DEVELOPING A PHOTOVOLTAIC MPPT SYSTEM. Thomas Bennett. A Dissertation Submitted to the Faculty of. The College of Engineering and Computer Science

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1 DEVELOPING A PHOTOVOLTAIC MPPT SYSTEM by Thomas Bennett A Dissertation Submitted to the Faculty of The College of Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, FL August 2012

2 Copyright by Thomas Bennett 2012 ii

3

4 ACKNOWLEDGEMENTS This may take a while! I would like to thank my advisor and chair of my dissertation committee, Dr. Ali Zilouchian, for all of his advice and support during the past few years. As hard as I tried, I was unable to wear away his patience. I would also like to thank Dr. Roger Messenger for introducing me to PV systems and helping me during the course of this work; Dr. Hanqi Zhuang, for his help in my engineering education while I was still in the mathematics department; Dr. Zvi Roth, for always being open to my questions and offering advice; Dr. Amir Abtahi for his help in the fuel cell lab, as well as lending me some PV modules to use in this work; Dr. Maria Petrie, for all her advice, as well as giving me the opportunity to travel abroad; Hank Vansant, for the many years of helping me with equipment in the lab and Dr. Chris Beetle for putting together the thesis class file for L A TEX, saving me a lot of formatting time while working on this dissertation. I also want to thank my many friends in the engineering, mathematics, and physics departments for helping me enjoy my time here. Lastly, I want to give a shout out to The Flying Spaghetti Monster. iv

5 ABSTRACT Author: Title: Institution: Dissertation Advisor: Degree: Thomas Bennett Developing a Photovoltaic MPPT System Florida Atlantic University Dr. Ali Zilouchian Doctor of Philosophy Year: 2012 Many issues related to the design and implementation of a maximum power point tracking (MPPT) converter as part of a photovoltaic (PV) system are addressed. To begin with, variations of the single diode model for a PV module are compared, to determine whether the simplest variation may be used for MPPT PV system modeling and analysis purposes. As part of this determination, four different DC/DC converters are used in conjunction with these different PV models. This is to verify consistent behavior across the different PV models, as well as across the different converter topologies. Consistent results across the different PV models, will allow a simpler model to be used for simulation and analysis. Consistent results with the different converters will verify that MPPT algorithms are converter independent. Next, MPPT algorithms are discussed. In particular, the differences between the perturb and observe, and the incremental conductance algorithms are explained and illustrated. A new MPPT algorithm is then proposed based on the deficiencies of the other algorithms. The proposed algorithm s parameters are optimized, and the results for different PV modules obtained. Realistic system losses are then considered, v

6 and their effect on the PV system is analyzed; especially in regards to the MPPT algorithm. Finally, a PV system is implemented and the theoretical results, as well as the behavior of the newly proposed MPPT algorithm, are verified. vi

7 DEVELOPING A PHOTOVOLTAIC MPPT SYSTEM List of Tables xiii List of Figures xiv 1 Introduction Introduction and Motivation Photovoltaic (PV) Systems PV modules System Types Maximum Power Point Tracking (MPPT) Problem statements and Objectives Contributions Organization of Dissertation Developing an MPPT PV System PV Modeling Temperature and Irradiation Effects Comments Converters DC/DC Converters Comments Control Algorithms Hill Climbing methods vii

8 2.3.2 Open Circuit Voltage and Short Circuit Current Fuzzy control Other Issues Partial shading and mismatch System with nonideal components Conclusion Modeling a solar cell Sample Modules No Resistor Model One Resistor Models Two Resistor Model Effects of resistors Conclusion Modeling the converter system Buck Converter System equations Results at STC Boost Converter System equations Results at STC Other topologies NonInverting BuckBoost Ćuk Converter Resistor/cap load Constant current source input viii

9 4.4 Conclusions The Control Algorithm STC Control P&O vs IncCond Single sample versus averaged samples Decreasing step size Control under changing conditions P&O vs IncCond Proposed algorithm Step Step Step Step Optimizing Algorithm parameters Finding α K and V S Proposed Algorithm - General Comparison to other Algorithms in the Literature MPPT with a Boost Converter Conclusions Advanced Modeling and Control Modeling With System Losses Diode loss Resistive Losses All System Losses Together ix

10 6.2 Effects on Control Algorithm Fixed Irradiance Changing Irradiance Conclusions Implementing the MPPT Circuit The IV curve Circuit Software and data acquisition (DAQ) Results and comparison to mathematical model Partial shading Series Parallel Buck Converter, no control Circuit Software and DAQ Issues Overcome Results and Conclusions MPPT Buck Converter Circuit and Software Method Results and Conclusions Results from outdoors IV Curve Partial shading Control Conclusions x

11 8 Conclusions Summary of work and contributions Future Work A Math of Buck converter system A.1 Solving the differential equations A.1.1 When δ = A.1.2 When δ = A.2 Attempting to solve with Fourier series B Matlab (pseudo)code B.1 P&O algorithm B.2 inccond algorithm B.3 Proposed algorithm - step B.4 Proposed algorithm - step B.5 Proposed algorithm - step B.6 Proposed algorithm - step B.7 Finding alpha, a simple example B.8 Finding alpha using PV values C SP Module Results D Results of other modules to proposed MPPT algorithm E Boost Model with losses E.1 Diode E.2 Transistor resistance E.3 Inductor resistance E.4 All together xi

12 F Labview Programs Bibliography xii

13 LIST OF TABLES 3.1 Two PV modules used for testing Parameter values for different PV modules and models Comparison of NRM MPP vs desired MPP PV voltage for different duty cycles based on diode forward voltage PV voltage for different duty cycles based on MOSFET on resistance PV voltage for different duty cycles based on inductor resistance Characteristics of 10 W SunTech module used in experiments D.1 A number of PV modules with their model and algorithm parameters, as well as their MPPT Buck converter efficiency for the Ropp input using stated parameters E.1 PV voltage for different duty cycles based on diode forward voltage E.2 PV voltage for different duty cycles based on MOSFET on resistance. 153 E.3 PV voltage for different duty cycles based on inductor resistance xiii

14 LIST OF FIGURES 1.1 An IV and power curve for a typical PV module A grid connected system with battery backup An IV and power curve for a typical PV module A circuit representation of a PV module Buck converter Boost converter (Noninverting) buckboost converter Ćuk converter Perturb and Observe MPPT algorithm Increasing power, and being left of MPP and being right of MPP Incremental Conductance MPPT algorithm Three panels with Voc=30.7 volts. Two have Isc=8.6 amps, one has Isc=7 amps The IV curves for the different models for the BP module The IV curves for the different models for the SP module The Power curves for the different models for the BP module The Power curves for the different models for the SP module The effects of R p on two different modules The effects of R s on two different modules The effects of R s on two different modules The effects of PV resistance to the IV curve xiv

15 4.1 The TRM PV model connected to a buck converter A plot of the BP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = A plot of the BP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = A plot of the SP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = A plot of the SP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = The TRM PV model connected to a buck converter The TRM PV model connected to a boost converter A plot of the BP NRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values A plot of the BP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values A plot of the SP NRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values A plot of the SP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values The behavior of two converters with the BP NRM The behavior of a buck converter with an RC load Buck converter using 8A current source Results of P&O and IncCond methods using a fixed voltage step size of 0.3 volts An example of IncCond behaving better than P&O Results of P&O and IncCond methods using a fixed voltage step size of 0.1 volts Results of P&O method using one voltage/current sample P&O with changing step for α = 0.2, 0.5, and 0.8 (bottom to top), with V m also shown xv

16 5.6 Changing Irradiation Input for MPPT testing Results of P&O algorithm with Ropp input Results of IncCond algorithm with Ropp input Results of proposed algorithm, step 1, with Ropp input Results of proposed algorithm, step 2, with Ropp input Results of proposed algorithm, step 3, with Ropp input Results of proposed algorithm, step 4, with Ropp input Efficiencies for different α and V S values for MPPT algorithm step Efficiencies for different parameters for MPPT algorithm step Errors for various α values for a simple tracking algorithm Average error for different alpha values using the BP module Efficiencies for different K values based on α = 66% and V S = A curve showing how V m changes for different starting V m values, 2 a values, and an 80% drop in irradiance Results of proposed algorithm using a boost converter Main loop of proposed algorithm Flag 0 of proposed algorithm Flag 1 of proposed algorithm Flag 2 of proposed algorithm The effects on PV voltage due to diode forward voltage Voltage versus duty cycle of BP NRM, for V f = 0.5, R t = 0.6, and R i = Comparison of steady state behavior between ideal buck converter, and one with losses, for a duty cycle of 80% A plot of the SP TRM buck circuit for C=L=.1, duty cycles of 40%, 60% and 80%, V out = 24, and for V f = 0.01V, R i = 0.01Ω and R t = 0.01Ω Output of BP module with system losses for STC xvi

17 6.6 Results of proposed algorithm with large system losses; R t = 0.6Ω, R i = 0.1Ω, V f = 0.5V How voltage changes with respect to irradiance for various duty cycles for the BP NRM Results of proposed algorithm with 10 BP modules in series Voltage versus duty cycle of BP NRM, for V f = 0.5, R t = 0.6, and R i = Inside - PV module/lamp configuration IV curve tracing circuit Labview Program for tracing the IV curve of a PV module Suntech IV and power curves Partial shading effects of two Suntech Modules PV-buck converter circuit Diode voltage - original setup Diode voltage - after adding 47µF capacitor across drain and source of MOSFET Diode voltage - after tripling inductor Voltage versus duty cycle Hardware used for implementation Proposed MPPT algorithm tested inside Suntech IV Curve outside Suntech Power Curve outside Partial shading effects of two Suntech Modules Suntech Power Curve outside Proposed MPPT algorithm tested outside C.1 Efficiencies for different α and V S values for MPPT algorithm step C.2 Efficiencies for different parameters for MPPT algorithm step xvii

18 C.3 Average error for different alpha values using the SP module C.4 Efficiencies for different K values based on alpha = 66% and V S = C.5 Results of proposed algorithm for SP module using the buck converter. 148 C.6 Results of proposed algorithm for SP module using the boost converter. 149 E.1 Voltage versus duty cycle of BP NRM, for V f = 0.5, R t = 0.6, and R i = F.1 Block diagram of PV buck circuit F.2 Block diagram of PV buck circuit with MPPT part I F.3 Block diagram of PV buck circuit with MPPT part II xviii

19 CHAPTER 1 INTRODUCTION 1.1 INTRODUCTION AND MOTIVATION The need for sustainable energy sources is well known, and has received much attention and funding in recent years [1]. There are many competing and complementary technologies, such as solar, geothermal, wind, hydro, biomass, as well as the use of fuel cells. Solar energy, and photovoltaics in particular, have one of the highest potentials. See [2], for a discussion of different energy sources, as well as the motivation for selecting solar energy. A photovoltaic system is comprised of a few main components. This includes the photovoltaic module, (DC/DC or DC/AC) converters, battery storage, and the electrical grid. It is the converter that is used to control the output of the photovoltaic module. It has the potential to maximize this output by using a maximum power point tracking algorithm. The topic of maximum power point tracking was an attractive topic, due to its involvement with renewable energy, control theory, and electronics. The ultimate goal of this work is to help in achieving higher efficiencies for the maximum power point tracking converters that are part of photovoltaic systems. As will be seen, there was a fair bit of room for improvement in the literature. It should be confessed, that a comparison to industry progress is much harder to make. Still, it is strongly believed that the information presented in this dissertation will be useful in furthering the progress of this topic. 1

20 1.2 PHOTOVOLTAIC (PV) SYSTEMS This section will give an overview of a typical Photovoltaic (PV) system, including how PV cells work, and the power conditioning equipment typically used in conjunction with them PV modules PV modules (often called solar panels - though one should not confuse them with thermal solar panels, such as solar hot water heaters), can be found in many sizes. The larger ones, roughly 3ft x 5ft, that are seen on buildings, generate around watts (for a solar irradiance of 1000W/m 2 ). They are typically connected together in series and parallel to create a PV array at a desired power. The modules themselves are made up of cells that are mostly in a series configuration. This increases the voltage, rather than the current (which increases when connected in parallel). This is beneficial in reducing resistive losses, or wire sizes in the system. It will be seen later that resistive losses in the converter may also negatively effect the maximum power point tracking algorithm. However, as will also be seen, series configurations cause less desirable behavior when it comes to mismatch (such as partial shading). How PV cells work The description given here will concern itself with the popular mono- or poly- crystalline type photovoltaic cells. There are other types of photovoltaics, such as thinfilm and dye-sensitized solar cells (the latter does not even require a pn junction [3]) which would require further explanation. Crystalline photovoltaic cells convert sunlight to electricity using the same technology that is used in standard semiconductor 2

21 electronics, namely pn junctions. A more detailed discussion can be found in [4, 5]. Basics of pn junctions can also be found in a typical electronics book such as [6]. However, a brief description is given presently. A typical semiconductor material, such as silicon, has four electrons in its outer shell and the atoms form a crystal lattice structure. When this semiconductor is doped with a substance with five electrons in its outer shell, such as phosphorous, this new material is called an n-type material (due to having one more negative charge). Doping with an element that has 3 electrons in its outer shell, such as with boron, creates a p-type material. When these two types are brought together 1, the n type gives off some extra electrons across the boundary (pn junction) over to the p type. This happens as a result of diffusion (thermal energy causes the electrons to move). Typically, one would expect the random movement of electrons to even out and so no net charge transfer takes place. However, these electrons from the n-type, will get captured in the crystalline structure of the p-type, where there was previously a hole. This electron becomes part of a valence bond in the crystal, and is effectively stuck there. As a result, a net positive charge forms at the n type side of the boundary, and a negative charge at the p type side, creating an electric field. The strength of this field depends on temperature. However, the electric field can not grow too strong, or it will overcome the strength of the valence bond, and the captured electrons would start to flow back. With an electric field in place, the photoelectric effect can now produce results. Typically, photons will free electrons from their atom, but these electrons recombine with their atom quickly, and so on average nothing changes. However, at the pn junction, within the electric field, freed electrons can be swept through the electric 1 Don t take this description as a description of how the junction is actually made. The two types are not generally made separately and then brought together. 3

22 field. The number of electrons, and so the current, depends on the amount of light (photons) being received. Closing this circuit allows for power generation (voltage due to electric field, and current due to photoelectric effect). From quantum theory, and particularly as a result of Planck s work in blackbody radiation, it is known that the amount of energy in a photon is proportional to the frequency of the light. Hence, for small frequency light, there is not enough energy to liberate an electron. The energy necessary to liberate the electron (and create what is called an electron-hole pair) is called the band gap energy (which is material dependent). If the energy of a photon is larger than this, an electron can be liberated. However, any extra energy is typically turned into heat (thermalized). If the energy is greater than 3 times the band gap energy it is possible to generate more than one electron-hole pair, and so prevent thermalizing. This is called carrier multiplication and involves the extra electron momentum being used to free another electron rather than just turning into heat[3]. See also [4] for a discussion of direct and indirect band gaps. There are other issues that are considered when creating a PV module out of the PV cells. This includes trying to capture as much light as possible by using anti-reflective coatings. One may also have multiple layers in the PV cell of different material to create layers with different band gap energies. See [4] for more on this and other important topics. See Figure 1.1 for a typical IV and power curve of a PV module System Types There are a variety of PV system topologies. For most of them, a converter is used (see Section 2.2). However, in some cases, where the load can accept a range of voltages, and the load itself is close to ideal (see 1.2.3), one may be able to get away with not using such a converter. A PV system (with a converter) may or may not 4

23 IV Curve Power curve Current Power Voltage Voltage (a) An IV curve of a typical PV module. (b) The Power curve of the same PV module. Figure 1.1: An IV and power curve for a typical PV module. have battery backup, and they may or may not tie into the electrical grid. An isolated system may even choose to keep the power at DC, and run DC only loads. Some of the more typical topologies are now given. Grid connected There are two main types of grid connected systems; ones with battery backup, and ones without. The ones without the battery backup are a bit simpler. The PV modules connect to an inverter, which converts the DC to AC. The AC side connects to the grid 2. The inverter will typically implement a maximum power point tracking (MPPT) algorithm, which will be discussed later. For the battery backup system, the PV modules connect to a charge controller (DC/DC converter), which then connects (and charges) a battery bank, which then connects to the inverter, and into the grid. For this system, most PV charge controllers implement an MPPT algorithm. The inverter of this type of system also usually has battery charging capabilities to charge the batteries from the grid if nec- 2 There are other components such as junction boxes, fuses, etc., but this covers the main components. 5

24 essary when the batteries are low, and the PV modules are producing no power. An example of this system type is shown in Figure 1.2. PV Module DC/DC Converter DC/AC Inverter Distribution Panel Electrical Grid Battery Bank Emergency Load Electrical Load Figure 1.2: A grid connected system with battery backup. Stand-Alone A stand-alone system is one that does not connect to a power grid; it is isolated. Similar to above, there can be battery backup or not (though without batteries, or some other type of backup, the system will not work when the PV modules are producing no power). Also, the final output of the system could be DC or AC. This allows for a DC/DC converter with MPPT (and possibly hooked up to batteries as well as) hooked up to the load. An inverter could also potentially be used in this setup. See [4] for a better discussion of PV systems Maximum Power Point Tracking (MPPT) Maximum Power Point Tracking (MPPT) refers to the process of maintaining the maximum power output of an energy source, when its power output changes in time. A PV module can increase its output greatly, when coupled with a converter that uses MPPT. Like a battery or fuel cell, a PV module has an IV curve of different current-voltage pairs where it can operate for a fixed environment (i.e. fixed irradiance and temperature). In fuel cells and batteries, using a higher current relates to using up more of the chemical fuel, and so maximizing output does not coincide with 6

25 optimizing (increasing the efficiency of) the system. This is not so for a PV module; having the maximum output power is desired, since only energy from the sun gets used (otherwise it would be converted to heat or reflected). Consider a typical IV curve of a PV module shown in Figure 1.1a. The power is zero when either the voltage or current is zero (i.e. at open-circuit voltage or short-circuit current). The function is concave down, zero on the endpoints, and positive. At some point, the percent increase in voltage is equal to that of the percent decrease in current. That is: Or equivalently: di i = dv v dp dv = di dv v + i = 0 Thus, there is a maximum power value. See Figure 1.1b for the power curve of the same PV module. IV Curve Power Curve Current Power Voltage Voltage (a) An IV curve at 0, 25, and 50 degrees C (right to left), and 1000, 700, and 400 W/m 2 (top to bottom). (b) The Power curve of the same PV module. Figure 1.3: An IV and power curve for a typical PV module. The problem now lies in trying to operate at the maximum power point (MPP) (V m, I m ). Any point on the curve can be obtained if we had a load that matched it, 7

26 i.e. R L = v i. The ideal load would then be R m = Vm I m. Unfortunately, due to changing atmospheric conditions, the IV curve, and hence the MPP changes. Consequently, we need R m to be variable. See Figure 1.3 to see how decreasing irradiance shifts the IV curve down, and increasing temperature shifts the IV curve to the left. In order to track the MPP, a converter with a controllable duty cycle is typically used (either DC/DC or DC/AC). The output voltage is typically fixed (or nearly so), allowing the input (PV) voltage to be changed by the duty cycle of the attached converter. However, trying to operate the PV array at V m presents some challenges. In particular, the IV curve is not only changing, but it is unknown, therefore V m is not known. This is the problem for MPPT. 1.3 PROBLEM STATEMENTS AND OBJECTIVES In this section some of the main questions, and the related objectives are given. More detail (and references) will be given in later chapters. A single diode, 2 resistor model for the PV module is often used for MPPT testing, as will be shown later. This model is represented by an implicit function. Removing the resistors would yield a model with an explicit function. For MPPT purposes, which makes use of closed loop control, a slightly less accurate model may yield similar results. This prompted the motivation to compare some of these variations of the single diode model. First, an attempt at finding mathematical solutions using this model with a buck converter was attempted (see appendix A). Unfortunately, the outcome of this was not too useful. However, even for modeling purposes, the simulation time could be reduced drastically using the simpler model. How do these models compare? To test the behavior a number of different DC/DC converters were used with the model to see how transient and steady state behavior of the system 8

27 depend on the circuit. Often times the MPPT algorithm is discussed independent of the converter itself. Because of this, the DC/DC converter behavior was also compared to see if the system behavior is independent of the particular converter used. If so, this would justify creating an algorithm independent of the converter (or inverter) it is to be used with. The incremental conductance MPPT method is usually claimed as being better than the perturb and observe method. However, this claim has never been well justified. Is the incremental conductance method better? How and why? Comparisons between different MPPT algorithms are not well conducted - no standard comparison method has been used. One algorithm could be better or worse depending on the input (irradiance/temperature) changes used for testing. A standard input for comparison has been recently proposed [7]. Using this standard, comparisons will be made, and a new algorithm will be proposed. The MPPT algorithm is usually explored independent of the converter used for testing the algorithm. Because of this, converter losses are usually not taken into account, but rather ideal converters (ideal switches and no resistive losses) are considered when modeling. Though real circuits are sometimes implemented in the literature, the effects of the realistic components are not discussed. What are the effects of realistic components on the system behavior? In particular, does the MPPT algorithm need to be modified to account for these effects? Even when circuits are implemented, they are often done more as a proof of concept, and the building of the circuit is discussed in little detail. The circuit built here will also be used as a proof of concept; verifying much of the previous theory. However, some of the issues that come up when trying to implement the MPPT converter circuit will also be discussed. Do any new issues occur that may affect the 9

28 MPPT control algorithm, or force one to rethink some of the theory? 1.4 CONTRIBUTIONS It is shown that a simplified PV module model yields similar results to a more standard model when used in conjunction with different converters. The difference between the incremental conductance, and the perturb and observe algorithms are explained and verified. The effects on MPPT algorithm sampling are explored. A new MPPT algorithm is proposed (and a method for optimizing is shown) in response to a suggested MPPT testing standard. The effects of system losses on the voltage-duty cycle relationship, and its effect on the MPPT algorithm are explored. A system was built, theory confirmed, and the proposed algorithm was tested. 1.5 ORGANIZATION OF DISSERTATION In Chapter 2, the stages of developing an MPPT converter for a PV system will be discussed, as well as current methods in use. This chapter will be used to set up some of the components necessary in the design of the MPPT system. It will also serve as a literature review. In Chapter 3, a discussion of modeling a PV module will be given. First, methods for finding model parameters will be given. After this, the IV and power curves of the different models will be compared. In Chapter 4, different converters will be used in conjunction with the different PV models developed previously. This will allow one to compare the converters, verifying system behavior is independent of the converter. It will also allow one to compare the different PV models in the context of a converter system. 10

29 In Chapter 5, MPPT control algorithms will be discussed. To start, the difference between two hill climbing methods will be given. Next, the effects of sampling times, as well as averaging samples, will be shown. Finally, a new algorithm will be proposed and compared to existing algorithms. In Chapter 6, system losses will be introduced to the PV-converter circuit to make the system more realistic. The losses will be analyzed, as well as the effects they have on the MPPT algorithm. In Chapter 7, a circuit is built and controlled using Labview software. This will be used to verify much of the theory previously discussed, as well as illustrate some of the potential issues when trying to translate theory to practice. Finally, Chapter 8 will conclude this work, and discuss possible future work. 11

30 CHAPTER 2 DEVELOPING AN MPPT PV SYSTEM In this chapter, the components required for developing a maximum power point tracking photovoltaic system are discussed. A review of the current literature will also be given. First, the mathematical model of a photovoltaic panel will be discussed. After this, a brief discussion of converters will be given, followed by using the converters in conjunction with the PV models. Next, a discussion about current MPPT control methods will be given. Finally, some other issues of interest will be discussed. 2.1 PV MODELING Perhaps the most popular model used to represent the PV module is the current source in parallel with a diode, with a parallel and series resistor. This is illustrated in Figure 2.1. i Ig Rp Rs v Figure 2.1: A circuit representation of a PV module. The equation of the circuit in Figure 2.1 is: 12

31 i = I g I s (e v+irs a 1) v + ir s R p (2.1) In this equation, i is the PV current, v is the PV voltage, R s is the series resistor, R p is the parallel resistor, I g is the light-generated current, I s is the diode s saturation current, and a = AkT/q, where A is the diode ideality factor, k is Boltzmann s constant, T is the temperature, and q is the charge of an electron. The derivative of this curve, with respect to voltage, is also useful, and is easily found to be: di dv = Is v+irs a e a + 1 R p 1 + IsRs e v+irs a a + Rs R p (2.2) The module specifications are given for standard test conditions (STC), and so I g, I s, a, and the resistors are all constant on the curve containing those points. There are 3 points from the spec sheet on the STC IV curve: (V oc, 0), (0, I sc ), and (V m, I m ). These points along with (2.1) yield three equations: 0 = I g I s (e Voc a I sc = I g I s (e IscRs a 1) V oc R p (2.3) 1) I scr s R p (2.4) I m = I g I s (e Vm ImRs a However, it is also know that dp = d(iv ) = 0 di dv with (2.2) yields another equation: I m V m I s a e Vm+ImRs a 1) V m + I m R s R p (2.5) = I V at (V m, I m ). This, along (1 R s I m V m ) 1 R p (1 R s I m V m ) = 0 (2.6) The papers [8, 9, 10] discuss different means of obtaining parameter values for this equation. These authors use 5 points on equations (2.1) and (2.2): 13

32 The open circuit voltage, V oc The short circuit current, I sc di = Im dv i=im V m dv di v=voc dv di i=isc The last 2 points require measurements that are not included on a typical PV spec sheet. In [9, 10] some approximations are made to make the variables easier to solve for (a numerical method is still required however). Another paper [11] attempts to solve for all the parameters using only values obtainable from a PV spec sheet. The first 3 points from above are used, but the last two are replaced by using (I m, V m ) as a point on the curve, and an approximation from [10]: di dv 1 I=Isc R p The point (I m, V m ) is certainly on a spec sheet. The other point however, uses an approximation. It is treated as an equality in their paper, and even in the original paper there is little justification Temperature and Irradiation Effects Previously, parameters were found based on STC (standard testing conditions). Under these conditions I g, I s, R s, R p, and a were constant. Equation (2.1) is generally expanded to accommodate changing irradiance and temperature, so that it can be used in more dynamic situations [4, 12, 11, 13, 14]. The saturation current is usually replaced with: [ ] T 3 ( [ 1 I s (T ) = I s0 exp α 1 ]) T 0 T0 T (2.7) 14

33 Where I s0 is the saturation current at STC, T 0 is the temperature at STC, and α = qe G ka, where E g is the band-gap energy of the semiconductor in use. Spec sheets provide the change in V oc and I sc with respect to temperature, yielding equations (2.8) and (2.9). V oc (T ) = V oc + k v (T T 0 ) (2.8) On top of temperature dependence, I sc also depends almost linearly on irradiance, yielding: I sc (T ) = I sc S S 0 (1 + k i (T T 0 )) (2.9) In the above, the k are the temperature coefficients and S and S 0 are the actual irradiance and the STC irradiance, respectively. Being able to change the IV equation will become important later, when the MPPT system is tested against changing environmental conditions. This will be discussed more then. Changing temperature is less important than changing irradiance in the context of this work, since it changes slowly relative to how quickly the MPPT algorithm samples and modifies the operating PV voltage Comments Models where one resistor [15, 16, 17], or no resistors [4, 12, 18] are used, can also be found. These simpler models are really just special cases of the two resistor model. However, they greatly reduce the mathematics needed, and so are considered separately. There are no papers dedicated to finding parameters for these simpler models, and so not much more can be said at present. These simpler models will be covered in more detail in Chapter 3. There are other models available. One example is the two diode model [10]. However, the PV model is being used in a closed loop 15

34 control system, and so the much more complex math is not justified. If the variations of the single diode model yielded significantly different results, then it might suggest one try to be even more precise. However, as will be illustrated, this is not the case. Consequently, a more complex model will not be considered in this work. 2.2 CONVERTERS Once a good PV model is found, the system needs to be set up. This involves the selection of a converter, as well as that of the load. For the converter, one could choose a DC/DC, or a DC/AC (inverter). For the DC/DC one could use the buck [12], the boost [19], the buck-boost [20], the ćuk, or a number of other converters. The converters that will be used are fairly simple, and a discussion can be found in [21, 22, 23], as well as many online references. For an AC system, one may use a DC/AC H Bridge type inverter [24] or a flyback inverter [25]. Using a DC/DC and a DC/AC together is also an option [26]. Usually not much emphasis is placed on the converters (or load) in the literature. In many ways the concepts of the control algorithm (to be discussed soon) will remain unchanged, and so the system is used as a means of demonstrating the control algorithm. In [19] a boost converter is used, but the authors state that the results obtained could be applied to any other converter topology as well. Other work discusses the control algorithm without even discussing a specific converter [27, 28, 29]. For the load, oftentimes a generic load is shown in the block diagram. It could be resistive, a battery bank, or otherwise. Consequently, not much is discussed in this section, just a passing acknowledgement. The load will be discussed more in Chapter 4 16

35 2.2.1 DC/DC Converters In Chapter 4, four different DC/DC converters will be used; the buck, boost, ćuk, and noninverting buckboost. Buck Converter The input and output voltage of the buck converter are related by v o = dv where d is the duty cycle, v is the input voltage, and v o is the output voltage, of the converter. The circuit is shown in Figure 2.2. One may notice that this circuit represents a low pass filter. Hence, for large enough capacitor and inductor values, only the DC component (the average - dv) is passed through. Later, when this is used in a PV system, capacitor and inductor values will be selected that are large enough to reduce ripple, but small enough for quick transients. Boost Converter The boost converter relationship is given by: v o = 1 1 d v The circuit is shown in Figure 2.3. The boost converter is effectively the buck converter run backwards. Noninverting Buckboost Converter The noninverting buckboost converter relationship is given by: v o = d 1 d v 17

36 i v Vo Figure 2.2: Buck converter. i v Vo Figure 2.3: Boost converter. i v Vo Figure 2.4: (Noninverting) buckboost converter. i v Vo Figure 2.5: Ćuk converter. The circuit is shown in Figure 2.4. The noninverting buckboost is the buck and boost converter cascaded together [21]. The (typical, inverting) buckboost converter is slightly modified in order to reduce the switches to one. However, the noninverting 18

37 will be used later, since it is not actually being implemented. Ćuk Converter The ćuk converter relationship is given by: v o = d 1 d v The circuit is shown in Figure 2.5. It is based on trying to cascade the boost and the buck converters [22]. This one, like the (typical, inverting) buckboost, is an inverting circuit, as suggested in the equation just given. The ćuk has a nice feature, in that a MOSFET source terminal is able to be connected to ground, making its gate drive circuitry easier to implement than some of the others [21]. When the buck converter is implemented in Chapter 7, one will see the, less than desirable, switch implementation. More information on these types of circuits can be found in many power electronics books, such as [21, 30] Comments As stated above, there isn t much to review when it comes to converters. In this paper only DC/DC converters will be considered. These will be the buck, boost, (noninverting) buckboost, and the Ćuk. More information will be given in Chapter 4. In particular, it will be seen that there are some issues that come up when applying a converter to a PV module. This seems to come from the fact that the PV module is closer to a current source than a voltage source. These issues do not seem to be covered in the literature, but will be discussed in this work. 19

38 2.3 CONTROL ALGORITHMS There have been many MPPT papers that focus on the control methods. This is not surprising, as MPPT is very much a control problem. However, many of them are slight modifications of existing methods. MPPT algorithms attempt to control the power of the PV panel. These algorithms are used in conjunction with (switching) converters, and so the control variable is the duty cycle. The main methods used for control will now be discussed Hill Climbing methods Perturb and Observe (P&O), and Incremental Conductance (IncCond) are perhaps the two most popular MPPT methods. For now a brief discussion will be given. In Chapter 5 a more detailed look will take place. Perturb and Observe The P&O method gets its name from how it works. The algorithm will change (perturb) the voltage of the PV panel (by changing the duty cycle), and then measure (observe) how the power changes. If the power increases the voltage will continue to be changed in this direction. If a change in voltage causes a decrease in power, the voltage will then be changed in the other direction. A condensed P&O algorithm is shown in figure The P&O method may sometimes move in the wrong direction. To illustrate this consider a scenario in which the voltage is increasing. Suppose going from v 1 to v 2 caused the power of the system to increase, and moving to v 3 decreases the power. 1 The open source program Dia was used to make the flowcharts in this dissertation. 20

39 Measure voltage and current v(n), i(n) YES p(n)-p(n-1)=0? NO p(n)>p(n-1) and v(n)>v(n-1) OR p(n)<p(n-1) and v(n)<v(n-1) NO YES Vset=Vset+ΔV Vset=Vset-ΔV Return Figure 2.6: Perturb and Observe MPPT algorithm. This could happen in two ways, as illustrated in the figures (2.7a) and (2.7b). Either v 2 < V m or v 2 > V m. In the scenario in Figure 2.7b, the voltage should be decreased, not increased, from v 2, because V m has been passed. Once it moves to v 3, it will then, without a decreasing voltage step size, move back to v 2, which is at a higher power, and so on again to v 1. For the scenario in Figure 2.7a, when the algorithm goes from v 3 to v 2 it should then turn around, but it will go to v 1 instead. This does not seem too terrible. In either case, the algorithm oscillates around 3 points. However, this incorrect tracking may be much worse when irradiance is changing (and so the IV curve is not fixed). The IncCond method is supposed to have better behavior in this regard. Incremental Conductance In [12], the authors state that the P&O method has a problem at steady state; namely that it continually oscillates around the maximum power point, since the 21

40 195 Power Curve 195 Power Curve Power v2 v3 Power v1 v2 194 v1 194 v Voltage (a) v 2 < V m Voltage (b) v 2 > V m Figure 2.7: Increasing power, and being left of MPP and being right of MPP. voltage is always being changed. However, they acknowledge that this can easily be remedied by decreasing the perturbation step size. This paper states that the main problem with the P&O method occurs during rapidly changing atmospheric conditions. The problem is that, for example, a decrease in power may be due to a decrease in irradiance, rather than because the operating point moved further from the MPP. Suppose, for example, that the voltage is increased, but is still to the left of the MPP. Then it should continue to increase. However, due to a sharp decrease in irradiance, the algorithm incorrectly decreases the voltage. (This will be seen more in Chapter 5.) To remedy this, they introduce the IncCond method, illustrated in Figure 2.8. The IncCond method claims to make use of the fact that dp/dv = 0 at the MPP, dp/dv > 0 to the left of the MPP, and dp/dv < 0 to the right of the MPP. Using dp/dv = d(iv )/dv = I + V di/dv, inequalities in terms of i and v are obtained. The derivatives are approximated numerically by sampling quickly enough. 22

41 Measure voltage and current v(n), i(n) di=i(n)-i(n-1) dv=v(n)-v(n-1) dv=0? YES NO YES di/dv=-i/v? di=0? YES NO NO YES di/dv>-i/v? di>0? YES NO NO Vset=Vset+ΔV Vset=Vset-ΔV Vset=Vset-ΔV Vset=Vset+ΔV Return Figure 2.8: Incremental Conductance MPPT algorithm. Comments Many papers incorrectly cite the P&O method as oscillating at steady state. Even though it seems this was never really an issue. Even a 1983 paper [24] shows it can be made to converge at steady state, as well as being acknowledged by [12], the paper that introduces the IncCond method. However, many articles erroneously state that P&O oscillates at steady state, but that the IncCond method does not. In [31] they state A problem with P&O is that it oscillates around the MPP in steady state operation. It is not that this is necessarily wrong. However, the implication is that this is not a problem with the standard IncCond algorithm. Usually this is acknowledged with regards to P&O, but not IncCond [19]. However, it is also an issue with the IncCond in its original form, as is sometimes acknowledged [32]. If one looks at both of the flowcharts, it is easy to see that neither algorithm, in their most 23

42 basic form, change the value of V. Another inconsistency with these methods comes in the discussion of the needed sensors. For example, [33] states 4 sensors are needed, which is more than for P&O, but they don t justify this claim. In [27] it is stated that the same number of variables are measured in both IncCond and P&O. In [19] they just say A disadvantage of the INC algorithm, with respect to P&O is in the increased hardware and software complexity; moreover, this latter leads to increased computation times and to the consequent slowing down of the possible sampling rate of array voltage and current. It would seem that the same number of sensors would be required. Power (or current) and voltage need to be measured for both of the algorithms to work, as seen in the flowcharts. It is true that the IncCond algorithm is slightly more complex, but this isn t really much of an issue since most microcontrollers should have plenty of memory to hold either algorithm. Also, as will be shown in Chapter 5, the MPPT algorithm should not be run at a high speed. The frequency of the switching, and perhaps even the sampling, will likely be much faster than the actual running of the algorithm. However, that should be done independent of the actual MPPT algorithm (as shown in the flow chart), and so the algorithm should have no effect on this. In Chapter 5, the IncCond method and P&O method will be compared in more detail Open Circuit Voltage and Short Circuit Current The open voltage method measures the open circuit voltage and operates at a voltage around 76% of that value [34]. This is based on the observation that the MPP voltage is almost directly proportional to the open circuit voltage. The short circuit method is similar, but uses current instead, operating near 85% of I sc [35]. Though easy to implement, and decently accurate, these methods have some obvious 24

43 drawbacks. For one, the V oc or I sc measurement would require the system to momentarily move away from where it is currently operating (presumably near the MPP). For better tracking this should be done often, but that has the obvious downside of being away from the MPP often. No power is produced at V oc or I sc. Depending on the intelligence of the implementation, it may not know to find a new value when environmental conditions change suddenly. Another downside is that the percentage of V oc (or I sc ) to operate at, is only a good approximation. It is neither exact, nor is it necessarily constant for the life of the PV module. This may require updating of the values used Fuzzy control Fuzzy controllers may also be used [36, 37]. A look up table is used for control decisions. A detailed look up table would require that it be specific to the PV modules (and configuration) being used. Also, due to PV degradation and mismatch (due in part to degradation, but also dirty modules), as well as the various temperature and irradiance conditions the table would have to represent, this approach has the potential for high complexity. If the table repopulates its values (based on measuring the IV curve for example), this time would be time lost in producing power. Though some simplified version may be appropriate, in general, fuzzy control does not seem worthwhile in MPPT algorithms. 2.4 OTHER ISSUES There are other concepts that have been touched upon in some of the papers found in the literature which will be discussed here. 25

44 2.4.1 Partial shading and mismatch Since no two PV modules are exactly alike for a given environment, even if they are the same brand and model, the MPP will differ. Suppose that two PV modules are connected in parallel, and one has its MPP at V m1, and the other at V m2, where V m1 < V m2. Then both (and probably neither) of the PV modules will be operating at their MPP. Likewise, cells in series will have problems with I m. For this reason, PV modules with very similar characteristics are usually sold together (a given production run of a PV model may be broken up into different bins of modules that better resemble each other). 9 IV Curve 600 Power Curve Current 5 4 Power Voltage (a) IV Curve Voltage (b) Power Curve. Figure 2.9: Three panels with Voc=30.7 volts. Two have Isc=8.6 amps, one has Isc=7 amps. Besides physical differences, modules may mismatch due to environmental reasons. In particular, if one module is getting more light than another, or if a module is partially shaded, they will likely not match well. In Figure 2.9 three modules, one which has a lower I sc value than the other two, are put in series with bypass diodes. Now the power curve has two peaks. For more mismatches, it is possible for the power curve to have many different peaks. (The case of when they are in parallel is less 26

45 severe since a large change in irradiance has only a small effect on V oc values. This will be shown later.) The control algorithms previously discussed can get stuck at a non-global maximum, greatly reducing the potential output power. To fix this, routines that check for the global peak will need to be implemented in conjunction with the main MPPT algorithm. This is done either by occasionally tracing out the IV curve, or responding to a change in power [38, 39, 40]. Notice the location of the larger peak in Figure 2.9b (this is generated using the BP NRM introduced in the next chapter, with values given in the caption). If only one panel is slightly off, the maximum still occurs close to 76% of V oc. For slight differences between many panels, the non-global peaks will be small. So, if the step size used in a hill climbing method is kept large enough, these local peaks may not be an issue at all. In general one would want to avoid any place where partial shading will occur. If there is a slight issue of partial shading, enabling a global IV scan occasionally can help. Otherwise, to be more optimal, one may wish to tune the algorithm to the location. With these thoughts in mind, the issue of partial shading will not be discussed in regards to MPPT. It will be explored again later though, in Chapter System with nonideal components Another important aspect to consider is the effect of using real components. In particular, a real circuit will not have an ideal switch. A transistor will be used which has delays, leakage current, and on resistance. A diode is often used as well, which will have a forward voltage value, as well as leakage current. Some of these issues are discussed in power electronics books, such as [21]. In the context of PV systems, there doesn t seem much discussion. In Chapter 6 these issues will be explored in the 27

46 context of a PV converter system. 2.5 CONCLUSION This chapter gave a short review of the components of a PV MPPT system. Many of these concepts and issues will be covered in more detail in the subsequent chapters. 28

47 CHAPTER 3 MODELING A SOLAR CELL This chapter will introduce methods for obtaining model parameters for four variations of the single diode model introduced in Chapter 2. These variations are the No Resistor Model (NRM), Series Resistor Model (SRM), Parallel Resistor Model (PRM), and the Two Resistor Model (TRM). Methods that make use of some of Matlab s built in functions to solve for model parameters will be introduced. The results of the different models are then compared to see if there are any large differences. If not, it may be justified in using the simplest model, the NRM, when modeling the PV system circuit. First, two modules are selected that will be used for the modeling done in this chapter, as well as some of the subsequent chapters. After this, the models will be considered. Since the NRM is the simplest model to analyze, it will be considered first. After this, the two one resistor models, SRM and PRM, will be considered before returning to the popular TRM. The IV curves of the various models used in this chapter for the BP and SP modules are given in figures 3.1 and 3.2. The power curves are given in figures 3.3 and SAMPLE MODULES The two modules that will be used for most of the work in this chapter, and subsequent chapters, are the 195 watt BP-SX3195, and the 305 watt Sunpower-305. The specs are given in Table

48 Model V oc (V ) I sc (A) V m (V ) I m (Ω) BP-SX SP Table 3.1: Two PV modules used for testing. These two modules have a substantial difference in their current to voltage ratio, allowing them to be useful PV representatives (the size is less important since everything can be scaled). They also have fairly different fill factors, as can be easily verified. 3.2 NO RESISTOR MODEL The No Resistor Model, NRM, is the model obtained by removing the series and parallel resistors from the model shown in Figure 2.1 (i.e. set R s = 0 and R p = ). For the NRM, equations 2.1 and 2.2 are simpler. The IV equation, for example, is i = I g I s (e v a 1) (3.1) For this model, the four equations based on the spec sheet are: 0 = I g I s (e Voc a 1) (3.2) I sc = I g (3.3) I m = I g I s (e Vm a 1) (3.4) I s = ai m V m e Vm/a (3.5) However, for this model, only 3 parameters are needed (a, I g, and I s ), and so one of the equations needs to be dropped. The last equation, 3.5, was dropped for what follows. After combining these equations to solve for a, the following is found: 30

49 I m = I sc I sc e Vm/a 1 e Voc/a 1 (3.6) Note that an exact solution for a is not obtained. This is true regardless of which equation was dropped. However, using some of Matlab s built in functions, such as fsolve or fzero, a can be solved for, and so too all the other parameters. (A good approximation which allows an exact equation exists, a Vm Voc ln(1 I m/i sc), but the implicit equation above was easy enough to solve for.) See Table 3.2 for the derived parameter values. Model I g (A) I s (A) a(v ) R p (Ω) R s (Ω) BP NRM e SP NRM e BP PRM e SP PRM e BP SRM e SP SRM e BP TRM e SP TRM e Table 3.2: Parameter values for different PV modules and models. 9 BP IV Curve 9 BP IV Curve Current Current NRM PRM SRM TRM Voltage (a) BP IV curves Voltage (b) A close up of the IV curves. Figure 3.1: The IV curves for the different models for the BP module. 31

50 6 SP IV Curve 6 SP IV Curve 5 Current Current NRM PRM SRM TRM Voltage (a) SP IV curves Voltage (b) A close up of the IV curves. Figure 3.2: The IV curves for the different models for the SP module. 200 BP Power Curve 195 BP Power Curve Power (watts) 100 Power (watts) 185 NRM PRM SRM TRM Voltage Voltage (a) BP power curves. (b) A close up of the Power curves. Figure 3.3: The Power curves for the different models for the BP module. In Table 3.3, the difference between where the spec sheet says the MPP should be and where the maximum is in the NRM due to dropping equation 3.5 is shown. This model appears to have done fairly well. This error in V m can be seen more easily in figures 3.3b and 3.4b. For both of these modules, it was confirmed that (V oc, 0), (0, I sc ), and (V m, I m ) (from the spec sheet) were points on the curve. 32

51 300 SP Power Curve 306 SP Power Curve Power (watts) Power (watts) NRM PRM SRM TRM Voltage (a) SP power curves Voltage (b) A close up of the Power curves. Figure 3.4: The Power curves for the different models for the SP module. Model V m (V ) I m (Ω) Max Power (W) BP-Actual BP-NRM SP-Actual SP-NRM Table 3.3: Comparison of NRM MPP vs desired MPP. 3.3 ONE RESISTOR MODELS Since 4 equations can be derived, and a one resistor model has 4 parameters, these seem like worthy models to consider. For simplicity, the model with the parallel resistor (and without the series) will be called the Parallel Resistor Model, PRM, and the one with the series (and without the parallel) resistor will be called the Series Resistor Model, SRM. Dropping the series resistor maintains an exact IV equation, as well as I g = I sc. So, this is a good one to start with. First, the PRM is considered: The equations needed here are similar to the ones derived previously. There are four 33

52 equations, and four parameters to solve for. Combining all four equations into one equation in terms of a was done, and the equation was solved using Matlab. The results are recorded in Table 3.2. For the BP module a negative resistor value was obtained: R = Ω. It is also interesting to note, that with a negative R value, the IV curve has a positive slope for low voltage values, as shown in figure (3.1). Another approach was also used to verify the results. Since the NRM worked well, the NRM parameters were taken as a starting point, and then R p was decreased from infinity, and new parameter values were obtained. The error of V m and I m were observed to see if these errors go to zero for a given R p value (recall that there was an error in the NRM as shown in Table 3.3). It is important to keep in mind that for each change in the resistor value, all the other parameters are changed as well. See Figure Errors for different Rp values in BP PRM Errors for different Rp values in SP PRM Im Error (A) Im Error Vm Error Vm Error (V) Im Error (A) Im Error Vm Error Vm Error (V) Rp (ohms) (a) Effects of R p on BP module Rp (ohms) (b) Effects of R p on SP module. Figure 3.5: The effects of R p on two different modules. By increasing V m for the BP module, one is able to cause the slope of the IV curve to decrease again. Increasing I m will cause it to decrease again. It would likely be possible to find the ratio of V m /V oc to I m /I sc (perhaps also as a function of the fill factor) in which the slope becomes zero. However, this is not treated here. 34

53 0.5 Errors for different Rs values in BP SRM Errors for different Rs values in SP SRM 1 Im Error (A) 0 Im Error Vm Error 0 Vm Error (V) Im Error (A) 0 Im Error Vm Error 0 Vm Error (V) Rs (ohms) (a) Effects of R s on BP module Rs (ohms) (b) Effects of R s on SP module. Figure 3.6: The effects of R s on two different modules. Next, the SRM is considered: This time two equations in two variables (a and R s ) were used, as solving for one variable was difficult. The results are recorded in Table 3.2. Note that this time the SP module yielded a negative resistor value. Also, similar to last time, a second method was used, where the effects of R s on the V m and I m errors were observed. This time R s was increased (and decreased) from zero. The results are shown in figure (3.6). 3.4 TWO RESISTOR MODEL The Two Resistor Model, TRM, is the model with both series and parallel resistors, as shown in figure (2.1). As stated previously, this is a popular model and is used in most of the papers on MPPT. The equations for this model were given in the previous chapter. Combining equations (2.3), (2.4), and (2.5), (such as by solving for I g in (2.3), I s in (2.4), and then plugging into (2.5)) yielded the following equation: ( IscRs Voc c 1 1 e a ) ( ) Vm+ImRs Voc + c2 1 e a = 0 (3.7) 35

54 where c 1 = V m V oc + (R s + R p )I m c 2 = V oc I sc (R s + R p ) Equation (3.7) has three unknowns: a, R s and R p. Based on the values given in [8, 11, 9], typically 0 < R s < 1 and 100 < R p < Based on this and the values of the PV specs, one should expect that c 1 is positive and c 2 is negative. One should also see that the first exponential term has an exponent that is negative since typically V oc > R s I sc. From this, it must also be that the 2nd exponential term is negative, or there is no solution. This gives V m + I m R s V oc < 0 = 0 < R s < V oc V m I m This may be useful in obtaining an upper limit for R s, helping to guarantee a solution. At this point, one can then use a range of R s values starting at 0 (which is the PRM), and increasing R s towards its maximum (not quite to its max, otherwise the parameter values become extreme, since R s is approaching the point where there is no solution). A plot of R s versus R p is shown in Figure 3.7. At this point there are an infinite number of solutions. However, to better distinguish this model from the NRM, PRM, and SRM, a point was taken where R s and R p are both significant. See Table 3.2 for the values chosen for each module. Clearly more information is needed in order to obtain a less arbitrary solution Effects of resistors It is pretty well known how the resistors affect the shape of the IV curve of a PV panel [8]. An increase in R s causes an increase in the slope of the IV curve near V oc. When R s is input into the NRM, to become the SRM, v becomes v + ir s. In other 36

55 500 Rp vs Rs for BP TRM 1400 Rp vs Rs for SP TRM Rp (ohms) 0 Rp (ohms) Rs (ohms) (a) Potential Rp,Rs pairs for BP TRM Rs (ohms) (b) Potential Rp,Rs pairs for SP TRM. Figure 3.7: The effects of R s on two different modules. words, R s acts as a variable horizontal shift, which depends on i. This, and the fact the R s was positive for BP, but negative for SP in the SRM, can also be used to explain why the BP and SP NRMs have their V m shifted in different directions from what the spec sheet dictates. In other words, based on the NRM parameters, and the spec sheet, one can determine whether R s will be positive or negative in the SRM. A decrease in R p causes a decrease in slope closer to I sc. To illustrate the effects better, the NRM was taken and resistance was added. See Figure 3.8. The effects by R s and R p on the slope will become important when explaining the difference in behavior of the NRM and the TRM in the next chapter. 3.5 CONCLUSION In this chapter, variations of the single diode model were considered. For the most part, the resulting IV and power curves were quite similar. However, there were some unexpected results. In particular, the single and double resistor models may yield negative resistor values. It was seen that a negative resistor, besides seeming unnatural in itself, could cause the slope of the IV curve to increase near short circuit 37

56 9 BP TRM IV Curve 9 BP TRM IV Curve 8 8 current Rs=0 Rs=0.5 Rs=1.0 current Rp=inf Rp=100 Rp=30 Rp= voltage (a) Effects of Rs voltage (b) Effects of Rp. Figure 3.8: The effects of PV resistance to the IV curve. current. This shows that one may actually do better in fitting the simpler NRM to the PV spec sheet than a more complex model. For the TRM case, negative resistor values were possible, though one could choose a point on the R s /R p curve that gave more realistic values. In fact there is a five parameter PV model (TRM) that is used with the System Advisor Model (SAM) by the National Renewable Energy Lab (NREL) that will yield realistic values, based on some further assumptions. The SAM program also contains another advanced model that is based on empirical information. This model is rather complex though, and would require a fair bit of work in trying to make it useful for converter/mppt purposes that will be seen in future chapters. At this point the PV model work is not complete. It is still necessary to see how these different models behave in the context of different converter topologies. This will be seen in the next chapter. 38

57 CHAPTER 4 MODELING THE CONVERTER SYSTEM In this chapter, the different PV models developed in the previous chapter will be used in conjunction with four common DC/DC converters to see how both the PV models, as well as the converters, compare. There were some slight differences in how the IV curves of the different PV models looked, but for the most part they were, as expected, similar. However, it seemed worthwhile to see how they compare when used in conjunction with a converter, as would be used in an MPPT system. For most of this work, simple buck and boost converters with a fixed voltage output will be used. Limited results from the noninverting buckboost, and the Ćuk will also be shown. Several papers use an RC load, rather than a fixed voltage. This type of load will also be considered briefly. Due to the large number of potential figures, only the NRM and TRM will be illustrated, with the SRM and PRM giving similar results. 4.1 BUCK CONVERTER To simplify the analysis, a fixed voltage output was assumed. In the typical buck converter circuit, as was shown in Chapter 2, there is no capacitor in parallel with the input voltage. During the time when the switch is off the input provides no power. However, for a PV module, a capacitor in parallel with the PV module is needed, so that even when the switch of the converter is off, the PV module is providing power. Otherwise, when the switch is off, the PV module would not be providing energy 39

58 from the sun. Only one PV module was used for these simulations. The results obtained here should also apply to a system where the converter covers an array of modules (just scale). The complete system, using the two resistor model, is given in Figure (4.1). Note that the switch occurs in zero time in what follows. A softer switch will be used in the actual implementation, as will be seen. i Rs L il Ig Rp C v Vout Figure 4.1: The TRM PV model connected to a buck converter System equations The following equations represent the circuit pictured in figure (4.1), where δ is 0 or 1 depending on the switch. i δi L = C dv dt δv v o = L di L dt (4.1) (4.2) When δ = 0, equation 4.1 yields the approximation v(t + h) = v(t) + h i(t) (4.3) C and 4.2 yields the exact equation i L (t + h) = i L (t) v out L h (4.4) 40

59 From the IV equation one is able to obtain i(t + h). However, for the TRM this means a lot of computation, since it would have to solve an implicit function every time. Another method would be to find the current by using the derivative of the IV equation allowing an explicit equation for i(t + h). The potential problem with this is that after a while the (i, v) points that are obtained might slowly deviate from the IV curve. To remedy this, at the end of every period, the implicit IV equation was used to make sure the points were on the curve. For one 30 second simulation, it took this method roughly 30 seconds compared to the almost 1500 seconds required for the implicit method. The results were compared, and were indistinguishable from each other. Hence, the latter, quicker method, was used for the TRM calculations. (The same could be said about the SRM as well.) The NRM and PRM did not have this problem, since their IV equations are not implicit. Due to the capacitor and inductor in our circuit, v(t ) = v(t + ), i L (T ) = i L (T + ), where T represents a point in time when the switch was turned on or off in the circuit. Due to the relationship between PV voltage and current, one also has that i(t ) = i(t + ). The equations for when δ = 1 are similar. The equation relating the input and output voltage for a buck converter is ideally given by v = v out /d (where d is the duty cycle). This will be seen to be true for duty cycles that yield PV voltages near the maximum power voltage (as will be seen below). The problem with this equation is that a small enough duty cycle will violate it, since the PV voltage is capped. In [41], the authors compare the buck, boost, and buckboost converter topologies and state a similar result, but in terms of resistance values (they used an RC load). In fact, with the circuit as it is, the battery would start draining and current would actually flow into the module. In the implementation of 41

60 the buck, as well as other converters, a diode will be assumed, which prevents back flow. In practice one could use a blocking diode 1 if the converter implementation does not make use of a diode. In short, the PV current will not be allowed to drop below zero in this model or the others (or allow battery discharge). The capacitor and inductor sizes should be small enough to allow for quick transients, but large enough to prevent large fluctuations of voltage and current when switching. Equations 4.1 and 4.2 can be used to get a handle on these values. A very high frequency switch would also reduce fluctuation, and thus allow for even smaller capacitor and inductor values. For this paper the period of the system is T s = seconds, to allow for quicker simulations. In other words, the switch is off (δ = 0) for (1 d)t s seconds and on (δ = 1) for dt s seconds. For the BP module, v out = 12 volts, and for the SP module v out = Results at STC In figures 4.2, 4.3, 4.4, and 4.5 one can see the effects of the capacitor and duty cycle on transient and steady state behavior. The inductor had a similar effect as the capacitor, and so changes to its value are not shown here. Note that for low voltages, the steady state has a large ripple. For voltages too close to V oc, the current is very near zero, making it unlikely that the converter will stay in continuous mode. So, the PV voltage must keep some distance from V oc. However, one can get rather close to V oc and stay in continuous mode. This is due to the sharp decrease in current near the open circuit voltage, which means that there is significant current up until the voltage is very close to V oc. This is evident in the figures of BP, where for a duty cycle of 40%, the voltage goes to 30 volts, and stays 1 Blocking diodes are not used as often in practice at present due in large part to better quality control. Still, one could potentially be used here if necessary. 42

61 BP NRM with Buck Converter BP NRM with Buck Converter PV Voltage 20 PV Voltage time (s) (a) L=0.1 henries, C=0.1 farads time (s) (b) L=0.1 henries, C=0.01 farads. Figure 4.2: A plot of the BP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = 12. BP TRM with Buck Converter BP TRM with Buck Converter PV Voltage 20 PV Voltage time (s) (a) L=0.1 henries, C=0.1 farads time (s) (b) L=0.1 henries, C=0.01 farads. Figure 4.3: A plot of the BP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = 12. below V oc, which is 30.7 volts. Another interesting result is that the TRM converges quicker for lower voltages, and the NRM for higher voltages. Consider Figure 4.4. The bottom curve oscillates between about 24 and 36 volts. Increasing either C and/or L by a factor of 100 only increased the period of the 43

62 65 SP NRM with Buck Converter 65 SP NRM with Buck Converter PV Voltage PV Voltage time (s) (a) L=0.1 henries, C=0.1 farads time (s) (b) L=0.1 henries, C=0.01 farads. Figure 4.4: A plot of the SP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = SP TRM with Buck Converter 65 SP TRM with Buck Converter PV Voltage PV Voltage time (s) (a) L=0.1 henries, C=0.1 farads time (s) (b) L=0.1 henries, C=0.01 farads. Figure 4.5: A plot of the SP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and v out = 24. oscillation. Also, decreasing the period T s by a factor of 100 seemed to have no noticeable effect at all. Notice that the curve is stopping at 24 volts. This is the minimum allowable value based on v out (due to blocking diode). It is interesting to note that it doesn t appear to be clipped at 24 volts though, but rather is a nice sinusoidal curve. Increasing the duty cycle from 80% to 90% yields a curve (not 44

63 shown) that oscillates between 24 and 29 volts. This curve is also sinusoidal, rather than appearing to be clipped. It is important to note that the decrease in amplitude of the oscillations when going from 80% to 90% is due to the blocking diode. Otherwise, in general, the lower the voltage, the larger the amplitude of the oscillation. Due to the blocking diode preventing the battery from discharging (and so send current left across the inductor), one might expect clipping. In both plots of Figure 4.4, the bottom curve oscillates between 24 and 36 volts. If the blocking diode is removed, it will be oscillating between 10 and 50 volts at the end of 30 seconds. Hence, the DC value is the same. So, the circuit is neither clipping, nor acting like a clamped circuit. Consider another example of this type using the BP NRM. Using a duty cycle of 80%, output voltage of 6 volts, but large (C=L=1) capacitor and inductor values, the simulation was allowed to run for a long time. See figure (4.6) for 6000 seconds of simulation time. In fact, though not shown, even after 24 hours the curve is still converging. So larger capacitor and inductor values to try to smooth out the horrible transients at low voltages will not solve the problem of large oscillations at low voltage. The current source-like behavior of the PV module at low voltages is the reason for this behavior. In a current source will be used to illustrate this further. 4.2 BOOST CONVERTER Just like for the buck system, a fixed DC output is used. In this case, the output needs to be a higher voltage. Here though, an inductor is not placed on the output, in series with the fixed voltage, as two inductors switched into series would dictate potentially two different currents, and so wouldn t work. See Figure

64 BP NRM with Buck Converter PV Voltage time (s) Figure 4.6: The TRM PV model connected to a buck converter. Ig Rp Rs v Vout Figure 4.7: The TRM PV model connected to a boost converter. 46

65 4.2.1 System equations The system equation for this circuit is simple: v (1 δ)v o = L di dt (4.5) Note that this is a first order differential equation, unlike the buck converter. One may expect the curves to look different. For the BP module, v out = 36V, and for the SP module v out = 72V. Also, the inductors used for the SP module are larger, as indicated in the figures. This was due to the poor convergence for the given duty cycles of 30%, 50%, and 70%, used for both modules Results at STC The results for the boost converter are shown in figures 4.8, 4.9, 4.10, and Like the buck system, the behavior is far worse at lower voltage values. Also, like the buck system, the NRM is slightly better for larger voltages. Note that the inductor values were different between the BP and SP modules. For the buck circuits the same values were used for both modules, but this time it was necessary to change the values in order to obtain results that weren t too unstable. The boost converter equation V = (1 d)v out holds for the convergent cases, as one would hope. Notice that for the values chosen for this converter system, the inductor seemed to have an effect on convergence as seen in Figure 4.8. This held true even when the duty cycle was changed from 70% to 80%. For the buck system the inductor and capacitor helped, though it seems more obvious with the boost converter. Still, there seems to be a point at which the operating voltage point gets so low that no reasonable capacitor or inductor value can smooth out the curve, as was shown in figure (4.6) previously. 47

66 BP NRM with Boost Converter BP NRM with Boost Converter PV Voltage PV Voltage time (s) (a) L=1 henry time (s) (b) L=10 henries. Figure 4.8: A plot of the BP NRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values. BP TRM with Boost Converter BP TRM with Boost Converter PV Voltage PV Voltage time (s) (a) L=1 henry time (s) (b) L=10 henries. Figure 4.9: A plot of the BP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values. 48

67 SP NRM with Boost Converter SP NRM with Boost Converter PV Voltage PV Voltage time (s) (a) L=10 henry time (s) (b) L=100 henries. Figure 4.10: A plot of the SP NRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values. SP TRM with Boost Converter SP TRM with Boost Converter PV Voltage PV Voltage time (s) (a) L=10 henry time (s) (b) L=100 henries. Figure 4.11: A plot of the SP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values. 4.3 OTHER TOPOLOGIES NonInverting BuckBoost This circuit is just a buck and boost converter cascaded together, with the inductor being shared. There are two switches that act at the same time. The typical (in- 49

68 verting) buckboost is able to use just one switch, but requires the signal be inverted. For the noninverting buckboost, the equation is given by the product of the buck and boost. The circuit is discussed in [21]. v = 1 d d v out The behavior was similar to the other converters: bad oscillation at low voltages. So, only one run is included, as seen in 4.12a. Using a duty cycle greater than 50% yielded a low, and very oscillatory, output, and so this is not shown. BP NRM with NonInverting BuckBoost Converter BP NRM with Cuk Converter PV Voltage PV Voltage time (s) (a) Noninverting BuckBoost with NRM BP; duty cycles of 30%, 40% and 50%; C=L=0.01; and V o =12V time (s) (b) Ćuk with NRM BP; duty cycles of 30% and 40%; L=L2=1H; C=0.01F, and V o =-12V. Figure 4.12: The behavior of two converters with the BP NRM Ćuk Converter It is like the buckboost, except it is a boost followed by a buck (boost-buck rather than buck-boost). It is also a well known converter, and more information can be found in [22, 21, 30]. This is like the (inverting) buckboost, and its equation is v = 1 d d v out 50

69 This converter uses two inductors, and they were larger than previously used values, being 1H each. Also, the capacitor was only 0.01 farads. Smaller inductors lead to very bad behavior even at larger voltages, which is why in Figure 4.12b, the duty cycle went no higher than 40%. Note that the output voltage was made negative for this run, v out = 12V Resistor/cap load Many papers ([19, 12] for example) use a capacitor in parallel with a resistor for a load, rather than a fixed voltage source. A buck converter with this setup was also employed, and the results observed. In general it has better low voltage behavior than that of the fixed voltage source. One still has that v = vout, but now both v and d v out can change. Using v out = i out R L, and i out = i, one obtains d v = ir L d 2 For larger d and smaller R L values, one can obtain a low voltage. See Figure 4.13 for some curves with different parameters. Consider one is at a terrible low voltage steady state, such as experienced in the previous sections. What happens when the constant voltage output is replaced with an RC load? When the PV voltage goes up, the output voltage also goes up. As output voltage goes up, so does the output current (by the same proportion - due to resistor). This causes the input current to also try to go up, causing the PV voltage to go down. In other words, the PV module voltage is not as free to swing up or down in this type of circuit. Consequently, oscillatory behavior will be reduced. The larger the output capacitor, the longer it takes. A fixed voltage output is similar to a capacitor with infinite capacitance (and proper starting voltage). 51

70 30 BP NRM with Buck Converter and Resistor/Capacitor Load 30 BP NRM with Buck Converter and Resistor/Capacitor Load 25 d=.5, R= d=.5, R=1.5 PV Voltage d=.5, R=.5 PV Voltage d=.5, R= d=.9, R=.3 5 d=.9, R= time (s) (a) BP NRM Buck, with C=C2=L= time (s) (b) BP NRM Buck, with C=L=0.1 and C2=1 Figure 4.13: The behavior of a buck converter with an RC load Constant current source input The behavior of the converters at low voltage, where the current was nearly constant, made the idea of trying one of the converters with an actual constant current source seem worthwhile. In Figure 4.14, the PV panel was replaced with an 8A current source (this value was chosen based on the BP module). This should make the converters behavior at low voltage (nearly constant current) more understandable. These converters work with voltage sources, not current sources. 4.4 CONCLUSIONS The TRM did better at lower voltage, because the slope was greater (in magnitude). The NRM behaved more like an ideal current source. However, at higher voltages the NRM did better. This is due to the closer to vertical slope near V oc, making it more of an ideal voltage source. However, the difference here was less pronounced, and so the NRM seems to be the one with slightly worse behavior. However, the behavior around 52

71 100 Constant Current Voltage time (s) Figure 4.14: Buck converter using 8A current source. V m seemed to be quite similar. Since this is where the system will be operating (it isn t difficult to program the MPPT algorithm to prevent low voltages, and this will be done in the algorithm proposed in the next chapter) the difference between the models is quite small. It was shown that for non-constant voltage loads, the oscillation at lower voltages decreases. It will also be shown later that losses in the converter system when considering real switches, also decreases this oscillatory behavior. With all these considerations in mind, as well as the NRM being easier to implement (and quicker to simulate), it seems that the NRM is preferred in MPPT testing than the more popular TRM. 53

72 CHAPTER 5 THE CONTROL ALGORITHM In this chapter many issues related to control will be discussed and simulated. To start, the differences between P&O and IncCond will be compared and explained, first at STC and then again for changing irradiance conditions. Issues such as how to sample for the algorithm, as well as step size changes will also be discussed. The P&O, IncCond, and a proposed method will be compared using a suggested testing standard, [7]. The results will mostly be considered using the NRM Buck system. However, the boost converter will also be shown. 5.1 STC CONTROL To begin with, a number of comparisons will be made at STC. In other words, the MPP is fixed, and so the algorithms need only converge to a point on a fixed IV curve, rather than deal with changing conditions, and so a changing MPP P&O vs IncCond In typical P&O and IncCond algorithms, the step size (and how it might change) are not discussed. So, for now, the voltage step size will be kept constant at 0.3 volts. Changing the step size will be discussed later. It is easy to see that dp dv is replaced with < and = are the same): > 0 is equivalent to di/dv > i/v (the cases where > 54

73 0 < dp dv = d(iv) dv = v di di + i dv dv > i v (5.1) Since the P&O algorithm looks at dp/dv > 0 and the IncCond looks at di/dv > i/v, they appear to be equivalent. The P&O algorithm checks on increase (or decrease) in power relative to changes in voltage, and the IncCond method checks whether change in current over change in voltage (i.e. a change in conductance; which is where the name of the algorithm likely comes from) is greater (or less or equal) than the negative of current over voltage. However, when the two algorithms were compared, their behavior is seen to be different. This comparison can be seen in Figure 5.1 (see appendix B.1 and B.2 for the simulation code of the algorithms). 30 BP NRM with Buck Converter and control 195 BP NRM with Buck Converter and control 28 P&O 194 PV Voltage PV Power P&O IncCond 22 IncCond time (s) (a) PV voltage output time (s) (b) PV power output. Figure 5.1: Results of P&O and IncCond methods using a fixed voltage step size of 0.3 volts. The original paper [12] does not explain the differences (nor any other papers that were considered in this research), but an explanation will be given presently. Suppose the P&O algorithm moved from position 1, to position 2 (v 1 to v 2 ). Then that algorithm is comparing the powers, P 2 vs P 1, at voltages 2 and 1. More exactly, one is comparing v 2 i 2 to v 1 i 1. Suppose one wishes to see if the change was positive. 55

74 That is, the following is checked: v 2 i 2 v 1 i 1 > 0 Compare this to the IncCond case, where the following inequality is checked: di dv = i 2 i 1 v 2 v 1 > i 2 v 2 2i 2 v 2 i 1 v 2 i 2 v 1 > 0 The fact that the derivatives are not being used, but rather numerical methods, is what makes the algorithms different. In the original paper [12], it is stated Hence, the PV array terminal voltage can be adjusted relative to the [MPP] voltage by measuring the incremental and instantaneous array conductances (di/dv and I/V, respectively).... The derivatives were approximated using the differences between the voltages and currents at positions 1 and 2. Many other papers seem to reuse this explanation, as if P&O and IncCond were not equivalent in the differential form. One should use i/ v rather than the derivative, since it is the difference equation form that sets the two algorithms apart, not the differential form. Now that the equations have been seen to appear differently, it will be shown mathematically that the IncCond is better in turning around once the MPP is passed. To do this, a comparison is made for the cut off case of the P&O algorithm; the case where v 1 v 2, but P 1 = P 2. It is this case where the algorithm is at the threshold of moving the voltage left or right. Suppose that v 2 > v 1, as the other case is similar. If the voltage were a pinch more to the left, power would have increased, and so voltage would have been increased after v 2 ; if it were a pinch to the right, power would have decreased, and so voltage would have been decreased after v 2. How does the IncCond algorithm fare for this case? In this case, v 1 i 1 = v 2 i 2, v 2 > v 1 56

75 Now, looking at the power curve, it should be clear that v 2 must be to the right of the MPP, and v 1 to the left (remember, at this point STC is assumed). However, the P&O algorithm does not know this. What about the IncCond method? For it to work better, it would be necessary that i/ v < i/v. Some simple algebra (requiring the substitution of v 1 i 1 = v 2 i 2 ), yields equation 5.2, which is clearly true. i 2 i 1 v 2 v 1 < i 2 v 2 (v 1 v 2 ) 2 > 0 (5.2) For the other case (v 2 < v 1 ) the math is quite similar. Now, v 2 v 1 is negative, so when multiplying be sure to flip the inequality sign, and the result is the same. This explains the slight difference between P&O and IncCond. This is a strong equality. Hence it is possible for a situation like that illustrated in ref 2.7b to move left, instead of right, at v 2, even if the power is greater at v 2. This explains why the IncCond turns around quicker than the P&O algorithm in Figure 5.1. Note, that the IncCond method does not always turn around when it should, but only better than P&O. Consider Figure 5.2. The power at v 2 is more than that at v 1, and so the P&O algorithm will perturb once more to the right (which in this case would cause it to go far off the plot - though it should be stated that a 3 volt step might be a bit larger than normal). What about the IncCond algorithm? It is easy to calculate i v =.293 and i 2 v2 =.283. Hence i v < i 2 v2, and so the voltage will move left. Assuming the step size doesn t change, it will go back to v 1. Since, i 1 v1 =.358, it will then move back to the right. In other words, the IncCond will bounce around v 1 and v 2, whereas the P&O algorithm will bounce around v 1, v 2, and v 3 (not shown) causing more power loss. It is important to note that if v 2 were slightly smaller, but still at a larger power, then IncCond would have failed just as P&O did. If one looks at Figure 5.1, they may notice that the IncCond seems to go more 57

76 Power Curve Power X: 23 Y: v1 vm X: Y: v2 X: 26 Y: Voltage Figure 5.2: An example of IncCond behaving better than P&O. left of the MPP than the P&O method. As if the advantage in one direction may be a trade off for the opposite direction. This is not the case. This is due to transient behavior (and sampling), as well as how small current and voltage values are handled. One could tweak the algorithm a bit more to fix this if one wished, such as by allowing more time to reach steady state, or using a smaller step size. See Figure 5.3 for how the two compare using a step size of 0.1 volts, as well as removing the averaging of samples to reduce the effect of transient behavior. Notice that the IncCond voltage doesn t go as high above V m as the P&O voltage does at steady state. It is interesting to note that without these changes to step size and sampling, the curves would have been the same. In other words, the advantage of the IncCond is not that pronounced. Conclusions for Control at STC For non-varying conditions, the two algorithms performed similarly, and both oscillated around the MPP. The IncCond method has a slight advantage, and could po- 58

77 26 BP NRM with Buck Converter and control 195 BP NRM with Buck Converter and control 25.5 P&O PV Voltage 25 PV Power IncCond time (s) P&O time (s) (a) PV voltage output. (b) PV power output. Figure 5.3: Results of P&O and IncCond methods using a fixed voltage step size of 0.1 volts. tentially oscillate slightly less due to turning around quicker once passing the MPP. However, for a small step size, the difference would be fairly negligible. It remains to be seen how they compare for changing conditions. There are also some issues, such as i/ v = i/v (dp = 0) when not at the MPP for the IncCond (P&O) method, due to approximating derivatives. However, these are small issues that are easily accounted for Single sample versus averaged samples In the description of the algorithms, little is said about how the sampling is done. Papers claim to be comparing measured voltage and current values to the prior measured values, but not how these values are obtained. In the above, averages were used, and were a second apart. In particular, 0.5 seconds was given for the transient to end, and then voltages and currents were recorded for 0.5 seconds at 1kHz. These values were then averaged. (Originally, checking the standard deviation was also done to make sure one was at steady state before changing the voltage, but this proved not 59

78 to work well during changing irradiance conditions.) In this section, only one sample will be used to calculate the voltage and current used within the algorithm. This will make use of voltage and current sampled at 10ms, 100ms, and 1s. The results for the P&O algorithm (the IncCond are similar) are shown in Figure BP NRM with Buck Converter and control ms 10ms PV Voltage Vm 100ms time (s) Figure 5.4: Results of P&O method using one voltage/current sample. Why do the shorter sampling times yield worse results? Suppose one is trying to increase the voltage. Then sampling too soon reads a smaller voltage, due to transient delay, than what the actual steady state value is. Perhaps now the voltage is set to the right of the MPP, but the actual value was still left of the MPP. So, the voltage is increased again; perhaps a couple more times. Now the voltage is finally sampled to be to the right of the MPP, and so the algorithm starts trying to go back. The voltage is decreased, but it is still swinging up, and so the power decreases. So, now it tries increasing again, etc. This type of behavior can result in going far past the MPP before everything is aligned enough to begin coming back. A slightly slower sampling may cause the values to be sampled nearer the overshoot, i.e. beyond where 60

79 the voltage is set, rather than premature. It may now try to turn around sooner than it should. Once again resulting in going the wrong way a bit, though not quite as bad as sampling too soon. It appears that sampling every second yields the best results. In fact, those results yield slightly better results than the averaging used previously. Averaging might still be preferable in actual system, where noise could interfere, depending on sensitivity of equipment used. Perhaps a smaller interval of averaging could also be considered. The best approach will depend on the frequency of the switch, as well as L and C values. For now though, the averaging method of 1/2 second will continue to be used Decreasing step size The oscillation around the MPP could easily be overcome by decreasing the voltage step size, v, previously set to 0.3 volts. It is fairly easy. Suppose that one wishes to increase the voltage from v 1 to v 2 = v 1 + v. For a buck converter, the following is true: Solving for d 2 yields v 2 = v 1 + v v out d 2 = v out d 1 + v d 2 = v out /(v out /d 1 + v) Decreasing the step size is now just a matter of changing v before using it; setting v new = α v old, where 0 < α < 1 is one approach. In Figure 5.5 three different α values are used. The P&O and IncCond methods yield identical results. Note that if α 0.5, it is highly unlikely that the system would converge to the MPP. This should be clear from the fact that Σ 1 2 n = 1. Obviously this is no good. Using a method where α is a variable, or only decreases the step size when the voltage changes direction, would probably be more sensible. These will be discussed more later. 61

80 30 BP NRM with Buck Converter and control PV Voltage time (s) Figure 5.5: P&O with changing step for α = 0.2, 0.5, and 0.8 (bottom to top), with V m also shown. 5.2 CONTROL UNDER CHANGING CONDITIONS In this section, only irradiance changes will be considered. Temperature changes are slower to occur, and so are fairly negligible. In what follows, the irradiance changes will be modeled by changing the I g parameter of the NRM. There are many papers that compare algorithms [12, 31], but the method of comparison is usually not well-defined, and there is no standard method for comparison. Recently, a standard method for testing the algorithms was presented [7]. This proposal suggests the irradiance input shown in Figure 5.6. The times when it changes are at the following seconds: 60,140,200,204,264,266,386,388,448,452,512,592; starting at 0 and ending at

81 1100 Ropp Input 1000 Irradiance (W/m 2 ) time (s) Figure 5.6: Changing Irradiation Input for MPPT testing P&O vs IncCond For now, the step size will be kept constant for both algorithms, to allow a fair comparison between the two. Changing step sizes will be considered more later. In the paper introducing the IncCond method, [12], it was claimed that this algorithm was developed to deal with poor tracking of the P&O algorithm during changing irradiance conditions. One can show mathematically that the IncCond is better during changing conditions. The mathematics are the same as in the STC case, except this time v 1 is on IV curve 1, and v 2 is on IV curve 2, where, assuming conditions changed during this time, the curves are not the same. The results of the two algorithms is given in figures 5.7 and 5.8. The P&O algorithm has an efficiency of 98.81%, and the IncCond algorithm gets 98.85% efficiency. The efficiency is calculated starting at 60s, since initial conditions 63

82 30 BP NRM with Buck Converter and P&O method 200 BP NRM with Buck Converter and P&O method PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.7: Results of P&O algorithm with Ropp input. should not count against the algorithms used. It is seen that the IncCond method is better, but not substantially so. Still, if there is no disadvantage to using the IncCond method in place of the P&O method, it should be done. 30 BP NRM with Buck Converter and IncCond method 200 BP NRM with Buck Converter and IncCond method PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.8: Results of IncCond algorithm with Ropp input. Notice that the algorithms move in the wrong direction near the beginning. This is due to the fact that increased irradiance might give an increase in power, even if moving in the wrong direction. In other words, the increase in energy due to increase 64

83 in irradiance is greater than the decrease in energy due to moving away from V m. Even though the IncCond is better at recognizing when it passes the MPP, it is not perfect. Both also fail to track the decreasing ramp that occurs around 500s. In the next section, an algorithm will be created to be more intelligent than those illustrated here. 5.3 PROPOSED ALGORITHM The algorithms used above were not in the most implementable form. They didn t discuss things such as initial conditions, or how to overcome some of the issues such as step size, or moving in the correct direction during changing conditions. This is what makes comparisons a bit difficult. In this section a proposed method is developed in stages. This algorithm will be based heavily on the ideas present in the P&O and IncCond algorithms. Some of what was stated here was actually used in the above implementations of the P&O and IncCond runs. This was because the basic algorithms themselves did not have enough information to implement them. This includes things such as using averaging for the samples, as well as initial condition settings. For the initial conditions in what follows, the algorithm will begin at V oc. This is easily reached by using a duty cycle of 0. Then the algorithm will move to 85% of V oc, and then 82% of V oc. From there it will move into the main algorithm, starting near V m. The flow chart of the completed algorithm is broken up into figures 5.20, 5.21, 5.22, and 5.23, included at the end of the chapter. 65

84 5.3.1 Step 1 To start, a power comparison (like P&O) was used due to easier implementation. Later it will be modified to behave more like the IncCond method. Some of the first things that will be done are having the step size decrease as the voltage approaches V m. Earlier this was done in an obviously non-intelligent fashion. Namely, the step size decreased all the time by a fixed percentage. Here, the step size will only decrease when the algorithm changes direction. So, if the algorithm is moving to the right, it will continue to do so using the same v. However, as soon as power decreases, and the voltage starts going left, v will be decreased (and similarly in the other direction). To start with, it will decrease by 50%. How much to decrease by, as well as what v should start as, will be discussed more in Section 5.4. For now, the step size will start at 0.3 volts, to be more consistent with the algorithms tested above, which used that value. This beginning step size will be represented by the variable V S to make discussions easier in the future. What about when the step size gets close to zero? What happens if the irradiance suddenly changes, and it is extremely slow in following? A couple methods have been tried. To begin, when v < 0.05V, the step size will no longer decrease. It will now follow the typical P&O algorithm, just oscillating back and forth with a very small step size. As noticeable in the power curves previously shown, being off by this much voltage is negligible, not to mention probably well within any error in measurement. Consequently, it is reasonable to oscillate at such a small step size. Furthermore, this allows slowly changing irradiance on a clear day to be corrected for. The problem is then when conditions change rapidly. To account for this, if during this phase, the voltage changes in the same direction more than a predetermined number of times (this will be dependent on α, but for now it will be set to 3 since α = 50%), the step 66

85 size gets set to the original v = V S value. To better understand what was discussed here, the code for the algorithm has been included in Appendix B.3. Once the algorithm is initialized, only flags 0 and 1 are used. Once the voltage gets small, as discussed above, the flag is set to 1, and it behaves like the P&O algorithm. When it moves in the same direction a few times, v is made large again, and flag 0 is used. The results are shown in Figure BP NRM with Buck Converter, Proposed Step BP NRM with Buck Converter, Proposed Step PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.9: Results of proposed algorithm, step 1, with Ropp input. This worked fairly well, but its efficiency is worse than that of the others; 97.68%. This is due to the slight delay when irradiance changes, due to waiting a few cycles before increasing the step size again. It still has similar issues to the other algorithms though, such as around time it did not move. The reason is that while oscillating at v <.05, when the voltage goes left, power decreases. So, it goes right. Power also decreases in that direction. So, it goes left, etc. Also, it still tracks in the wrong direction at the beginning. These are issues that were inherent in the other algorithms as well, and will be addressed in the next step. 67

86 5.3.2 Step 2 To correct for the issues last time, a couple things were done. These corrections require a new flag setting, 2, and so this is discussed first. While the flag is set to 2, there are two possible actions that could be taken, and these actions might oscillate between each other so long as irradiance is changing. In the first setting (f2chk=0), the duty cycle is kept the same, and f2chk is set to 1 for the next cycle. So, at the next cycle it will be seen whether the power changed for a fixed duty cycle (and so fixed voltage). The voltage step is then set to v = 2K P P M (5.3) The K is a constant that represents roughly by how much the voltage should change for a given percent change in power; P M is the maximum power of the module; and the factor of 2 is to make up for the lost cycle due to using the same duty cycle (voltage) twice in a row. This equation will be discussed a little more in the next section. To start, some rough values were given; P M was set to 200, and K was set to 4 (voltage changes almost 4 volts when going from percent of max irradiance). When the v is calculated in equation 5.3, if it is less than 0.05 volts, the flag is set back to zero, and the voltage is perturbed by its original step size (in this case 0.3 volts). Otherwise the flag remains set to 2, and f2chk is set to 0 again. Now, the use of flag 2 will be shown. First, tracking in the wrong direction will be addressed. If power increases (decreases) by more than 5%, while at flag 1, voltage will be increased (decreased) by V S, and the flag set to 2, and f2chk set to 0. This will see if irradiance is still changing, and then change the voltage based on equation 5.3. The other issue to correct for is the tracking behavior during a decreasing (ramp) 68

87 irradiance. Now, if the voltage changes direction more than three times consecutively, it will be assumed that the irradiance is decreasing, and so the voltage will be dropped by v, where v is calculated between the latest power measurement, and the power measured the first time the voltage direction changed, using equation 5.3. The flag is now set to 2, and f2chk to 0. In both cases, once the irradiance seems to have stopped changing, the voltage is perturbed, and the flag is set to 0, to allow convergence to the MPP. The results are shown in BP NRM with Buck Converter, Proposed Step BP NRM with Buck Converter, Proposed Step PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.10: Results of proposed algorithm, step 2, with Ropp input. This time the algorithm gave 99.64%. This is certainly an improvement. See appendix B.4 for the code Step 3 For this step, the power comparisons were changed to be comparisons between i/ v and i/v for flag 0. In other words, the incremental conductance method was incorporated. The efficiency obtained is still 99.64%, and the results are shown in Though the efficiency didn t seem to go up in this run, other runs (slightly modified 69

88 parameters) have shown a slight improvement. See appendix B.5 for the code. 30 BP NRM with Buck Converter, Proposed Step BP NRM with Buck Converter, Proposed Step PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.11: Results of proposed algorithm, step 3, with Ropp input Step 4 Though the previous setup seems pretty good, there are a couple small changes that were made based on some issues that could use improvement. For one, notice that after steady state irradiance conditions are obtained, and flag 2 goes back to flag 0, the voltage perturbation was always to the left. Now, it will move in the direction that the irradiance just moved by keeping track of two new variables; inc and dec. The result of this is seen, for example, around the 140 and 450 second marks (compare to previous step). Another thing that happened during some runs, was that, due to transient, flag 2 thought it had returned to constant irradiance conditions, even when it did not. Now, an extra check will be given. This was done by adding an f2chk=2 option under flag 2. There is a trade off here, since there is now a longer delay. However, it does prevent the flag from changing incorrectly while irradiance conditions are still changing. See the appendix for more details. The efficiency of this newest run was 99.72%, and the results are seen in Figure

89 30 BP NRM with Buck Converter, Proposed Step BP NRM with Buck Converter, Proposed Step PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.12: Results of proposed algorithm, step 4, with Ropp input. Though the results appear better, one can not conclude that this algorithm works better than the standard based on the few runs shown above. For example, for the P&O and IncCond results shown previously, they both happened to track in the wrong direction at the beginning. This was merely because the voltage so happened to be moving left exactly when the irradiance changed. If this change was delayed by a second or two, it would have started in the correct direction. Also, if the step were smaller it would have possibly followed it very well (granted it would have been slower in the sharper ramps). Consequently, it is hard to state anything conclusively based on what has been shown so far. In the next section, the proposed algorithm will be discussed further, and some optimizations made and robustness shown. After that, the final version can be given with a little more confidence. See appendix B OPTIMIZING ALGORITHM PARAMETERS Though the last algorithm behaved fairly well, it wasn t necessarily optimized as far as some of the parameters went. In Figure 5.13, efficiencies for a range of α and V S values are shown. Also, in Figure 5.14 the efficiencies are shown for α, averaged over 71

90 the V S values, and V S, averaged over the α values. Efficiency for different alpha and VS values 1 efficiency alpha VS Figure 5.13: Efficiencies for different α and V S values for MPPT algorithm step Efficiency for different alpha values 1 Efficiency for different VS values efficiency efficiency alpha (a) Efficiencies of α for average of V S values VS (b) Efficiencies of V S for average of α values. Figure 5.14: Efficiencies for different parameters for MPPT algorithm step 4. The highest efficiency was at a V S value of 0.55 volts, and an α value of 66%; 72

91 yielding 99.9% efficiency. Unfortunately, these results can not be taken without further consideration. This is due to the fact that it was run for only one setup, and some behavior is just chance; similar to how P&O and IncCond would have correctly tracked the first ramp, if the ramp was delayed for 1-2 seconds. In fact it is this type of behavior, as well as limited run time, that likely explains the ill-formed efficiency plots. Hence, it seems a better look should be taken. For the SP module, the values obtained were similar for α, even having a max near 66% on the α curve. However, V S did better closer to 1 volt. This is not surprising given that the SP module has a V oc nearly double that of the BP module. See Appendix C for the results of the SP module for the tests in this section. In the following subsections, an attempt at generalizing these values will be made, as well as a discussion about the optimal K value Finding α Consider a number 0 < x < 1 that one wishes to approach (like one wishes to approach V m ). If one is moving towards the number, one continues to approach using the same step size. If one passes the number, one then goes in the opposite direction, and the step size is α times its previous value, just as was done in approaching V m in the algorithm discussed above (for flag=0). Suppose at each state one took the error (distance between x and actual position), squared this error, and added it up for N steps. This was done, for a range of α values, and a range of x values. For each α value, the sum of squared errors were averaged for the various x values. To imitate the PV MPPT system a little better, the step size was not allowed to get smaller than The program used is given in Appendix B.7, where the range was from 0 to 3, rather than 0 to 1, to better match PV voltage changes. The plot obtained is shown in Figure 5.15a. 73

92 The best value seems to be around 60-70%. However, there are some properties of actual V m tracking that are not properly reflected here. In particular, the algorithm does not know to turn around once V m is passed. It could continue on, as illustrated earlier in this chapter. First suppose that the power curve were symmetric. Then 1 volt left of V m would have the same power as 1 volt to the right of V m. So, if the algorithm moved from V m 1 to V m +1, the step size would be 2 volts. However, since the power is the same at both positions the algorithm would potentially do nothing (depending on how this possibility was implemented in the controller). In particular, it would be necessary that Position-V m is greater than 50% of the step size, in order to turn around. The results of this are shown in Figure 5.15b, where the curve is less smooth than before, but still roughly the same, with the minimum closer to 70%, and a slight increase in error near 65%. In the more realistic case, the slope of the power curve is steeper to the right. So, rerunning the previous program, but this time implementing that turn arounds don t happen unless the position is 40% past x while going right, or 60% past x while going left, gives the output shown in Figure 5.15c. Notice this caused the curve to shift slightly to the right. However, there is one more consideration. The error should be larger on the right, since the power curve is steeper there. Being 1 volt larger than V m yields a smaller power than being 1 volt smaller. Multiplying all errors to the right of x by 3 yields the result shown in figure 5.15d. One last test will be given. This time the algorithm will be done, but using the BP module and its V m and power values to calculate the error. See Appendix B.8 for the code. The results of this are shown in figure It is seen that a value a little less than 70% seems to be ideal. There are some additional things to consider. For one, the difference in error 74

93 250 Error for different alpha values 250 Error for different alpha values Total Error Total Error alpha (a) Sum of errors for different alpha values for a simple case alpha (b) Turns around 50% of step size past V m both ways. 250 Error for different alpha values 300 Error for different alpha values Total Error Total Error alpha (c) Turns around 40% or 60% of step size past V m alpha (d) Turns around 40% or 60% of step size past V m, and 3 times larger error to right. Figure 5.15: Errors for various α values for a simple tracking algorithm. between left of V m and right of V m decreases as one approaches V m, meaning a smaller α value might make sense if the irradiance didn t change much, and V S were small. However, for the input being considered this is not the case. In general, one might expect that a better fill factor would imply a larger inequality between the power left and right of V m. One need also consider picking a value that seems to have some robustness. In particular, there are a number of values that seem to yield almost the same result, as 75

94 seen in Figure 5.14a for example. Even though 66% yielded the best value for the BP and SP modules, it seems to have a slight decrease around that value in many of the plots in Figure From those plots, a value of 60% seems like it may be slightly better. In Table D.1, in the appendix, these two α values, as well as a value of 50%, are compared for a handful of different PV modules. Though the results were close, a value of 66% is what will be used in what follows. 26 Error for different alpha values Average Error alpha Figure 5.16: Average error for different alpha values using the BP module K and V S The proportionality constant between percent change in power to change in voltage, K, (previously given as 4) should also be considered in more detail. The results of a range of K values for the BP module are shown in Figure 5.17 (SP results are shown in Appendix C). The BP module is best for a K value near 4.4, and SP around The value of 4.4 is not far off from the value of 4 used previously. So, it wasn t a bad guess. Note that 4.4 volts is almost 18% of V m for the BP panel, For the SP module, a value of 5.75 is seen to be optimal, which is a little more than 10% of V m. This will now be investigated in more detail. 76

95 1.002 Efficiency for different K values efficiency K Figure 5.17: Efficiencies for different K values based on α = 66% and V S = 0.5. Decreasing irradiance by 80%, decreases power by slightly more than 80%. This is clear for the NRM module, where an 80% decrease in irradiance corresponds to an 80% drop in all current values. However, the V m value also decreases slightly. If it were exactly 80%, it would mean that V m did not change. If one assumes the NRM, and that the ratio of I m to I g remains constant (roughly 90%), then the following equation can easily be derived from the IV curve and its derivative: V m2 e Vm2 Vm1 a V m1 = I g2 I g1 (5.4) If one has a available, one could see how V m changes due to a change in I g (irradiance). In Figure 5.18, a plot shows how V m changes due to a decrease of 80% in irradiance. As can be seen, the larger the starting V m, the smaller the change. This would explain why the SP module voltage changed by about 10%, but the BP changed by about 18%. Also, one may see that changes in a affect the curve (though the shape remains unchanged). On top of that, for a more realistic model, such as the TRM, V m will likely decrease more. This is because the slope of the IV curve decreases 77

96 1 How Vm changes when irradiance decreases 0.95 Vm2/Vm a=2 a= Vm1 Figure 5.18: A curve showing how V m changes for different starting V m values, 2 a values, and an 80% drop in irradiance. more when left of V m. Note that due to the decreasing change in percentage of V m as V m1 increases, the voltage range is fairly similar for a PV panel having a V m of closer to 20, as one closer to 100. However, this exact range changes based on a, and so using a fixed voltage would probably not work so well. It is also important to notice that the curve is roughly logarithmic, and is concave. Because of this, it is probably better to approximate the voltage, linearly, by using a slope of more than the actual change. For example, suppose for a PV module that at a 20% irradiance V m = 50, and at a 100% irradiance, V m = 55. If one were to draw a straight line from (.2I g, 50) to (I g, 55) on a plot of V m versus I g, that line would be beneath the actual curve the whole time, since it is concave. Because of this, it would be better to use a line of a larger slope to better represent V m versus I g. Recall that for the proposed algorithm, once irradiance stops changing, the algorithm gets perturbed one more time in its current direction. For this reason, one might also wish K to be a little less. With these things in mind, and after doing some simulations, a value of K = 16%V m was decided on. This value for K is based on a 78

97 PV system of only one module. Hence, in a more typical system (using a string of modules generating a much higher voltage), a value smaller than 16% should be used. In general, it will depend on the input range of the converter. In the next chapter, when a string of modules are used, this value will be changed. The V S values that performed best for the BP and SP modules were roughly 1/8 of the calculated K value, as seen in Figure 5.14b. This is a reasonable starting step size (the step size does decrease in time), as it is reasonably large, without being close to K, which roughly represents the complete range of voltage values for different conditions. 5.5 PROPOSED ALGORITHM - GENERAL Using the values obtained in the previous section, the final version of the algorithm has been set. It is the same as the algorithm given during the 4th step, only now it has parameters that are less arbitrary, and are fairly optimal. These parameters are obtained with α = 66%, K = 0.16V m, and V S = K/8. For the BP module, this gave an efficiency of 99.88%. Even though the efficiency could be made higher, it did not seem wise to optimize the setup for the particular input discussed previously, but rather try to generalize it a bit. This is what was done. Keep in mind that some parameters might reasonably be changed to better suit different environments (cloudy Florida versus clearer Arizona for example). See Appendix D for the results of a handful of PV modules that were tested with this algorithm for the buck converter. They all yielded well above 99% efficiency. The smallest panel and the largest panel (in terms of power) had the worst results due to K better representing the average of these panels. However, once an actual converter 79

98 system is created, and so the voltage range for the input is determined, this value can be modified slightly. For now it is pretty good. 5.6 COMPARISON TO OTHER ALGORITHMS IN THE LITERATURE There are a number of algorithms that are some variation on the P&O or IncCond algorithm. In this section, some comparisons will be made between the proposed algorithm just presented and some that have already been proposed. The idea of a changing step size is nothing new, and there are a number of methods to do this, as will now be discussed. One method is to change the step size all the time. This was illustrated previously in the simple case where it was changed by the same percentage each time. However, that was a simple example. A more practical approach is basing the step size changes on power changes. In [42], the duty cycle changes based on D(k) = D(k 1)+a Slope, where D(k) is the duty cycle at iteration k, the slope is 1 or -1 depending on if power increased or not, and a is given by a(k) = M( P )/a(k 1). In [43], a slightly more straightforward version is given: P (k) P (k 1) D(k + 1) = D(k) ± M V (k) V (k 1) where they claim M can be evaluated as M = D max V max / P max. They also mention just using: D(k + 1) = D(k) ± M P (k) P (k 1) The results are rather limited, and one can t conclude if M was optimized. It is very similar in [32], except the equation for M in terms of maximum duty cycle, voltage, and power changes, is given as an upper bound. It is interesting to point out that duty cycle is used often in these papers, rather than the voltage. Though the equation can 80

99 likely be changed fairly easily, it does seem to make some of these methods converter dependent. There are other papers; some are used with P&O [28, 43, 42], and others are used with IncCond [32, 44]. However, it seems no optimal solution is ever given for M. Tuning M is seen as a major hindrance [20, 43]. In [19] an attempt at optimizing the step size is made based on a small signal model. Unfortunately, this value is heavily dependent on PV parameters and so not too generic, and it also can track incorrectly [20]. In [45], the voltage per power to duty scalar, M, is made variable. In [20], a fixed percentage is used (like in the previous section). Namely, the step size changes by a fixed percentage, in this case 50%, whenever the voltage changes direction. No explanation is given for choosing 50%. When the irradiance changes (presumably one is already at the MPP and steady state), the program tries to catch it. It isn t explained what the response to such a change is. One can assume the step size is increased again in some fashion. (This is the problem with many conference papers. They always seem incomplete.) A method for coping with changing irradiance conditions is also given in [46, 47]. This method, after further consideration, is similar to part of the algorithm proposed in the previous section. In this method, the authors propose sampling twice in every iteration. For example, if the P&O algorithm was run every second, there would be a sample of v and i every half second. The power change during the first half second would be considered to be due to both the change in voltage and irradiance. While the power change of the second half second would be assumed to be due entirely to irradiance changes. Using these measurements, one could solve for the change due only to the perturbation. The assumption is that the irradiance changes fairly linearly. In order for this to work though, it seems one would have to assume that the transient due to the perturbation is over before each (1/2 second) sample was taken. This means that one is running their algorithm fairly slowly (when irradiance 81

100 isn t changing), since it is waiting well beyond steady state for the main algorithm to run. In this chapter s proposed algorithm, during flag 2, the duty cycle is kept constant, so that one can see the effects of changes in irradiance. So in this respect the algorithms have a similarity. However, this was only during a flag setting of 2 in the proposed algorithm. In [46, 47] the voltage step size is based on this change at all times, it seems. 5.7 MPPT WITH A BOOST CONVERTER As stated before, it is often implied or even claimed outright, that the MPPT algorithm is independent of the converter used [19]. This section is included to verify that the control algorithm could also be implemented using a boost converter. The main difference is that the relationship between the duty cycle and the PV voltage is now different. In particular, the buck converter equation for changing the duty was: d 2 = v out /(v out /d 1 + v) For the boost converter this was changed to: d 2 = d 1 v/v out Also, to set the PV voltage to 85% of V oc, d = 1.85V oc /v out was used. Other changes are that L was changed from 0.1 to 1 H, and v out was set to 36 volts instead of 12. Otherwise, everything was the same. The α, K, and V S values were obtained as discussed previously. The results are shown in figure The efficiency was 99.89%. 5.8 CONCLUSIONS In this chapter, control algorithms for MPPT were discussed in more detail. The differences between the P&O and IncCond methods were shown, and explained. As 82

101 27 BP NRM with Boost Converter and control 200 BP NRM with Boost Converter and control PV Voltage PV Power time (s) (a) Voltage time (s) (b) Power. Figure 5.19: Results of proposed algorithm using a boost converter. can be checked by comparing the algorithms in the appendix, the P&O algorithm is simpler than the IncCond algorithm. This is even more true when the step size starts changing. This was why the IncCond method type comparison wasn t used until step 3 in the proposed algorithm. The effects of sampling were also discussed, as well as why fast sampling could cause large errors. A new MPPT algorithm was proposed, and its parameters optimized. It was then compared to some other hill climbing algorithms. Though there were similarities, which is far from surprising given the simple idea behind hill climbing, the proposed method is noticeably different. In some ways a bit more complicated as it has different modes in which it runs in; namely flag 0, 1 and 2. The proposed method, for flag 2, had power change for a fixed voltage, unlike most of the previously proposed methods. When the flag is at 0, it is changed by a percentage, 66%, that has been shown to be fairly optimal, unlike [20], which just used 50%. Flag 0 also imitates the incremental conductance, where di/dv is compared to i/v. However, under flag 1, the powers are compared (like P&O). The reason was that for small perturbations, di and dv might be small, and so any noise, or error in measurement 83

102 due to, say a transient, could cause significant changes in the ratios. Consequently, the power comparison seemed a little more robust. Lastly, it was shown that this algorithm could be used with other converters, by using it with a boost converter. 84

103 Figure 5.20: Main loop of proposed algorithm. 85

104 Figure 5.21: Flag 0 of proposed algorithm. 86

105 Figure 5.22: Flag 1 of proposed algorithm. 87

106 Figure 5.23: Flag 2 of proposed algorithm. 88

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