Non-linear techniques for control of boost initial stage
Robert Balog, August 2001
The purpose of this report is to detail the development of a boost stage (input stage) for the University of Illinois’s Future Energy Challenge 2001 Low Cost Fuel Cell inverter.
The proposed power source for the low cost inverter system is a fuel cell. Based on the information provided by the National Energy Testing Laboratory, we can assume the fuel cell to have a power curve as shown in Figure 1.
Figure 1 Fuel cell power curve
The fuel cell voltage is expected to be in the range of 42V to 52V based on load current and normal operating conditions. However, the open cell voltage can be as high as 72V if the pre-load fails and must be designed for or protected against in the inverter design. Inherent in the fuel cell is a high intolerance for current ripple at low frequency, for example 60Hz or slower load transients. The fuel cell electrical equivalent model has a rather large capacitance shunting the device. This suggests that the fuel cell can in-fact tolerate ripple if the frequency if high enough.
The NETL published the following guidelines for current ripple in their test fuel cell :
· 120 Hz ripple: < 15% from 10% to 100% load, not to exceed 0.6 A for lighter loads.
· 60 Hz ripple: < 10% from 10% to 100% load, not to exceed 0.4 A for lighter loads.
· 10 kHz and above: < 60% from 10% to 100% load, not to exceed 2.4 A for lighter loads
· >120 Hz to <10 kHz, limit linearly interpolated between the 120 Hz and 10 kHz limits.
· Transients below 60 Hz represent "load following" action of the system, and should track the Maximum Available Current signal from the fuel cell to within 1% for purposes of both fuel cell integrity and efficiency.
Since the fuel cell voltage is nominally in the 48V range, the voltage will have to be strictly boosted to either a 340V bus for use with an inverter or a intermediate voltage of perhaps 60V to coincide with a 30 cell, 5 battery, pack. Following the battery pack the design has previously been proposed to include a forward converter coupled with an ac-ac cycloconverter and the output filter.
Figure 2: Proposed inverter system
Therefore the load for the boost stage is the battery and the forward converter stage of the inverter system which together behave more like a constant power load rather than a resistive load. The proposed boost converter will be operated in current control mode rather than voltage control mode. The boost converter is ideally suited for interfacing the inverter system with the fuel cell. Based on the load conditions, the boost stage can be commanded to draw a specific amount of current from the fuel cell with a ripple well defined by the frequency, size of the inductor, and duty ratio.
The inverter system will need to respond to dynamic load conditions. However, the chemistry of the fuel cell cannot respond instantaneously to these load dynamics. The function of the battery bank is to provide momentary ride through capability while the fuel cell adjusts to the new operating point. On average, the state of charge (SOC) of the battery bank has no net change: any energy supplied to the load by the battery during a transient must be replenished from the fuel cell once the chemistry has reached steady state condition. Therefore, the current command for the boost converter must include the average load current as well as the battery charging current:
The proposed boost converter topology is shown in Figure 3.
Figure 3: Proposed Boost Converter
The converter equations, ignoring series resistance, are:
Applying KVL and KCL to the boost converter, illustrates that there are three possible states or circuit configurations. Since we intend to operate the boost converter in current control mode we must allow for the situation where a low average inductor current causes the circuit to operate in discontinues conduction mode (DCM). Each one of these configurations can be expressed as a second order differential system of equations. Then, weighting the contribution of each switching configuration, a complete expression can be written for each of the state variables:
The steady state solution for this set of equations can be expressed for each switch configuration and thus yields an equilibrium point for each configuration. The following plot illustrates the trajectory of the state variables for a given set of initial condition and no switching action.
RL=0.05Ω (inductor resistance)
The equations for the boost converter can be solved to find the load line for the converter. While the ideal converter, neglecting series resistance, has an output voltage that is proportional to 1/(1-D) and grows to infinity as D ® 1, a real inverter has series resistance in the inductor that causes the output voltage to "fold back" as the inductor current increases as shown in Figure 4. The figure also includes the two equilibrium points for the two converter states. The parameters used for the simulation are:
Since the load-line of a real converter is not a one to one mapping, the solution will have two answers. Based on the previously calculated expected operating point, we limit the solution set to the section of the load line from 0% to 50% duty ratio characterized by monotonically increasing output voltage in Figure 5.
Figure 4: Load-line for boost converter
Figure 5: Boost converter load-line
Given that the boost converter is a second order system, each circuit configuration has an equilibrium point that can be found by solving the converter equations for that particular switch configuration.
The general concept in graphically constructed boundary control is to place the control law such that the phase portrait is divided into sections or regions with the “on” and “off” equilibrium points residing in different regions in the phase plane. As the system evolves and the state variables approaches an equilibrium point, the trajectory crosses the graphical boundary and initiates a switch action. The new system begins to evolve and the trajectory moves toward the equilibrium point.
For a rigorous treatment of graphical boundary control, consult the thesis by Muyshondt .
The following pages contain Simulink simulations of various non-linear control techniques. The first few examples use pure boundary control laws. The switch action is based solely on the state variables. The advantage of such a system is that the controls are relatively simple to implement and switching is done on an “as needed” basis. If the load suddenly is dropped, the switching interval or period correspondingly increases. A disadvantage in this scheme is that there is no enforced switching frequency. We will see that the operating mode of the converter is based completely on the values of the energy storage components (boost inductor and output capacitor.) If the values are small enough, the converter operates in "sliding mode" where the switching frequency is fast and the state variables have small deviations about the boundary control law. If the values of the inductor and capacitor are large, the circuit operates at a lower frequency and the state variables generally have larger ripple components.
The first of the following simulations assume zero energy storage as initial conditions. Following this case a simulation is presented based on the initial condition expected in the fuel cell, battery buffered, inverter system.
Figure 6: Simulink model for hysteretic peak current controller
The Simulink simulation for the hysteretic controller illustrates how switch action is initiated based on the value of the inductor current. For inductor current less than 17.75A the boost switch is turned on causing the inductor current to ramp up. The capacitor voltage drops as the capacitor discharges to support the load current. When the inductor current reaches 19.75A the boost switch is turned off, the wheeling diode forward biases turning on. The capacitor voltage rises as the inductor current supports the load and recharges the capacitor.
The phase portrait in Figure 7 shows the trajectory for the first switching cycle as the inductor current increases and then after the boost switch turns off and the energy is transferred into the capacitor and the capacitor voltage rises. After the first switching cycle, the state trajectory oscillates about the boundary as it slides toward the steady state operating point. The magnitude of the deviations from the boundary interface is bounded by the hysteretic limits. The switching becomes a function of the load conditions and the size of the energy storage elements. As the hysteretic band is reduced, the frequency of the switching cycle increased.
True sliding mode control has no hysteretic band and hence switching frequency approaches infinity as the controller state moves along the boundary. Practically, the switching frequency is limited by the bandwidth of the control circuit. In the case of power electronics, it is often desirable to limit the frequency of switch action in order to reduce switching losses.
Figure 7: Phase portrait for hysteretic sliding mode control
Figure 8: Capacitor voltage for hysteretic sliding mode
Figure 9: Inductor current for hysteretic sliding mode
From Figure 8 and Figure 9 we can observe the large excursion that the state variables undergo during the first switching cycle of the converter. Further, the converter requires a number of switching cycle to reach the steady state operating point.
Recall that the phase portrait trajectory is controlled by the values of the energy storage elements. It should be possible to choose the values of the inductor and capacitor such that within one switching cycle the converter is operating at the steady state operating point. We can call this condition “one cycle startup.”
The Simulink model for this converter is the same as in previous example and is shown in Figure 6. The values of the inductor and capacitor for this simulation where chosen by trial and error or a so-called manual “iterative” method. However, analytical methods have been found based on the concept of “time reversal method” to calculate the “off” trajectory backwards from the operating point and find it’s intersection with the “on” trajectory computed from initial values. 
From the phase portrait in Figure 10, we see that the converter is at the steady state limit cycle by the second switching period.
Figure 10: Phase portrait for hysteretic “one-pulse” startup
Figure 11: Capacitor voltage for hysteretic “one-pulse” startup
Figure 12: Inductor current for optimal “one-pulse” startup
Figure 11 and Figure 12 show the state variable settle into the steady state limit cycle by the second switching period. The maximum values of the state variables are significantly reduced compared to the “sliding mode” simulation resulting in less stress on the switching components and faster settling after a transient condition.
The previous simulations where based on a pure hysteretic controller. Since there was no constraint on frequency, the converter experience variable switching frequencies surrounding a transient. The next model uses a fixed frequency clock to control the FET turn on and a peak inductor current boundary to turns the FET off.
From the equation for voltage gain, we determine the duty ratio to be 20%.
Chose a clock frequency fclock=10kHz.
Calculate inductor size for a ripple of 6A:
Calculate capacitor size for a ripple of 2V:
These values assume a linear rise and fall for the capacitor voltage and inductor current. While this is not entirely accurate, it provides reasonable results.
Chose current limit based on load current and ripple current. Since we are measuring the inductor current, output current must be reflected to average inductor current.
The Simulink model is shown in Figure 13. A set-reset flip-flop is used to toggle the FET main switch. The master clock sets the flip-flop while the current limit resets the flip-flop. The duty ratio of the clock is set to low value, in this case 1% so that it does not interfere with the action of the current limit reset[b5]. A reset dominant SR latch is desirable.
Figure 13: Simulink model for the clocked peak current controller
Figure 14: Phase portrait for clocked peak current control
Figure 15: Capacitor voltage for clocked peak current control
Figure 16: Inductor current for clocked peak current control
By controlling the turn on of the switch, the frequency of the switching action is controlled and the duty ratio is automatically determined by the current limit control. Inductor and capacitor size are still easy to calculate. The plot of inductor current and q(t) indicate a “reset dominant” flip-flop is desired to prevent the clock from turning on the FET if the inductor current is above the threshold – thus at startup or other transient conditions, the converter can operate in “pulse skipping” mode.
The previous simulations represented the case with no initial energy storage. In the intended fuel cell inverter application, the boost converter is buffered by a 60V nominal battery bank. Therefore, the output capacitor has an initial value of the battery voltage.
The following simulation repeats the Simulink model in Figure 13 using the calculated inductance and capacitance with the battery voltage as the initial condition of the capacitor.
Figure 17: Phase portrait for clocked peak inductor current control with battery buffer initial conditions and specification derived component values
Figure 18: Capacitor voltage for clocked peak current control with initial conditions
Figure 19: Inductor current for clocked peak current control with initial conditions
The chemistry behind the fuel cell manifests itself as a long time constant with respect to changes in available power from the cell. If an attempt to draw more power from the fuel cell is made prior to adjustment of the cell’s chemistry, damage to the cell can result. Therefore, an intermediate energy storage device is required. One such topology has a battery stack buffering the inverter from the fuel cell as shown in the figure below.
Figure 20: Inverter with battery model
A battery is a fairly complicated device with many parameters such as capacity, dead-cell voltage, discharge impedance, self-discharge impedance, and shunt capacitance. In order to simplify the simulation, a model for a sealed lead acid battery was presented in and is shown in Figure 20. The battery is modeled as a capacitor for energy storage, a dc offset voltage, and a series resistance to limit the short circuit current.
The value of RS is taken as 80mΩ per cell as suggested in . The calculated equivalent series resistance of the pack is:
The typical “dead cell” voltage for SLA technology is about 1.75V. Therefore the total offset voltage:
Lastly, the energy storage capacitor can be calculated. First we calculate the maximum battery pack voltage:
The maximum capacitor voltage must be the difference between the maximum expected battery voltage and the dead-cell voltage:
Using the relationship for a capacitor we can calculate the required value:
A Simulink model is shown in Figure 21.
Figure 21: Simulink based battery model
Adding the model for the battery expands the system to a third order (three state variable) system.
The discussion of peak current boundary control has been limited to control law that is a function only of inductor current. In the phase portrait with inductor current on the vertical axis, this control law is simply a horizontal line as shown by the dotted line in Figure 22. With the addition of the battery and the output of the boost converter, we must consider battery charging requirements in the control law.
As batteries reach full capacity, the charging current naturally tapers off to a “trickle charge” level of only a small fraction of the C rating. To improve the stability of our boundary controller we account for the state of charge (SOC) of the battery by introducing the battery voltage into the control law. The solid line in Figure 22 is our new boundary control law as a function of both commanded current and battery voltage. Regardless of the commanded current, as the batteries charge and the voltage increases to the maximum value the inductor peak current will back off to preventing “boiling” of the batteries.
Figure 22: Graphical boundary control with current taper
We choose to begin the taper when the battery pack is at 64V or about 2.13V per cell indicative of a almost fully charged battery.
The boundary control law can we expressed analytically as:
Figure 23 shows a simulation of the boost converter with battery model using straight current control and tapered current control. In order to speed the simulation, a storage capacitor value of .1F was chosen rather than the calculated 1200F for the 7Ah batteries used with the prototype converter. Without the tapered current control, the boost converter will continue to supply current into the battery and battery damage can occur.
Figure 23: Simulation of boundary control with and without current taper
See Appendix for schematics of hardware and Figure 26 and Figure 27 for photos of the prototype.
Figure 24 is a scope image of the boost converter delivering approximately 200W into a resistive load. The input voltage is 48V supplied by a Kenwood power supply. The average input current was measure to be 4.56A. The boost converter is operating at an output voltage of 60V. The following table summarizes the traces:
Vsense – inductor current when FET is on
Figure 24: Operation of boost converter
The duty ratio of the converter needed to satisfy the boundary controller is 31%.
The current taper function was tested with no additional load on the boost converter. The current command was set to approximately 0.6A. The battery voltage prior to charging was 65.3V. Figure 25 is a plot of the battery voltage, current and computed SOC obtained from the prototype converter. As the battery voltage increases, the boost output current is in-fact reduced.
Figure 25: Experimental charging profile with taper control
The final hardware design of the clocked peak current boundary controlled boost converter was tested at 500W into an electronic load. The efficiency was calculated to be 92% at this power level. This was achieved without special regard to component selection, for example, the boost FET was a IRF520 and the diode was a fast rectifier but not Schottky. The current taper feature of the boundary control automatically reduced the peak current limit as the battery pack SOC increased to taper off the charging current.
Figure 26: 1.5KW Fuel Cell inverter input stage with batteries
Figure 27: Boost converter and control circuitry