In this lecture we will introduce another important application, power electronics inverters for photovoltaic power systems. In a photovoltaic or PV power system, the power is obtained in DC form from a PV array. On the other hand, it is desirable to deliver that power to the AC utility grid in the AC form. On the AC side, the grid voltage is ideally sinusoidal and it is best to deliver power at unity power factor, so that the AC current, feeding the AC grid line, is ideally sinusoidal as well. In the single phase case, the product of AC voltage and AC current on the output side of the system, gives us this expression for average power in front. and an AC component in a cosinusoidal form at twice the line frequency. You may recall we have discussed very similar case in the low harmonic PFC rectifier. The objective of this box in between, which is really the power electronics converter we are interested in, or commonly referred to as inverter, is to take DC power produced by a PV array and convert that into AC power delivered to the AC power grid. So we will briefly review this application first and then show how average current mode control can be used to control DC to AC inverter. A very brief introduction to photovoltaic power systems, we'll start by looking at a PV cell. A single cell in crystalline silicone PV systems, is typically of 100 to 200 centimeters square size. An equivalent circuit model for the cell is shown down here. The cell behaves essentially as a p-n junction which is biased by a current proportional to the solar irradiation. As an example of numerical values, we will have here characteristics of a PV cell shown in two forms: output power produced as a function of voltage across the cell, Vpv is on the horizontal axis, and also output current as a function of the voltage across the PV cell. In the output current as a function of voltage, we notice that the highest current will be produced when the voltage is equal to zero, which is called a short-circuit current, or I sub SC. When the cell is open-circuited on the other hand, the current is equal to zero, but the voltage has a maximum value. The product of the two gives us the output power, the output power is going to be equal to zero on both ends, at short-circuit and the open-circuit. Somewhere in between there is a voltage where the output power has a maximum value. As a numerical example, we have here an open-circuit voltage of about 0.7 volts, that would be a fairly typical value, and a short-circuit current for a PV cell of this size is about 6 amps. Now it is important to note that these values are often given for what is called standard test conditions, which means the solar irradiation of 1,000 watts per square meter, which means a full sun, and the particular temperature of 25 degrees C. Now going back to this note that we have a particular operating point for which the power delivered from the cell has a maximum. Let's look at that point. So that point is called the maximum power point, and for the same numerical example that maximum power point would occur at a voltage level of 0.56 volts, the current level of 5.9 amps, corresponding to output power for a single PV cell of about 3 watts. Now of course, this single cell as a source is very limited both in voltage and in power, and so it is desirable to find ways to scale this up in both voltage and power to make the system practical, and that's done simply by stacking cells in series, and then converting that into what is called a PV module and then stacking modules in series or parallel to scale the power up. So, as an illustration, here is a typical 72-cell PV module and the characteristics of that PV module would simply scale in voltage with respect to what we have seen for single cell. The maximum power point power for a single module would now be more respectable 240 watts, and this module would take about 1.2 square meters of space. The curve shown here for power versus PV voltage for the module are shown for different values of solar irradiation, from full sun of 1,000 watts per meter square, down to one-fifth of that, or 200 watts per meter squared. Now, scaling this up further, a typical rooftop type PV system of about 5 kilowatt rating, could consist, for example, of 3 parallel strings of 7 such modules in series. So large PV systems are constructed in general by stacking modules in series, and then having strings of modules connected in parallel. Characteristics now for the full systems are shown here. They're now scaled in both current and voltage for different values of the solar irradiation. Now the point of this brief introduction to the solar PV systems was simply to realize that the output of a PV array is a DC voltage and DC current. and whatever follows that PV array is desirable to operate the array at a maximum power point, to extract the maximum electrical power for a given solar irradiation. The system that would do that can be done in many different ways. A typical PV architecture includes what is called a centralized inverter, that would be a single power electronics converter that interfaces the PV array to AC utility grid. The inverter functions are to operate the PV array at the maximum power point, to generate AC current in phase with AC utility grid voltage, to balance the power between the P_PV, to DC value produced by the PV array, and the time varying pac(t) value that is provided to the grid, which means that a single phase system would have to include an energy storage element, typically in the form of a capacitor. Of course, such system has to have high power conversion efficiency, high reliability, and low cost as usual for any power electronics. Realization of the system can be done in many different ways. I will show you here one particular example that consists of two sub-converters, so the entire “converter” really consists of two boxes. One is a boost DC-DC converter in front followed by a single-phase DC to AC inverter on the output side. The function of the boost DC-DC converter is to set the operating point of the PV array to the maximum power point, and to efficiently step up the output voltage of the PV array to a higher DC voltage across this energy storage capacitor C. The DC to AC inverter efficiently generates output AC current which is in phase and following the waveshape of the AC output voltage, which ideally is sinusoidal, and the single-phase DC-AC inverter is balancing the average power delivery from the PV array to the grid, which, taking efficiencies into account, is shown right here. Finally, the energy storage capacitor provides the function of balancing the difference between the instantaneous power Pac(t) and the average DC power taken from PV array. A system of this type must meet all these conditions right here and in addition, must meet various site specific standards such as IEEE 1547, which, for example, requires the output current to have low total harmonic distortion of less than 5%, has requirements for anti-islanding, and a number of other issues related to how to interface PV systems to the AC utility grid. Now going further into details, here is a diagram of the power stage of this inverter that consists of a boost front end followed by a four transistor or what is called full bridge inverter, with a control block diagram shown at the bottom. So let's examine this a little bit in some detail, the boost DC-DC converter has the purpose of setting the operating point of the PV array to the maximum power point and it does that continuously, so that under varying conditions, in terms of solar irradiation, the maximum power point can be maintained under all those operating conditions. So there is a box that is called maximum power point tracking box in the controller for the boost converter that computes the power produced from the PV array and makes adjustments in the set point for the PV array voltage until the maximum power point is in fact extracted from the PV array. The control loop over the boost DC-DC converter is often done in the form of controlling the input voltage of the DC-DC converter, not the output. So the input voltage is sensed compared to a reference that comes from the MPP tracking, through a voltage loop compensator and a pulse width modulator. The boost converter is controlled to set the input voltage of the PV array to the reference value determined by the maximum power point tracking. The DC to AC bridge inverter has the purpose of efficiently generating AC current, in phase with the AC grid voltage and ideally with a sinusoidal waveshape, and to balance the average power delivery from the PV array to the grid. The control system for the inverter is a bit more complicated. The reference for the current control loop, which is really the one that we're going to explain in a little bit more detail in the next lecture, is determined so that it has a sinusoidal waveshape. The output current is sensed, compared to that reference, passed through the average current mode control compensator, and that determines control signals for the four switches so that the current at the output follows the reference. The reference has an amplitude that is determined in the voltage control loop over the energy storage capacitor, but has a waveshape that is determined by synchronization with the AC grid. So the reference is sinusoidal, and is obtained by synchronizing that sinusoid with a waveform that is obtained by sensing the grid voltage. And there are a number of other protection functions that include over and under voltage and frequency with respect to what is called anti-islanding, shutting down the inverter in case the grid voltage goes out of bounds in terms of frequency or voltage. Finally, the energy storage capacitor is what balances the instantaneous power delivered to the AC grid with respect to the DC power delivered by the PV array. So our objective in the lecture that follows will be to discuss details of the average current mode control of the output current of the DC to AC inverter.