Modeling and Robust Control Design for Distributed Maximum Power Point Tracking in Photovoltaic Systems. Audrey Kertesz

Transcription

1 Modeling and Robust Control Design for Distributed Maximum Power Point Tracking in Photovoltaic Systems by Audrey Kertesz A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright 2012 by Audrey Kertesz

2 Abstract Modeling and Robust Control Design for Distributed Maximum Power Point Tracking in Photovoltaic Systems Audrey Kertesz Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2012 Photovoltaic installations in urban areas operate under uneven lighting conditions. For such a system to achieve its peak efficiency, each solar panel is connected in series through a micro-converter, a dc-dc converter that performs per-panel distributed maximum power point tracking (DMPPT). The objective of this thesis is to design a compensator for the DMPPT micro-converter. A novel, systematic approach to plant modeling is presented for this system, together with a framework for characterizing the plant s uncertainty. A robust control design procedure based on linear matrix inequalities is then proposed. In addition to designing for robust performance, this procedure ensures the stability of the time-varying system. The proposed modeling and control design methods are demonstrated for an example rooftop photovoltaic installation. The system and the designed compensator are tested in simulations. Simulation results show satisfactory performance over a range of operating conditions, and the simulated system is shown to track the maximum power point of every panel. ii

3 Acknowledgements I gratefully acknowledge my supervisors, Bruce Francis and Olivier Trescases, for their wisdom, guidance and support over the past two years. I am deeply indebted to Professor Francis for his immense knowledge and boundless patience, and to Professor Trescases for sharing his creative ideas and practical expertise. Each offered a unique perspective on my research, and both have been wonderful mentors to me. I thank my colleagues in the control and power electronics groups for their camaraderie, and for their willingness to be pestered with questions. My particular thanks go to Shahab, for sharing his technical expertise, and to Karla, for countless fascinating discussions. This work would not have been possible without the unconditional support of my family. I wholeheartedly thank my parents, for their love and understanding, and my fiancé, for making my life easy when the going was difficult. Finally, I thank NSERC, OGS, Alberta Education, and the University of Toronto ECE Department for providing financial support. iii

4 Contents 1 Introduction 1 2 Background Maximum Power Point Tracking MPPT algorithms Direct perturb and reference command MPPT Distributed MPPT Micro-converters and micro-inverters Control Challenge Double loop control structure Statement of Objective Running Example System Components Block Diagram Power Converter Model Linearization Parameter values Solar Panel Model iv

5 3.3.2 Linearization Parameter fitting Inverter Principle of operation Model Linearization Parameter values Plant Model SISO System Model Plant uncertainty Load Model Derivation A module s output impedance Simplifying the series output impedances Neglecting the inverter dynamics Plant Model Uncertain parameters Disturbances Compensator Design Robust Control Theoretical background Polytopic Covering Covering the module parameter uncertainty set Control Synthesis Controller structure Linear matrix inequalities v

6 5.4 Practical Design Example Direct synthesis Single plant synthesis Analysis of the obtained controller Discussion Simulations Tracking and disturbance rejection Simulation of DMPPT system operation Conclusions and Future Work Limitations and Future Work Bibliography 91 A Supplementary proofs 99 A.1 Local power optimization is equivalent to global power optimization A.2 Error bound of averaged PWM A.3 Unimodal characteristic of solar arrays A.4 Controllability of the augmented system B Converter design 109 B.1 DMPPT module boost converter B.2 Inverter C Algorithms 116 C.1 Photovoltaic parameter fitting C.2 Polytopic covering in 2D vi

7 List of Tables 3.1 Converter parameters Fitted panel parameters Inverter parameters Constraint equations Load parameter uncertainty Polytopic covering Single plant synthesis parameters Sample test conditions for simulations B.1 Components selected for micro-converter C.1 Datasheet values for the SW 240 mono solar panel vii

8 List of Figures 1.1 Cumulative installed grid-connected and off-grid PV power in reporting countries [1] (a) I-V and (b) P-V curves of the SW 240 mono solar panel [2] A simple grid-connected solar array Maximum power point tracking feedback loop Two MPPT architectures: (a) direct perturbation and (b) reference command to compensator Distributed maximum power point tracking: (a) a simple MPPT system showing multiple series-connected PV panels, (b) a micro-converter system, and (c) a micro-inverter system Block diagram representation of the micro-converter system A boost converter with capacitive input filter Block diagram illustration of the averaging approximation: (a) signal flow in the physical device, (b) introduction of the averaging operator, and (c) final model Circuit diagram model of a solar cell A simple grid-tie inverter and its control system A capacitor decoupling an ideal DC power source from an ideal AC power sink viii

9 3.7 Simplified inverter model Frequency domain model of the simplified inverter Block diagram representation of the simplified inverter model, neglecting the sinusoidal disturbance: (a) nonlinear and (b) small-signal models Block diagram of a single DMPPT model Double loop control structure of a DMPPT module The load of a DMPPT module Model of the load impedance of a DMPPT module An open-loop DMPPT module Block diagram of a compensated DMPPT module An ideal DMPPT module An ideal DMPPT module with output capacitor (a) Output impedances Z out of modules sharing a common string current, (b) worst case approximation error of 6 Z out,k in Monte Carlo experiments The load impedance and its constituent terms: (a) typical operating conditions, (b) worst-case operating conditions Small-signal schematic of plant model Uncertainty region of the converter and panel in terms of high-level parameters: (a) physical constraints, (b) boost ratio constraints, (c) panel constraints, (d) all constraints Projection of P 1 R 4 into its 2D coordinate planes DMPPT module with severed ESC loop Integral control with full state feedback Real part of the slowest closed-loop system eigenvalue, plotted against the open-loop zero position Worst case agreement of Z out (s) and Z apr (s) ix

10 6.1 Compensated DMPPT module simulation model Reference tracking simulation results: (a) d 1 and (b) v s String current disturbance rejection simulation results: (a) d 1 and (b) v s Irradiance disturbance rejection simulation results: (a) d 1 and (b) v s Illustration of variable time P&O Distributed MPPT simulation results Maximum power point tracking of module A.1 Passive sign convention A.2 Solar cell I-V characteristic B.1 Inverter control system: a) schematic diagram, b) block diagram C.1 Illustration of the 2D optimal covering algorithm C.2 Output of the 2D polytopic covering algorithm x

11 Chapter 1 Introduction Solar energy shows great promise as a renewable energy resource; it is clean, abundant, and inexhaustible. In the space of ninety minutes, enough sunlight strikes the earth s surface to fuel the world s energy needs for a full year [3]. Photovoltaics (PV) are semiconductor devices that convert solar energy into usable electrical energy. Recent years have witnessed a dramatic increase in the world s installed photovoltaic capacity, illustrated in figure 1.1. This trend is expected to continue as the production costs of solar panels fall. Photovoltaic power systems fall into three broad categories: off-grid, centralized gridconnected, and decentralized grid-connected installations. Grid-connected systems account for over 95% of current PV power generation capacity [1]. In these installations, harvested solar power is fed directly into the electrical utility grid. A key benefit of grid-connected PV is that peak power output tends to coincide with peak electricity demand, offsetting daily and seasonal fluctuations in electricity consumption. Decentralized grid-connected installations can be built close to population centers. Small-scale rooftop and building-integrated photovoltaic installations are increasingly found in urban areas, thanks in part to government-sponsored incentives [1]. Urban solar installations pose unique engineering challenges. A typical solar panel 1

12 Chapter 1. Introduction 2 Figure 1.1: Cumulative installed grid-connected and off-grid PV power in reporting countries [1]. has a terminal voltage of around 30 V. For grid-connected applications, it is usual to connect several panels together in series to increase the terminal voltage of the array. However, mismatches in the level of incident solar radiation, or irradiance, received by series-connected solar panels can decrease the efficiency of the installation. In urban environments, uneven shading conditions, reflections, panel surface debris, and differences in panel orientation make such mismatches unavoidable. The string mismatch problem is the focus of this thesis. As we shall see, power electronic devices play a critical role in ensuring that the maximum available power is harvested from any photovoltaic installation. Several researchers [4 9] have proposed solving the mismatch problem by introducing micro-converters, individual per-panel dc-dc power converters. The resulting smart solar modules operate autonomously to correct the effects of mismatch [8]. However, some of the control challenges inherent in this solution have been widely overlooked. Micro-converter controllers must be robust to uncertain operating conditions, and

13 Chapter 1. Introduction 3 must contend with the dynamic coupling between series-connected modules. Control design methods that have been previously applied to this problem are ad hoc, and may not address these challenges explicitly. In this thesis, systematic modeling and control design procedures for per-panel dc-dc converters are developed. It is intended that these or similar techniques will be applied by power electronics designers for solar applications. The original contributions of the thesis include: 1. A technique for modeling the apparent load of a single micro-converter connected in a grid-connected string of micro-converters. 2. A framework for modeling the plant uncertainty for the purpose of micro-converter control design. 3. Applying LMI-based control design techniques to the micro-converter control problem. The thesis has been written so as to be accessible to readers versed either in control theory or power electronics. As such, the reader will encounter some familiar concepts explained in detail; this is for the benefit of readers from a different area of expertise. Chapter 2 provides background information on solar power and the role of power electronic devices in PV installations. The concept of mismatch is fully explained, the function of micro-converters is discussed, and a brief literature review is provided. At this point, it is possible to define our control problem more concretely. The chapter ends by introducing a running example of a rooftop solar installation that will be used to illustrate modeling and control design throughout the thesis. Models of each of the PV and power electronic devices that make up a small scale gridconnected solar installation are derived in chapter 3. These models are used in chapter 4 to devise a simplified plant model appropriate for control design. Key challenges are

14 Chapter 1. Introduction 4 modeling the converter load from the perspective of a single module and creating a structured description of the system s uncertainty. The proposed control synthesis procedure is described in chapter 5. The procedure uses modern robust control techniques and draws on the theory of systems subject to uncertain time-varying parameters. The resulting controller is then tested in full system simulations, the results of which are presented in chapter 6.

15 Chapter 2 Background The reader is assumed to be familiar with the basics of power electronics: dc-dc switched mode converter topologies, pulse width modulation (PWM), and duty cycle. An introduction to these topics is provided in [10]. A reader unfamiliar with power electronics may also consult chapter 3, in which mathematical models of the devices are derived. 2.1 Maximum Power Point Tracking A solar panel is modeled as a memoryless circuit element. The I-V characteristic of a solar panel is highly nonlinear, as figure 2.1 illustrates. A panel s I-V curve depends on the irradiance, measured in W/m 2, and the temperature at which it operates. These characteristics also change somewhat over the lifetime of the device. As shown, the power produced by the panel is highly dependend on its position on the I-V curve. The operating point (voltage and current) at which the panel achieves its maximum power is called the maximum power point (MPP). As evident in figure 2.1, the position of the MPP depends on the irradiance and temperature of the panel. For optimal energy harvesting, a solar panel should always be operated at MPP. Figure 2.2 shows a high-level diagram of a simple grid-connected PV installation. 5

16 Chapter 2. Background 6 I (A) P (W) W/m W/m W/m 2 60 C 25 C C 25 C 1000 W/m W/m W/m (a) 40 V (V) (b) 40 V (V) Figure 2.1: (a) I-V and (b) P-V curves of the SW 240 mono solar panel [2]. Arrows beginning on filled and open circles represent voltage and current measurements respectively 1. The PV array consists of identical solar panels connected in series and parallel. The array s I-V characteristic is a scaled version of figure 2.1. The system s load is a dc-ac converter, or inverter, that interfaces the DC solar power source to the AC utility grid. The photovoltaic source and its load are connected through a dc-dc converter. Photovoltaic array DC-DC converter DC-AC converter inverter Utility grid i v MPPT control u Figure 2.2: A simple grid-connected solar array. The role of the dc-dc converter here is analogous to an ideal AC transformer: It transforms voltages and currents to match the source to the load. The conversion ratio of the dc-dc converter, analogous to the turns ratio of an AC transformer, must be selected 1 The diagram shows a block diagram together with an electric circuit. By convention, a block drawn in heavy lines represents an electrical device, and a heavy line connecting two such blocks represents an electrical connection. A block drawn in thin lines is a block diagram component, and a thin arrow represents signal flow.

17 Chapter 2. Background 7 v i v v i MPPT control u Plant Figure 2.3: Maximum power point tracking feedback loop. such that the PV array operates at its MPP. This is the task of the maximum power point tracking (MPPT) controller, which adjusts the converter duty ratio to optimize the PV array power. The maximum power point tracking controller takes the PV array current i and voltage v as its inputs, and produces a control signal u, the duty cycle of the dc-dc converter MPPT algorithms Figure 2.3 shows the MPPT controller in a feedback loop. The plant block models the PV installation of figure 2.2 from input u to output v. We take for granted that this plant has stable dynamics. The nonlinear relationship between the PV array s voltage and current is shown explicitly. An enormous body of work on MPPT control exists in the power electronics literature [11,12]. The majority of algorithms use a periodic sampling approach and are implemented digitally, although some continuous-time MPPT controllers are reported [13, 14]. The simplest and most widely used MPP tracker is perturb and observe (P&O), a discrete hill-climbing algorithm. This MPPT controller climbs the photovoltaic array s P-V curve by manipulating the converter duty cycle u. This is possible because the equilibrium map from u to v can be shown to be monotonic [15]. The input signals v and i are sampled at t = k t, k = 0, 1, We will use the convention v(kt) = v[k] for the sampled signals. The PV array power is computed for

18 Chapter 2. Background 8 each sample; p[k] = v[k]i[k]. At every time step, u changes by a fixed constant u. The direction of the change is determined by the change in power since the last sample. For k 1, u is determined by the equation u[k + 1] = u[k] + sgn ( ) p[k] p[k 1] u, (2.1) u[k] u[k 1] where sgn( ) is the sign function, with sgn(0) := 0. For the algorithm to be effective, the wait time, t, must be sufficiently long to allow the circuit transient to settle to a new equilibrium. The optimization of the P&O parameters u and t are discussed in [16]. Many proposed improvements to the P&O algorithm employ time-varying u and t to improve the resolution and speed of the algorithm. The MPPT algorithm is a simple one-dimensional application of extremum seeking control (ESC). Extremum seeking controllers are studied rigorously in the control literature; see for example [17 19]. Like P&O, these algorithms require that the dynamical system being optimized be stable and that its dynamics be fast relative to that of the extremum seeker Direct perturb and reference command MPPT The architecture of the MPPT control block in figure 2.3 can have one of two structures [20], which are illustrated in figure To avoid confusion between the MPPT block and the MPPT algorithm, we will henceforth refer to the MPPT algorithm as the extremum seeking controller. In figure 2.4a, the ESC controls the converter duty cycle directly, as in equation (2.1). In figure 2.4b, the ESC instead outputs a reference voltage v ref. A compensator adjusts the converter duty cycle u to track the reference signal. This MPPT structure employs a 2 Signals entering a summation junction are positive unless indicated with a negative sign.

19 Chapter 2. Background 9 i v i v p ESC u p ESC v ref e Compensator u (a) (b) Figure 2.4: Two MPPT architectures: (a) direct perturbation and (b) reference command to compensator. control double loop, as the ESC acts on the closed loop system formed by the compensator and the rest of the system. The double loop structure has been advocated by several authors. Femia et al. [21] discuss its advantages. Consider figure 2.3 with the MPPT control block of figure 2.4b, and sever the ESC from the loop. With a well-designed compensator, the system from v ref to v will have much faster dynamics than the system from u to v. This allows the ESC to employ a shorter interval t, so the system converges more rapidly to optimal power. A second advantage concerns the inverter. As explained in section 3.4, the inverter introduces a disturbance at 120 Hz into our MPPT system. This disturbance causes an undesired oscillation in v, which may confuse the ESC, delaying its convergence to the MPP. Once reached, voltage oscillations about the MPP will also reduce the harvested power [22]. However, the 120 Hz disturbance is attenuated by a compensator having sufficiently high bandwidth. From the perspective of inverter design, this improved disturbance rejection is beneficial because the system can tolerate a larger amplitude disturbance. This frees the inverter designer to use a smaller DC link capacitor, the reduction of which has been a focus of recent literature [22,23]. The DC link capacitor is an expensive and failure-prone inverter component; by reducing the needed capacitance, a designer can select a superior capacitor technology. Although the tracker can command either the panel voltage or current, a system that

20 Chapter 2. Background 10 issues a voltage reference will show less sensitivity to irradiance changes [6]. 2.2 Distributed MPPT Consider figure 2.5a, which depicts n series-connected solar panels in a simple gridconnected PV installation. In an urban environment, these n panels may not all receive the same irradiance. Differences may arise due to partial shading, different panel orientations, or reflections from nearby buildings. Since these panels are series connected, they share a common current. If the panels I-V characteristics are not identical, then some panels will be forced to operate away from their respective maximum power points. The power harvested using a single dc-dc converter with centralized MPPT is less than what could be achieved if each panel were locally optimized and the resulting panel powers summed. Depending on the installation, it is estimated that 10-30% of the available energy yield is lost due to mismatch [24, 25] Micro-converters and micro-inverters Figure 2.5b shows the same installation with a dedicated dc-dc converter assigned to every panel. This micro-converter system configuration was first suggested by Walker and Sernia [4]. Many researchers in the power electronics community have since contributed Solar panel 1 Solar panel 2 DC-DC converter DC-AC inverter Utility grid Solar panel n (a)

21 Chapter 2. Background 11 Solar panel 1 DC-DC converter 1 Solar panel 2 DC-DC converter 2 DC-AC inverter Utility grid Solar panel n DC-DC converter n (b) Solar panel 1 DC-DC converter DC-DC 1 converter 1 DC-AC converter DC-DC 1 converter 1 Solar panel 2 DC-DC converter DC-DC 2 converter 2 DC-AC converter DC-DC 2 converter 2 Utility grid Solar panel n n DC-DC converter DC-DC n converter n DC-AC converter DC-DC n converter n (c) Figure 2.5: Distributed maximum power point tracking: (a) a simple MPPT system showing multiple series-connected PV panels, (b) a micro-converter system, and (c) a micro-inverter system. [5 9] and commercial versions have recently been brought to market [26,27]. If the microconverters are lossless and capable of achieving any positive conversion ratio, then it can be shown that local per-panel optimization will recover all of the energy otherwise lost due to mismatch. A proof is presented in appendix A.1. Another solution to the mismatch problem, shown in figure 2.5c, assigns a dedicated dc-dc converter and dc-ac converter 3 to every panel in a configuration dubbed microinverter. This too has received much attention in the literature [28 30], and the concept has been commercialized [31, 32]. Compared to the micro-inverter architecture, the micro-converter architecture requires fewer components, has lower overall system cost, and is more efficient. However, 3 In micro-inverters, these two functions can be performed in single-stage, using an isolated topology.

22 Chapter 2. Background 12 the micro-inverter architecture offers some practical advantages: It is easier to install, eliminates the high voltage DC bus, and eliminates the central inverter, which makes the system modular and flexible. This thesis considers the micro-converter architecture. The ability to perform maximum power point tracking at the individual panel level is called distributed maximum power point tracking (DMPPT). It is worth noting that DMPPT could be performed at still finer levels of granularity than the panel level, a possibility discussed in [24]. 2.3 Control Challenge The building block of micro-converter distributed MPPT is the DMPPT module, which consists of panel, micro-converter, and controller. Distributed MPPT using micro-converters is a more challenging problem than central MPPT, because the series-connected modules are coupled. To illustrate, suppose that the system in figure 2.5b is operating with every panel at its respective MPP, when one of the panels is suddenly shaded. Its local MPP tracker responds by changing the conversion ratio of its micro-converter. The apparent load that is seen by each of the remaining modules changes as a result, and they are perturbed from their respective maximum power points. In the context of figure 2.3, a local MPP tracker will not see a monotonic equilibrium map from u to v. A multi-input, multi-output or distributed control structure could mitigate this effect. This would require that all of the modules communicate continuously, but dedicated wiring for this purpose would be costly and impractical. The possibility of power-line communication (PLC) in a micro-converter system is discussed in [5], and several commerical micro-converters transmit data wirelessly to a recording station [26,27]. However, these systems are designed for sporadic communication. A DMPPT system with autonomous, non-communicating micro-converters is modu-

23 Chapter 2. Background 13 lar, extensible, easy to install, and requires no additional wiring [8] Double loop control structure The solution to the problem of coupled DMPPT, proposed in [7] and [8], is to give the DMPPT module s local controller the double loop structure of figure 2.4b. In this case, the ESC output is v ref, now a reference voltage for that module s solar panel. Since the panel s MPP voltage is not affected by changes elsewhere, the local ESC is disassociated from the DMPPT system s complexities. The double loop control structure can confer this benefit only if the inner loop compensator is able to track the ESC reference voltage despite the time-varying dynamics of the module s apparent load. Literature review Few authors have discussed compensator design for the DMPPT module. Femia et al. [7] analyze the stability of a system of series-connected DMPPT modules having a double-loop control structure. The analysis neglects the extremum seeker and focuses on the coupled dynamics of the micro-converters connected in series. To the author s knowledge, no other paper has addressed this topic. For this analysis, [7] presents a boost micro-converter together with a type 3 analog compensator (a PID controller with two added high frequency poles). The details of the compensator design and performance are not discussed. The inverter is modeled as a Thévenin equivalent circuit having a small resistance; this model is consistent with the ideal voltage source model of the inverter that is common in the literature, as for example in [21, 33]. Linares et al. [8, 34] design a non-inverting buck-boost micro-converter. Their work is the first to explicitly state the benefit of a double loop control structure in decoupling the MPPT functions of neighboring DMPPT modules. A low bandwidth PI controller is

24 Chapter 2. Background 14 chosen for the inner loop compensator; the same compensator is used in both buck-mode and boost-mode operation. The selection of the controller parameters is not discussed in detail. Linares et al. use a dynamical inverter model in simulations. The inverter is modeled as a block that adjusts its current to maintain a fixed voltage across its terminals; this is achieved by integral control. One of the most important considerations for the inner loop compensator is robustness, since it must stabilize the system for multiple converter conversion ratios and operating conditions. The question of robustness is briefly addressed in [21], in which the double loop control structure is proposed for a central MPPT system (i.e., figure 2.2). As the dc-dc converter in this system is connected to a fixed DC voltage inverter, the plant contains only one uncertain parameter. Nowhere in the literature is the question of robustness and parameter uncertainty discussed for DMPPT systems, in which both the micro-converter s output voltage and conversion ratio vary in time. 2.4 Statement of Objective The objective of this thesis is to develop a systematic modeling procedure for the DMPPT system described, and to propose a method of control synthesis for the inner loop compensator. The compensator should be compatible with any extremum seeking scheme. For simplicity, we will assume that all of the series-connected DMPPT modules have the same solar panel model, and identical converters and controllers. The compensated DMPPT module must be able to make the solar panel s terminal voltage track the MPPT reference voltage. It must do so despite disturbances in the common string current, which result from the operation of neighboring modules and the inverter. The module must fulfill these objectives regardless of its operating point. The

25 Chapter 2. Background 15 compensator must therefore be robust to variable parameters, such as the conversion ratio, the output voltage, and the panel characteristic. A precise statement of the control specifications is deferred until section Running Example The modeling and control design procedures will be illustrated with a running example of a grid-tied rooftop PV installation. The installation consists of between six and ten solar panels 4, dedicated per-panel micro-converters, and a 2.5 kw single-stage inverter. The micro-converters have a boost topology, chosen in our example for simplicity. Boost converters are common in DMPPT applications [5 7]; however, many modern DMPPT module designs use a non-inverting buck-boost topology for improved power harvesting [8, 24, 27]. The non-inverting buck-boost converter s three operating modes (buck mode, boost mode and pass-through mode) allow it to achieve a wider range of conversion ratios than a boost converter while maintaining a high efficiency. The components of the example PV installation are described in detail in chapter 3. 4 Small installations of six to ten panels are common, since the resulting series string voltage of V is within the MPP range of typical two-stage inverters; see for example [35].

26 Chapter 3 System Components The first step in control design is to create a mathematical model of the plant. In this chapter, we derive models of each of the three electrical devices in a DMPPT system: the dc-dc converter, the solar panel, and the inverter. 3.1 Block Diagram We begin by expressing the high level circuit diagram of figure 2.5b in block diagram form. The system depicted in figure 2.5b is an interconnection of electrical subsystems, each of which is either a one-port or a two-port device. In order to reduce an electrical subsystem to an input-output block, we assign to each port an input (either current or voltage) and the corresponding output. In general the choice will be arbitrary; however, it may be motivated by exigencies of the interconnections, or by the structure of the subsystem itself. The procedure is analogous to the modeling of linear circuits as twoport networks. The resulting block diagram is shown in figure 3.1. The photovoltaic modules and the grid-tied inverter are one-port devices; the dc-dc converters are two-port devices. Each dc-dc converter has a control input, d i, around which the control system will be designed. In the following sections, we derive mathematical models for each of the three 16

27 Chapter 3. System Components 17 block types. PV 1 i s1 v s1 i o1 Converter 1 v o1 d' 1 PV 2 i s2 v s2 i o2 Converter 2 v o2 v string Inverter i string d' 2 PV n i sn v sn i on Converter n v on d' n Figure 3.1: Block diagram representation of the micro-converter system. 3.2 Power Converter A dc-dc power converter is an electronic power processing device that functions by commutating between two or more circuit configurations. Figure 3.2 shows a synchronous boost converter with a capacitive input filter. A boost converter has a voltage converter ratio M = vo v s 1. It is called synchronous because the switches S 1 and S 2 are controlled always to be complementary. Let u sw (t) be the switch position function, which takes only binary values {0, 1}. When u sw (t) = 0, S 1 is closed and S 2 is open, and when u sw (t) = 1, S 1 is open and S 2 is closed.

28 Chapter 3. System Components 18 + L + v s i s + vc 1 C 1 i L S 1 S 2 + v C2 C 2 i o v o Figure 3.2: A boost converter with capacitive input filter The goal of the system is to regulate one of the four port quantities (v s, i s, v o, i o ). The converter is controlled through u sw (t) Model Explicit two-port models of dc-dc converters are unusual in the power electronics literature, in which it is common to model the source as an ideal voltage source and the load as a resistor. Since a solar panel does not resemble an ideal voltage source, it is convenient to derive a two-port converter model. A similar approach is advocated by Suntio in [36]. We assign the port variables (i s, i o ) as inputs and (v s, v o ) as outputs, a choice made necessary by the input and output capacitors. Their voltages must be assigned as outputs if we are to obtain a proper state model of the device. The boost converter of figure 3.2 contains only ideal switches and reactive elements. In reality, transistor switches and reactive components are not perfectly lossless. A more realistic model of the converter includes a series parasitic resistance for every switch and reactance. We will neglect these parasitics in our model, but revisit them when the controller is tested in simulations; see section Nonlinear switching model To derive a model of the boost converter, we fix the positions of the switches and obtain the differential equation model of the resulting circuit using Kirchhoff s laws. The two models are then combined into a single model parametrized by u sw (t). The reference directions of the converter port currents and voltages are indicated in

29 Chapter 3. System Components 19 figure 3.2. When S 1 is closed and S 2 is open, C 1 dv C1 dt = i s i L L di L dt = v C 1 C 2 dv C2 dt = i o. (3.1) When S 1 is open and S 2 is closed, C 1 dv C1 dt = i s i L L di L dt = v C 1 v C2 C 2 dv C2 dt = i L i o. (3.2) The switching converter model is thus C 1 dv C1 dt =i s i L L di L dt =v C 1 v C2 u sw C 2 dv C2 dt =i L u sw i o. (3.3) The signal u sw (t) is generated by a controller, design techniques for which are discussed in [10] and [37]. We confine ourselves to the class of controllers for which u sw (t) is generated by a fixed frequency pulse width modulator (PWM) Averaging It is convenient to neglect the switching nature of the converter in our model. By applying the method of state space averaging, first introduced by Middlebrook and Cuk in [38], we replace the binary u sw (t) in (3.3) with the continuous signal d (t). The signal d (t) takes values on the closed interval [0, 1]; its relationship to u sw (t) will be explained shortly.

30 Chapter 3. System Components 20 Thus, we obtain the non-switching nonlinear model C 1 dv C1 dt =i s i L L di L dt =v C 1 v C2 d C 2 dv C2 dt =i L d i o. (3.4) The symbol d is chosen for consistency with the power electronics literature, where by convention the duty ratio of S 1 is called d and its complement d = 1 d. The use of state-space averaging has been extensively justified in the literature [37]. The averaged model is correct in the limit of infinite switching frequency; a rigorous treatment can be found in [39] and [40]. Practically speaking, the averaged model has limitations. An empirical rule of thumb is that the averaged converter model is valid up to half the switching frequency f s [10]. State space averaging is illustrated in figure 3.3. A controller generates a continuous signal u c (t), which is pulse-width modulated at frequency f s to generate the input u sw to the switching model (3.3) of the converter. We introduce the non-causal averaging operator T ave : u v, v(t) = 1 ˆ Ts+t u(τ)dτ, T s t where T s = 1 f s is the switching period. The fictitious averaging block is depicted in dotted lines in figure 3.3b. The converter is replaced with system (3.4), which has the same dynamics as the switching model but takes the continuous input d. In power electronics it is convention to equate d and u c as in figure 3.3c. In doing so, we approximate the averaging operator as the inverse of the PWM operator. This idea has intuitive appeal provided that the signal u c changes slowly relative to the sampling period T s. Our intuition is justified in appendix A.2, which proves that for uniform pulse width modulation, under mild assumptions on u c, d (t) u c (t) can be made arbitrarily small by choosing T s sufficiently small.

31 Chapter 3. System Components 21 Compensator u c PWM u sw Converter (switching) (a) Compensator u c PWM u sw d' Converter Average (nonswitching) (b) Compensator d' (c) Converter (nonswitching) Figure 3.3: Block diagram illustration of the averaging approximation: (a) signal flow in the physical device, (b) introduction of the averaging operator, and (c) final model Linearization System (3.4) exhibits a continuum of equilibria (V C1, I L, V C2 ) with D = V C 1 V C2 (0, 1]. A linearized model can be constructed by taking the Taylor series expansion about any such equilibrium. The resulting linear model 1 has state x = (ṽ C1, ĩ L, ṽ C2 ), control input u = d, deviation port currents w = (ĩ s, ĩ o ). and output y = (ṽ s, ṽ o ): C L 0 ẋ = 1 0 D x + V C2 u w 0 0 C 2 0 D 0 I L 0 1 }{{}}{{}}{{}}{{} K KA KB u KB w y = x } {{ } C (3.5) The converter s boost ratio, M = Vo V s, is equal to 1 D. 1 When discussing signals in a linearized system, we will adopt the following convention: For a signal v, its steady state value is denoted by V and its small-signal component is denoted by ṽ, where v = V +ṽ.

32 Chapter 3. System Components Parameter values The design parameters of the boost converter in our running example are presented in table 3.1. Their selection is explained in appendix B.1. Table 3.1: Converter parameters Parameter Value f s 250 khz L 40 µh C 1 10 µf C 2 40 µf I L,min 1.18 A The synchronous PWM switching described and modeled in this section is called continuous conduction mode (CCM). If I L < I L,min, we assume that the converter operates using a different switching pattern, as explained in appendix B.1. It is typical for power electronic devices to use a different switching mode for low power operation [41, 42]. In our running example, we will consider only converter operation with I L > I L,min. The design procedures described in chapters 4 and 5 can, if necessary, be modified to accommodate more complex mode boundaries definitions. 3.3 Solar Panel A solar panel is made up of PV cells, the basic building block of photovoltaics. A single PV cell produces a current of several amps at a voltage of around 0.5 V. In a solar panel, many PV cells are connected in series to provide a more usable terminal voltage. An introduction to the physics of solar cells can be found in [43].

33 Chapter 3. System Components Model The ideal photovoltaic cell is modeled as an ideal current source in parallel with a silicon diode [43, 44]. A more realistic model of the solar cell includes the parasitic effects of leakage currents (R p ) and resistive electrical contacts (R s ), as shown in figure I pv R p R s i + v Ideal cell Figure 3.4: Circuit diagram model of a solar cell. The relationship between the cell s current and voltage, easily derived via the Schockley diode equation, is ( ) q(v + Rs i) I pv I 0 (exp ak B T ) 1 v + R si i = 0. (3.6) R p Here I pv (A) is the current of the fictitious internal ideal current source, I 0 (A) is the reverse diode saturation current, T (K) is the absolute temperature of the cell, q is the fundamental charge, k B is the Boltzmann constant and a [1, 2] is the diode ideality factor. The current I pv is proportional to the irradiance, G (W/m 2 ), incident on the surface of the cell [43]. Model parameters for real solar cells are selected by curve fitting experimental I-V data. A solar panel is composed of n s series-connected solar cells. Typical I-V characteristics of a solar panel operating under different irradiance levels are shown in figure 2.1. The panel characteristic can be modeled using a modified version of equation (3.6) [44], in which a is replaced by n s a and R p and R s are interpreted as parasitic resistances at the 2 The dynamics of the PV cell s junction capacitance are assumed to be so fast as to be negligible.

34 Chapter 3. System Components 24 panel level, αg I 0 (exp ( ) ) q(v + Rs i) 1 v + R si i = 0. (3.7) n s ak B T R p In equation (3.7), the proportionality I pv G is made explicit in the constant α. Some authors [44, 45] also include an empirical temperature correction for I pv ; we neglect it here but note that it would not be difficult to incorporate. If the panel s cells are unevenly illuminated, then equation (3.7) can hold only approximately. Nevertheless, it can be shown that the P-V characteristic of the panel is unimodal under uneven irradiance, and therefore that extremum seeking methods of MPPT remain effective. A simple proof is given in appendix A.3. Note that this result does not hold if the panel includes bypass diodes, which are connected across substrings of cells in many solar panels. In this case, a multimodal power characteristic may result [46]; we will neglect this effect. Equation (3.7) describes a one-to-one relation between i and v, parametrized by G and T. However, the function f G,T : i v cannot be expressed in closed form. One can evaluate it numerically via the Lambert W function [47] Linearization The panel model of equation (3.7) is a memoryless nonlinearity. The implicit function can be linearized by taking the Taylor series expansion of equation (3.7), h(i, v, G, T ) = 0, about a point (I, V, G 0, T 0 ). This yields k i ĩ + k v ṽ + k G G + kt T = 0, where ki, k v, k G and k T are the Taylor coefficients evaluated at the equilibrium. The small-signal model of the panel is ĩ = k ( ) v kg k T ṽ G + T = Rpv 1 ṽ + ĩ pv. (3.8) k i k i k i If G and T remain constant, the panel s small-signal model is resistive; a negative sign appears because i and v were assigned using the active sign convention. Perturbations in G and T are modeled as a disturbance current ĩ pv in parallel with resistor R pv.

35 Chapter 3. System Components Parameter fitting A solar panel datasheet provides the MPP voltage V mpp and current I mpp, the short circuit current I sc, and the open circuit voltage V oc of the panel under industry standard test conditions (STC) of G = 1000 W/m 2 and T = 25 C. Since highly specialized equipment is required to replicate these conditions experimentally, the parameters of equation (3.7) must be determined using the manufacturer s provided data. The algorithm proposed by Villalva et al. [44] is widely used to fit the parameters of equation (3.7) to the datasheet values. However, when applied to the SW 250 mono, the algorithm returns a negative value for R p regardless of the initial choice of α [1, 2]. In order to produce a viable curve fit, the algorithm described in appendix C.1 was used to compute the fitted parameter values shown in table 3.2. Table 3.2: Fitted panel parameters Parameter Value Unit α Am 2 W 1 I o A R p 4112 Ω R s Ω a n s Inverter An inverter is a dc-ac power converter. We will consider a single phase, single stage inverter appropriate for a small scale PV installation. The inverter is a complex device, the design of which is complicated by the requirements of regional power utility standards. This section introduces one of the conceptually simplest inverter topologies. However, inverter design and control remain an active area of research.

36 Chapter 3. System Components 26 This section presents a simplified, topology-independent model of the inverter as seen from the DC side Principle of operation Figure 3.5 shows a simple grid-tie inverter, which consists of a DC-link capacitor and a switching power converter, connected to the utility grid. The objective of this inverter is to present itself to the grid as a unity power factor source 3. This requirement is tantamount to ensuring that the inverter s output current i out is sinusoidal and phaselocked to the grid voltage v grid. The utility grid is modeled as an ideal AC voltage source 4. The control system includes two sensors, a DC link voltage sensor and an inverter output current sensor, and actuates by modulating the duty cycle u of the power converter. DC-link capacitor + i string Bridge converter Utility grid + v grid i out v string u Inner loop controller e i V dc e v Outer loop controller i peak i ref cosωt resembles Figure 3.5: A simple grid-tie inverter and its control system. The following subsections describe the functions of the DC-link capacitor and of the 3 However, some modern inverters can be programmed to provide reactive power to the grid. 4 This model neglects grid disturbances. These would ultimately appear as output current disturbances in the micro-converter model, to which the compensator of chapter 5 is designed to be robust.

37 Chapter 3. System Components 27 switching converter. DC-link capacitor Consider the power balance of an ideal (lossless) grid-tie inverter. The output power waveform of the inverter oscillates at twice the grid frequency, since p out (t) = v grid (t)i out (t) = (V grid cos ωt)(i out cos ωt) = 1 2 V gridi out (1 + cos 2ωt). (3.9) Here V grid (V) is the peak value of the grid voltage waveform, I out (A) is the peak value of the inverter output current waveform, and ω (rad/s) is the grid frequency. The input power to the inverter comes from a DC source, the PV array. The decoupling of the DC power source from the AC power sink is performed by the DC-link capacitor. The capacitor alternately stores and releases into the inverter the deficit and surplus power delivered by the PV array. To analyze the DC-link capacitor in a simplified context, figure 3.6 shows a decoupling capacitor separating an ideal DC power source from an ideal AC power sink. An ideal power source is a fictitious element having the memoryless terminal characteristic i(t)v(t) = p(t). The reference directions for voltage and current are shown in the figure. i c P + v P (1 + cos2ωt) Figure 3.6: A capacitor decoupling an ideal DC power source from an ideal AC power sink. The system of figure 3.6 exhibits a periodic steady state when the average power drawn by the AC sink equals the power supplied by the DC source. From Tellegen s theorem we have v(t)i c (t) + P P (1 + cos 2ωt) = 0. (3.10)

38 Chapter 3. System Components 28 When the capacitor s terminal characteristic, i c = C v, is substituted into equation (3.10), the resulting differential equation can be solved analytically: C Cv v + P cos 2ωt = 0 ˆ t v vdt + P ˆ t 0 0 cos 2ωt = C [ v(t) 2 v(0) 2] + P sin 2ωt = 0. 2ω (3.11) An inverter is always designed to operate with its initial voltage v(0) 2 P, so we use ωc this assumption to solve equation (3.11). The capacitor voltage exhibits a periodic ripple, v(t) = v(0) 2 P sin 2ωt. (3.12) ωc We can approximate equation (3.12) by taking the Taylor series expansion of f(x) = x about x = v(0) 2, and treating P ωc sin 2ωt as a perturbation x. Since v(0)2 P ωc, the ripple waveform is approximately sinusoidal. The amplitude of the ripple is inversely proportional to the size of the decoupling capacitor: v(t) v(0) 1 P 2v(0) ωc sin 2ωt. (3.13) In a PV system, the ripple propagates through to the terminals of the PV modules. The PV voltage oscillation is undesirable because it may interfere with MPPT, and because oscillations around the MPP reduce the harvested power. For these reasons, we would prefer a large capacitor to minimize DC-link voltage ripple. However, the capacitor is one of the most expensive components of the inverter. The capacitor also has the shortest lifespan of any of the inverter s electronic components, and often requires replacement during the inverter s service life [22]. The trend in recent years has been towards smaller DC-link capacitors and more ripple-tolerant systems on both the inverter and PV sides [48, 49], which allows designers to use less failure-prone