string, converters are connected to each PV module. It can be seen that by connecting power converters, more power can be extracted and the P-V characteristic present only one maximum power point (MPP). The DMPPT topology that was used is presented in Fig. 2. A Buck converter is connected to every PV module in order to harvest more energy under partial shading conditions. The outputs of the converters are connected in series in order to create a standard DC bus voltage which is used in microgrids (Etz et al. 2012).
Fig. 2 DMPPT system.
Fig. 1 P-V characteristic with bypass diodes and with a DMPPT system. Additional control can help the algorithms reach the MPP faster. A comprehensive analysis of various MPPT topologies can be found in (Petreus 2011). All the disadvantages of the existing MPPT systems are eliminated by implementing an average current mode control (ACMC) on the solar panel current, Fig. 3. This method is proposed and developed (Petreus 2011, Petreus 2010) for a boost converter in low luminosity applications. A buck converter is chosen to step down the voltage of the solar panel voltage to the battery voltage level. The implementation of such a system (Petreus 2011) requires a specialized IC controller that provides access to all its internal operational amplifiers in order to implement the compensation network.
Fig. 3 Power converter for each solar panel with ACMC. The existing average current mode controllers are not at all suitable for this application as they cost too much, have high power consumption or do not allow access to the compensation network. The present paper demonstrates that digital control can improve the development of DMPPT systems, which are optimized by adding two additional loops: one to control the input voltage and the other one to control the average current through the solar panel. The easiest way to compensate a digital controller is to compensate it in the analog domain and then transform it in its discrete form. The paper proposes the compensation in the continuous time domain from the small signal model of a buck converter to obtain a stable system independent of the operating point on the solar panel characteristics. The transfer function is then
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digitized to obtain an implementation suitable for a digital controller. The small signal model for the buck converter that regulates the output voltage is described in (Erickson and Maksimovic 2001). This model is modified to correspond to a system that charges a battery from a PV module, . The battery is replaced by a short circuit (constant output voltage Vout), and the input of the model is represented by a dynamic resistance rin. This model is always valid because the converter works all the time in continuous conduction (synchronous buck).
b) Voltage loop Fig. 5 Small signal model. From (2), it seems that the stability of the system depends on the operating point of the solar panel (represented by the dynamic resistance rin). This disadvantage can be eliminated by choosing the crossover frequency of the voltage loop equal to the frequency of the zero introduced by the compensation network, Fig. 6. Fig. 6 is used to calculate the compensation network. The strategy is to calculate the MQ segment in two ways and then make them equal, keeping in mind that:
MN = PQ MQ = MN + NQ Fig. 4 Luminosity step response for the proposed system. The current provided by the solar panel is the current through the Buck transistor. To ensure correct functionality of the solar panel, a large value capacitor must be introduced to filter the ripple current. The simplified transfer function of the current loop can be calculated from Fig. 5a):
H c ( s) =
Vout (1 + sL ⋅ I L / Vout ) ⋅ Vp sL / Rs
(1)
where Vout is the output voltage, Vp is the amplitude of the saw tooth buck converter and Rs is the gain of the current sensor. For (1), a simple PI structure is enough to boost the gain at low frequencies and to boost the phase at the crossover frequency. The schematic of the small signal model for the voltage loop is presented in Fig. 5b). The transfer function of the voltage loop is: H v (s) = H v (s) =
ve ' ( s ) = H p ( s ) ⋅ H comp ( s ) ve ( s )
(1 + s ⋅ τ f ) Gdiv ⋅ rin 1 ⋅ ⋅ 1 + s ⋅τ i RS s ⋅τ f
(2)
where, Gdiv= R2/(R1 +R2) represents the gain of the input voltage divider, τf=Rf·Cf, and τi=rin·Cin, Hp(s) (power circuit transfer function) has the first two elements of the equation and Hcomp(s) (compensation circuit transfer function) has the last term.
(3)
The MQ segment can be calculated as the sum of two segments:
(
)
MQ = 20 log τ i / τ f + 20 log(Gdiv ⋅ rin / RS )
(4)
The MQ segment can be calculated directly from the 40deb/dec slew:
(
MQ = 40 log τ i / τ f
)
(5)
In order to reach the condition previously mentioned, the following condition must be satisfied:
(rin ⋅ Gdiv )/ RS = (rin ⋅ Cin )/ (R f
⋅Cf
)
(6)
Setting Rf, one can find the value of Cf by using:
(
C f = (RS ⋅ Cin ) / R f ⋅ Gdiv
)
(7)
The voltage loop makes the system stable and independent of the solar panel, no matter what the position of the operating point on the I-V curve (left or right side of the MPP) , or the irradiance variation. The difference between the analog controller and the digital controller is the additional term that introduces a delay in the model of the digital PWM modulator, (Van de Sype et al. 2004). A linear equation can be obtained by using a first order Padé approximation: PWM ( s ) =
1 1 − (s ⋅ D ⋅ Ts / 2 ) e − sDTs ≈ ⋅ Vp V p 1 + (s ⋅ D ⋅ Ts / 2 )
(8)
where D is the duty cycle of the converter and Ts is the sampling frequency. This is the case for a trailing-edge modulation. The cutoff frequency of the voltage loop is small compared with 1/(DTs) and both transfer functions(analog and digital, phase and amplitude) are identical long after the cutoff frequency, thus (8) does not influence (7), as seen in Fig. 7.
a) Current loop
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‘x7’ the perturbation step size increases to step6. The algorithm sees that ‘x7’ is getting further apart from the MPP and changes the direction. Now the search is locked between points ’x6’ and ’x7’. When there is a change of direction in the command signal, the algorithm has locked the MPP between two previous operating points. The algorithm detects that step6 is too large, so the step size is decreased. The step size is decreased when the current operating point does not change or when the operating point oscillates between two states. After the MPP is locked, the step begins to decrease because the above statements occur. Fig. 6 Compensation of the voltage loop. Flexibility and programmability allow the implementation of average current mode control with access to all three control loops (power, voltage and current) of the MPPT controller in order to make the stability of the system independent of the operating point. An analog controller with this type of compensation is difficult to implement, because all the DCDC regulators are application oriented.
The proposed algorithm computes the step size independently of the dP/dV value, thus making the algorithm suited for any PV module, without the need of modifying the gain of the current and voltage sensors and threshold levels.
Fig. 8 Proposed MPPT algorithm. 4. SYSTEM MODELING AND SIMULATION Because of the DMPPT system each block can be treated as a system with a single panel. The block diagram for the control of a synchronous buck converter used in a DMPPT system is illustrated in Fig. 3. The simulations have the same decision time for both MPPT algorithms (classical P&O and proposed P&O). The amplitude of the perturbation step size for the fixed P&O is 100mV and from 40mV to 400mV for the P&O with variable step size.
Fig. 7 Analog and digital voltage loop transfer functions 3. MPPT ALGORITHM The most popular algorithm found in many MPPT controllers on the market is the P&O algorithm. An improved version of the P&O algorithm, with variable step size, is proposed. The principle of this algorithm is based on the P&O with fixed step size and three weights method. This algorithm uses a novel method to calculate the step size. This method does not involve complex calculation to compute the slope of the P-V curve (dP/dV), like in (Petreus et al. 2010). The proposed algorithm has a fixed number of step sizes, step1step8, where step1 is the smallest and step8 is the largest. When the system is far from the MPP, the step size is high and gets smaller as it gets closer to the MPP. The algorithm is detailed in Fig. 8. ’x1’-‘x8’ represent the points through which the operating point passes when finding the MPP. Let us suppose that the initial start up of the operating point is in ‘x1’ and the perturbation step size is step4. The step size increases when the operating point changes in the same direction twice in a row. Between ’x1’-
The simulation results for the P&O with fixed step converge to the MPP in 2.6s as shown in Fig. 9. If a luminosity step change occurs, the time response of the PV is approximately 450ms as can be seen in Fig. 10.
Fig. 9 Convergence to the MPP for fixed step P&O.
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Fig. 10 Luminosity step change response for fixed step P&O. The starting point, the decision time for computing the new command signal and the irradiance levels are unchanged. The simulation result for the P&O with variable step, converges to the MPP in 1s, as shown in Fig. 11 and the time response to a luminosity step is 300ms (Fig. 12). The proposed algorithm has increased performances demonstrated through faster converging times to the MPP.
of step luminosity change by varying the short circuit current to known and repetitive values and enables the comparison between the two algorithms, under the same conditions. A TMS320F2808 digital controller from Texas Instruments was used. The solar panel gives a Voc of 25.5V and Isc of 2A, the load of the converter is a 12V lead acid battery, 45Ah and the components of the converter are L=23uH, Cin=1mF, and Fsw=100kHz. The experimental results for the P&O algorithm with fixed step converging to the MPP presents a time response of 60s (Fig. 13) and, for the variable step, presents a time response of 30s (Fig. 14). It can be seen that the proposed algorithm has increased performance demonstrated through a faster convergence to the MPP than the fixed step P&O. Furthermore, the experimental results for a luminosity step response for the P&O with fixed step takes 25s (Fig. 15) and 15s (Fig. 16) for the proposed algorithm. It can be seen that the proposed algorithm has increased performance, demonstrated through faster transient response when a luminosity step variation occurs. For the simulations, the decision time of the MPPT algorithm is 15ms and the MPP is reached in 1.5s. In the experimental results, the decision time is 300ms and the maximum power point is reached in almost 30s. Simulations have to be as short as possible in order to see the results in a short time, but the shading changes are a slow process in real life so a scaling was made for the decision time used in simulations. The short circuit current is varied from 1A to 2A, just like in the simulation results.
Fig. 11 Convergence to the MPP for proposed algorithm.
Fig. 13 Experimental results for convergence to the MPP of the P&O algorithm.
Fig. 12 Luminosity step change response for proposed algorithm. 5. EXPERIMENTAL RESULTS The experimental results obtained with the novel configuration and the proposed algorithm, are discussed in this section. A current source with diodes is used to build the model for the solar panel. This method allows the simulation
Fig. 14 Experimental results for convergence to the MPP of the proposed algorithm.
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Fig. 15 Experimental results for luminosity step response of the P&O algorithm.
Fig. 16 Experimental results for luminosity step response of the proposed algorithm. 6. CONCLUSIONS The new system presented in this paper resolves the problems of the existing maximum power point controllers. It improves the shading response and step load changes by making both the stability of the system and the MPPT algorithm independent of the solar panel. A MPPT configuration was developed, with its small signal model, and an improved P&O algorithm, unique through its feature of step size change, proved with simulation and experimental results. Digital control adds the flexibility to design a control structure for which there is no specialized IC for implementation. The performance of the system was tested with simulation and experimental results. REFERENCES Ansari, M.F., Chatterji, S. and Iqbal, A., (2010). A fuzzy logic control scheme for a solar photovoltaic system for a maximum power point tracker. International Journal of Sustainable Energy, 29(4), 245-255. Enrique, J.M., Andujar, J.M. and Bohorquez, M.A., (2009). A reliable, fast and low cost maximum power point tracker for photovoltaic applications. Solar Energy, 84(1), 79-89. Erickson, R.W. and Maksimovic, D., (2001). Fundamentals of Power Electronics 2nd edition, Kluwer Academic Publishers.
Etz, R., et al., (2012). Digital control for phase-shift converter. In: AQTR, 24-27 May, Cluj-Napoca, Romania, 62-67. Femia, N., et al., (2008). Distributed Maximum Power Point Tracking of Photovoltaic Arrays: Novel Approach and System Analysis. IEEE Transactions on Industrial Electronics, 55(7), 2610-2621. Parida, B., Iniyan, S. and Goic, R., (2011). A review of solar photovoltaic technologies. Solar Energy, 15(3), 16251636. Pastor, A.C., et al., (2011). Design of photovoltaic-based current sources for maximum power transfer by means of power gyrators. IET Power Electronics, 4(6), 674 – 682. Petreus, D., et al., (2010). A maximum power point tracker for a photovoltaic system under changing luminosity conditions. In: 2010 IEEE International Symposium on Industrial Electronics, 4-7 July, Bari, Italy, 556-561. Petreus, D., et al., (2011). A novel maximum power point tracker based on analog and digital control loops. Solar Energy, 85(3), 588-600. Petreus, D., Daraban, S., et al. (2011). A novel implementation of a maximum power point tracking system with digital control. In: ISIE 2011, 27-30 June, Gdansk, Poland, 977-982 Petrone,G. et al., (2012). An analog technique for distributed MPPT PV applications. IEEE TIE, 59(12), 4713-4722. Safari, A. and Mekhilef, S., (2011). Simulation and hardware implementation of incremental conductance mppt with direct control method using Cuk converter. IEEE TIE, 58(4), 1154-1161. Shaiek, Y., et al. (2013). Comparison between conventional methods and GA approach for maximum power point tracking of shaded solar PV generators. Solar Energy, 90, 107-122. Shenov, P.S. et al., (2013). Differential Power Processing for Increased Energy Production and Reliability of Photovoltaic Systems. IEEE TIE, 28(6), 2968-2979. Silvestre, S. et al. (2009). Study of bypass diodes configuration on PV modules. Applied Energy, 86, 16321640. Tsang, K.M. and Chan, W.L. (2011). Multi-loop controller for wide operating range single-ended primary inductor DC/DC converter. IET Power Electronics, 4(8) , 891898. Van de Sype, D.M., et al., (2004). Small-signal z-domain analysis of digitally controlled converters. In: IEEE 35th Power Electronics Specialists Conference, 20-25 June, vol. 6, 4299-4305 Villalva, M.G., de Siqueira T.G. and Ruppert, E., (2010). Voltage regulation of photovoltaic arrays: small-signal analysis and control design. IET Power Electronics, 3(6), 869-880. Zhang, L. and Bai, Y.F., (2008). On-line neural network training for maximum power point tracking of PV power plant. Transaction of the Institute of Measurement and Control, 30(1), 77-96.
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Novel Control for a Buck Converter used in a DMPPT System Stefan Daraban*, Dorin Petreus*, Daniel Moga ** *Technical University of Cluj-Napoca, Applied Electronics Department, Romania (e-mail: stefan.daraban@ ael.utcluj.ro, [email protected]). **Technical University of Cluj-Napoca, Automation Department, Romania (e-mail: [email protected]) Abstract: This paper is focused on the digital implementation of a voltage and a current loop for a buck converter used in distributed maximum power point systems (DMPPT). It presents the importance of an additional control to help the maximum power point tracking (MPPT) algorithm. The paper gives a detailed small signal analysis and describes the method for compensating the current and voltage loop, making the stability of the system independent of the operating point on the power voltage (P-V) characteristic. The paper also proposes an original Perturb and Observe (P&O) algorithm with variable step size to search a local maximum. The calculation of the step size is independent from the slope of the solar panel’s P-V characteristic, making the tracking performances independent of the PV module. Simulation and experimental results are finally provided to verify the performance of the system Keywords: DMPPT System, Power Supplies, Digital Control, Solar Energy, Battery Charging. 1. INTRODUCTION Solar power is one of the renewable energy sources with the highest potential. A lot of research work and significant investment funds are necessary before renewable energy can replace fossil fuels dominance. The average efficiency of commercial PV arrays is close to 20%, (Parida et al. 2011). The maximization of the power extracted from the solar panel is achieved by placing a high efficiency DC-DC converter between the PV array and the load. The converter matches the impedance of the load to the impedance of the solar panel for maximum power transfer (Pastor et al. 2011). Digital control offers many advantages, in particular in power management and renewable energy systems, where there are multiple power sources, loads, power buses and converter modules. Classical topologies of MPPT systems have at most one closed loop for the input voltage and the maximum power point tracking algorithm implemented (Safari and Mekhilef 2011). An analog version of a MPPT system for a boost converter is presented in (Enrique et al. 2009). This type of implementation is suitable for low cost systems but the performance decreases with the sudden change of irradiance and with aging. By far the most popular algorithm is the P&O, due to its ease of implementation and good performance. Adaptive algorithms can modify their perturbation step so that when the operating point is far from the MPP, the step size is big, and when it is close to the maximum power point, the step becomes small. Intelligent MPPT systems based on fuzzy logic and neuronal networks are discussed in (Zhang and Bai 2008, Ansari et al. 2010).
978-3-902823-32-8/2013 © IFAC
The major drawback of the above systems is that these algorithms fail to find the global maximum power point (GMPP) under partial shading conditions (Shaiek et al. 2013). Bypass diodes are mounted on each panel to eliminate the hot spot effect (Silvestre et al. 2009). These diodes introduce local maximum power points (LMPP) in the power voltage (P-V) characteristic. A hardware solution is to replace the bypass diodes with power converters, thus eliminating LMPP, (Femia et al. 2008, Petrone et al. 2012, Shenoy, et al. 2013). The performance of the algorithm can be enhanced by adding control loops. Practical implementations employing a control loop for the input voltage are presented in (Petreus et al. 2011,). The addition of a control loop on the input voltage helps the system to respond faster to luminosity changes, (Villalva et al. 2010), but it still has limitations in terms of response time of the system and over current protection. A loop can then be employed on the input current, thus providing increased immunity to output voltage disturbances. This also leads to a more accurate control of the output current, making it suitable for battery and supercapacitor charging. Furthermore, it introduces current limiting when a short circuit is present at the output. Additionally, the added loop simplifies the compensation network of the control loops using simple proportional-integrative (PI) controllers, (Etz et al. 2012, Tsang and Chan 2011). The present paper improves the solution proposed in (Petreus and Daraban 2011), by solving the problem of partial shading with a DMPPT system. This paper presents a control method and an algorithm that are independent of the characteristics of the solar panel. The operating point has a great influence on the stability of the
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system as will be pointed out later in the paper. An algorithm with variable step is proposed which helps in finding the MPP, for a solar panel with single maxima, in a short time. The step size is calculated independently of the characteristics of the solar panel, thus obtaining the same performance for different solar panels. 2. DIGITAL AVERAGE CURRENT MODE CONTROL The efficiency of the system is influenced by the power topology and the control structure. For the last twenty years engineers have been concentrating on harvesting solar energy just by developing complex MPPT algorithms. Additional hardware can improve energy harvesting.
The response of the system for a luminosity step change is presented in Fig. 4. The first I-V characteristic has its MPP in ‘A’ and the second I-V characteristic has its MPP in ‘C’. When step luminosity occurs from the first characteristic to the second characteristic, the operating point moves from ‘A’ to ‘B’. From this moment, the MPPT algorithm searches the MPP on the second characteristic and finds it in ‘C’. After a second luminosity step, the system is back on the first I-V characteristic and has the operating point in ‘D’. Another tracking begins and gets the operating point to the MPP located in ‘A’. This closed perimeter (‘ABCD’) shows the response of the power loop to an irradiance step change.
The P-V characteristics for two strings of PV panels are presented in Fig. 1. In order to avoid the hot spot effect for a string bypass diodes are connected to each panel. For the other string, converters are connected to each PV module. It can be seen that by connecting power converters, more power can be extracted and the P-V characteristic present only one maximum power point (MPP). The DMPPT topology that was used is presented in Fig. 2. A Buck converter is connected to every PV module in order to harvest more energy under partial shading conditions. The outputs of the converters are connected in series in order to create a standard DC bus voltage which is used in microgrids (Etz et al. 2012).
Fig. 2 DMPPT system.
Fig. 1 P-V characteristic with bypass diodes and with a DMPPT system. Additional control can help the algorithms reach the MPP faster. A comprehensive analysis of various MPPT topologies can be found in (Petreus 2011). All the disadvantages of the existing MPPT systems are eliminated by implementing an average current mode control (ACMC) on the solar panel current, Fig. 3. This method is proposed and developed (Petreus 2011, Petreus 2010) for a boost converter in low luminosity applications. A buck converter is chosen to step down the voltage of the solar panel voltage to the battery voltage level. The implementation of such a system (Petreus 2011) requires a specialized IC controller that provides access to all its internal operational amplifiers in order to implement the compensation network.
Fig. 3 Power converter for each solar panel with ACMC. The existing average current mode controllers are not at all suitable for this application as they cost too much, have high power consumption or do not allow access to the compensation network. The present paper demonstrates that digital control can improve the development of DMPPT systems, which are optimized by adding two additional loops: one to control the input voltage and the other one to control the average current through the solar panel. The easiest way to compensate a digital controller is to compensate it in the analog domain and then transform it in its discrete form. The paper proposes the compensation in the continuous time domain from the small signal model of a buck converter to obtain a stable system independent of the operating point on the solar panel characteristics. The transfer function is then
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digitized to obtain an implementation suitable for a digital controller. The small signal model for the buck converter that regulates the output voltage is described in (Erickson and Maksimovic 2001). This model is modified to correspond to a system that charges a battery from a PV module, . The battery is replaced by a short circuit (constant output voltage Vout), and the input of the model is represented by a dynamic resistance rin. This model is always valid because the converter works all the time in continuous conduction (synchronous buck).
b) Voltage loop Fig. 5 Small signal model. From (2), it seems that the stability of the system depends on the operating point of the solar panel (represented by the dynamic resistance rin). This disadvantage can be eliminated by choosing the crossover frequency of the voltage loop equal to the frequency of the zero introduced by the compensation network, Fig. 6. Fig. 6 is used to calculate the compensation network. The strategy is to calculate the MQ segment in two ways and then make them equal, keeping in mind that:
MN = PQ MQ = MN + NQ Fig. 4 Luminosity step response for the proposed system. The current provided by the solar panel is the current through the Buck transistor. To ensure correct functionality of the solar panel, a large value capacitor must be introduced to filter the ripple current. The simplified transfer function of the current loop can be calculated from Fig. 5a):
H c ( s) =
Vout (1 + sL ⋅ I L / Vout ) ⋅ Vp sL / Rs
(1)
where Vout is the output voltage, Vp is the amplitude of the saw tooth buck converter and Rs is the gain of the current sensor. For (1), a simple PI structure is enough to boost the gain at low frequencies and to boost the phase at the crossover frequency. The schematic of the small signal model for the voltage loop is presented in Fig. 5b). The transfer function of the voltage loop is: H v (s) = H v (s) =
ve ' ( s ) = H p ( s ) ⋅ H comp ( s ) ve ( s )
(1 + s ⋅ τ f ) Gdiv ⋅ rin 1 ⋅ ⋅ 1 + s ⋅τ i RS s ⋅τ f
(2)
where, Gdiv= R2/(R1 +R2) represents the gain of the input voltage divider, τf=Rf·Cf, and τi=rin·Cin, Hp(s) (power circuit transfer function) has the first two elements of the equation and Hcomp(s) (compensation circuit transfer function) has the last term.
(3)
The MQ segment can be calculated as the sum of two segments:
(
)
MQ = 20 log τ i / τ f + 20 log(Gdiv ⋅ rin / RS )
(4)
The MQ segment can be calculated directly from the 40deb/dec slew:
(
MQ = 40 log τ i / τ f
)
(5)
In order to reach the condition previously mentioned, the following condition must be satisfied:
(rin ⋅ Gdiv )/ RS = (rin ⋅ Cin )/ (R f
⋅Cf
)
(6)
Setting Rf, one can find the value of Cf by using:
(
C f = (RS ⋅ Cin ) / R f ⋅ Gdiv
)
(7)
The voltage loop makes the system stable and independent of the solar panel, no matter what the position of the operating point on the I-V curve (left or right side of the MPP) , or the irradiance variation. The difference between the analog controller and the digital controller is the additional term that introduces a delay in the model of the digital PWM modulator, (Van de Sype et al. 2004). A linear equation can be obtained by using a first order Padé approximation: PWM ( s ) =
1 1 − (s ⋅ D ⋅ Ts / 2 ) e − sDTs ≈ ⋅ Vp V p 1 + (s ⋅ D ⋅ Ts / 2 )
(8)
where D is the duty cycle of the converter and Ts is the sampling frequency. This is the case for a trailing-edge modulation. The cutoff frequency of the voltage loop is small compared with 1/(DTs) and both transfer functions(analog and digital, phase and amplitude) are identical long after the cutoff frequency, thus (8) does not influence (7), as seen in Fig. 7.
a) Current loop
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‘x7’ the perturbation step size increases to step6. The algorithm sees that ‘x7’ is getting further apart from the MPP and changes the direction. Now the search is locked between points ’x6’ and ’x7’. When there is a change of direction in the command signal, the algorithm has locked the MPP between two previous operating points. The algorithm detects that step6 is too large, so the step size is decreased. The step size is decreased when the current operating point does not change or when the operating point oscillates between two states. After the MPP is locked, the step begins to decrease because the above statements occur. Fig. 6 Compensation of the voltage loop. Flexibility and programmability allow the implementation of average current mode control with access to all three control loops (power, voltage and current) of the MPPT controller in order to make the stability of the system independent of the operating point. An analog controller with this type of compensation is difficult to implement, because all the DCDC regulators are application oriented.
The proposed algorithm computes the step size independently of the dP/dV value, thus making the algorithm suited for any PV module, without the need of modifying the gain of the current and voltage sensors and threshold levels.
Fig. 8 Proposed MPPT algorithm. 4. SYSTEM MODELING AND SIMULATION Because of the DMPPT system each block can be treated as a system with a single panel. The block diagram for the control of a synchronous buck converter used in a DMPPT system is illustrated in Fig. 3. The simulations have the same decision time for both MPPT algorithms (classical P&O and proposed P&O). The amplitude of the perturbation step size for the fixed P&O is 100mV and from 40mV to 400mV for the P&O with variable step size.
Fig. 7 Analog and digital voltage loop transfer functions 3. MPPT ALGORITHM The most popular algorithm found in many MPPT controllers on the market is the P&O algorithm. An improved version of the P&O algorithm, with variable step size, is proposed. The principle of this algorithm is based on the P&O with fixed step size and three weights method. This algorithm uses a novel method to calculate the step size. This method does not involve complex calculation to compute the slope of the P-V curve (dP/dV), like in (Petreus et al. 2010). The proposed algorithm has a fixed number of step sizes, step1step8, where step1 is the smallest and step8 is the largest. When the system is far from the MPP, the step size is high and gets smaller as it gets closer to the MPP. The algorithm is detailed in Fig. 8. ’x1’-‘x8’ represent the points through which the operating point passes when finding the MPP. Let us suppose that the initial start up of the operating point is in ‘x1’ and the perturbation step size is step4. The step size increases when the operating point changes in the same direction twice in a row. Between ’x1’-
The simulation results for the P&O with fixed step converge to the MPP in 2.6s as shown in Fig. 9. If a luminosity step change occurs, the time response of the PV is approximately 450ms as can be seen in Fig. 10.
Fig. 9 Convergence to the MPP for fixed step P&O.
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Fig. 10 Luminosity step change response for fixed step P&O. The starting point, the decision time for computing the new command signal and the irradiance levels are unchanged. The simulation result for the P&O with variable step, converges to the MPP in 1s, as shown in Fig. 11 and the time response to a luminosity step is 300ms (Fig. 12). The proposed algorithm has increased performances demonstrated through faster converging times to the MPP.
of step luminosity change by varying the short circuit current to known and repetitive values and enables the comparison between the two algorithms, under the same conditions. A TMS320F2808 digital controller from Texas Instruments was used. The solar panel gives a Voc of 25.5V and Isc of 2A, the load of the converter is a 12V lead acid battery, 45Ah and the components of the converter are L=23uH, Cin=1mF, and Fsw=100kHz. The experimental results for the P&O algorithm with fixed step converging to the MPP presents a time response of 60s (Fig. 13) and, for the variable step, presents a time response of 30s (Fig. 14). It can be seen that the proposed algorithm has increased performance demonstrated through a faster convergence to the MPP than the fixed step P&O. Furthermore, the experimental results for a luminosity step response for the P&O with fixed step takes 25s (Fig. 15) and 15s (Fig. 16) for the proposed algorithm. It can be seen that the proposed algorithm has increased performance, demonstrated through faster transient response when a luminosity step variation occurs. For the simulations, the decision time of the MPPT algorithm is 15ms and the MPP is reached in 1.5s. In the experimental results, the decision time is 300ms and the maximum power point is reached in almost 30s. Simulations have to be as short as possible in order to see the results in a short time, but the shading changes are a slow process in real life so a scaling was made for the decision time used in simulations. The short circuit current is varied from 1A to 2A, just like in the simulation results.
Fig. 11 Convergence to the MPP for proposed algorithm.
Fig. 13 Experimental results for convergence to the MPP of the P&O algorithm.
Fig. 12 Luminosity step change response for proposed algorithm. 5. EXPERIMENTAL RESULTS The experimental results obtained with the novel configuration and the proposed algorithm, are discussed in this section. A current source with diodes is used to build the model for the solar panel. This method allows the simulation
Fig. 14 Experimental results for convergence to the MPP of the proposed algorithm.
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Fig. 15 Experimental results for luminosity step response of the P&O algorithm.
Fig. 16 Experimental results for luminosity step response of the proposed algorithm. 6. CONCLUSIONS The new system presented in this paper resolves the problems of the existing maximum power point controllers. It improves the shading response and step load changes by making both the stability of the system and the MPPT algorithm independent of the solar panel. A MPPT configuration was developed, with its small signal model, and an improved P&O algorithm, unique through its feature of step size change, proved with simulation and experimental results. Digital control adds the flexibility to design a control structure for which there is no specialized IC for implementation. The performance of the system was tested with simulation and experimental results. REFERENCES Ansari, M.F., Chatterji, S. and Iqbal, A., (2010). A fuzzy logic control scheme for a solar photovoltaic system for a maximum power point tracker. International Journal of Sustainable Energy, 29(4), 245-255. Enrique, J.M., Andujar, J.M. and Bohorquez, M.A., (2009). A reliable, fast and low cost maximum power point tracker for photovoltaic applications. Solar Energy, 84(1), 79-89. Erickson, R.W. and Maksimovic, D., (2001). Fundamentals of Power Electronics 2nd edition, Kluwer Academic Publishers.
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