Simulation of a Bidirectional DC-DC Boost Converter Using a State-Space Averaged Model

Boost converters are electronic devices that are widely used to power devices that require a higher voltage than the input voltage supply can deliver. When designing boost converter systems, it is difficult to optimize component values without damaging components. Models and simulations of a boost converter are developed to give insight into how to design the system and prevent harm to the boost converter. In particular, a state-space averaged (SSA) model of a DC-DC boost converter that includes a secondary inductor and parasitic resistances is developed. This model is then further developed to match a hardware bidirectional DC-DC boost converter that operates in continuous conduction mode (CCM) and uses voltage-mode with current-mode control. The SSA model is used to simulate an actual hardware boost converter system. The SSA based simulation successfully matched experimental data of the hardware boost converter system at low (10kW) and medium (60 – 120kW) level power outputs, but only approximately matched the system behavior at high (250kW) level power outputs.

When designing a boost converter, there are many things to consider. High power outputs can require levels of current and voltage that can damage the system [2]. The voltage source or the load may not be ideal and can change in time [3]. There are parasitic resistances in the components that affect the system's ability to deliver the necessary power [4][5][6]. Secondary components may be required to stabilize the output of the converter [2].
To aid in designing a boost converter, simulations are developed to analyze variables in the system. However, simulations that fully recreate the boost converters can be computationally complex and taxing to run, and due to the non-linear nature of boost converters, provide little insight into how to improve the system. In these cases, averaging techniques that are much less taxing to simulate and can provide the general behavior of the system are used to model the converter [7]. A popular method of averaging boost converters is the state-space averaging (SSA) method [8][9][10][11][12][13]. This method averages the models over the possible operating conditions it can exist in, usually regarding time. These averaged models also offer a means to linearize the system using small signal analysis, which aids in developing a control system for the converter [1,14].
This work will focus on the function of boost converters and how to develop an SSA model that can be implemented as a simulation. First, a background on general, ideal boost converters and how to model those using SSA methods will be presented.
Then a complex hardware DC-DC bidirectional boost converter with parasitic resistances, secondary inductor, non-ideal voltage source, and multiple switching devices, will be introduced. An SSA model describing the hardware of the converter will be developed, first describing only its parasitic resistances and secondary inductor. This will then be used to develop a model that includes the non-ideal voltage source and multiple switching devices of the hardware in order to provide a complete description of the converter. This final SSA model will then be implemented as a simulation and a comparison between the model and the hardware will be conducted to determine the accuracy and limitations of the model.

-Boost Converters
A boost converter is an electronic device that produces a greater output voltage than that of its input voltage without generating power. These devices are widely used in electronic power supplies where the voltage source is not large enough to power the required load. Boost converters store and release energy using an inductor and a switching device to generate the higher voltage. Since these devices do not generate power, there is power lost in the conversion, although this can be minimized by implementing a control system.

-Simple Boost Converter Topology
The general topology of a simple boost converter can be seen in Fig. 1. There is a constant voltage supply (Vin) in series with an inductor (L). This inductor is connected to a diode (D) and a single pole single throw (SPST) switch. The switch connects back to the voltage supply, and the diode connects to a capacitor (C) in parallel with the load, which is modelled as a resistor (Rload). This resistor and capacitor then connect back to the negative terminal of the voltage supply.

-Boost Converter Function
The boost converter converts a low voltage into a higher voltage by using the switch to manipulate the current to store and release energy in the inductor. Inductors are components that resist a change of flow in current. The inductor resists a positive change in the rate of current flowing through it by storing energy in a magnetic field and increasing the voltage across the inductor. When the current through the inductor decreases, the inductor releases this stored energy in an attempt restore the decreasing current, becoming another supply of voltage for the output.
By using the switch, the boost converter causes the current in the inductor to increase and decrease. When the switch is closed, there is little resistance in the circuit from Vin, through the inductor and switch, and back to Vin. This allows the current output from Vin to increase, thus causing the inductor to store energy in its magnetic field. When the switch is opened, there is more resistance in the circuit due to the load.
This causes the current through the inductor to drop, therefore causing it to release that stored energy. This voltage from the inductor plus that of the input voltage is how the boost converter is able to supply a greater voltage to the load.

= +
Due to this relation, it is common to refer to dcl as the duty cycle (dc) of the system.
Since the boost converter depends so heavily on this switching action, a general calculation of the voltage across the load (Vload) can be obtained from the duty cycle and input voltage.
Because of the switching action, the voltage across the load can fluctuate slightly, however it is near constant due to the capacitor C providing voltage for the load when the switch is closed.

-Boost Converter Control
In many applications, the load of the boost converter is not constant, or the voltage supply is not constant [3]. In these cases, the duty cycle of the converter must be controlled to provide a consistent and stable output. There are two popular methods of control for boost converters using closed-loop (or feedback) control: voltage-mode control, and voltage-mode with current-mode control [1,2,14]. In these two methods, proportional-integral-derivative (PID) control is used. This work will focus on proportional-integral, or PI, control.

Voltage-Mode Control
In voltage-mode control, the voltage being delivered to the load (Vload), is monitored and used to calculate the correct duty cycle for the system. A drop in the load voltage from what is required relates to an increase in the duty cycle. A general block diagram for voltage-mode control can be seen in Fig. 2.

Figure 2 Block diagram of a system utilizing voltage-mode control.
The load voltage is measured and compared to a reference voltage (Vref), and the difference between these two signals is the error signal (e). This error is sent through the PI controller which calculates the duty cycle Where kp is the proportional gain, and ki is the integral gain. The gains of the PI controller are chosen to produce an appropriate duty cycle for the system.

Voltage-Mode with Current-Mode Control
In voltage-mode with current-mode control, both the load voltage and the current through the inductor are measured to calculate the correct duty cycle for the system. The current-mode control is used to provide information about the source and is a form of feedforward control. When used in conjunction with voltage-mode control, this method can provide better stability for the overall controller. A general block diagram for this method can be seen in Fig. 3.

Figure 3 Block diagram of a system utilizing voltage-mode with current-mode control.
Similar to voltage-mode control, the load voltage is measured and compared to a reference, creating an error signal (ev). However, unlike in pure voltage-mode control, this error is sent through a PI controller to calculate a reference current (Iref).
The current through the inductor (IL) is then compared to this reference current creating another error signal (ei), which is then used to determine the appropriate duty cycle via a PI controller. Equations for the reference current and duty cycle are shown below = + ∫( ) Where kpv and kiv are the proportional gain and integral gain for the voltage-mode controller, and kpi and kii are the proportional gain and integral gain for the currentmode controller.

-Boost Converter Conduction Modes
Boost converters have two main modes of conduction. Discontinuous Conduction Mode (DCM) and Continuous Conduction Mode (CCM) [5][6][7]. These modes describe the behavior of the current through the inductor. This work will focus on boost converters operating in CCM, but both modes are explored below.

Discontinuous Conduction Mode (DCM)
In DCM, the current through the inductor is not maintained for a period of time during one switching cycle. This means that the current through the inductor remains at 0A for a portion of the switching period when the switch is open. A waveform of the inductor's current can be seen in Fig. 4.

Continuous Conduction Mode (CCM)
In CCM, the current through the inductor never remains at 0A for a significant portion of time, although it may pass through 0A during a switching period. A waveform of the inductor's current can be seen in Fig. 5.

-Averaging Techniques
In power electronics, especially that of converters, systems can produce signals that fluctuate rapidly in time [8,13]. While important in their proper functioning, these fluctuations can make it difficult to explore how the system acts in order to gain an understanding of how to improve their design. To find the general behavior of the system, averaging methods have been developed for these situations. State-space averaging (SSA) is a popular method for averaging a system with signals that undergo small, and rapid changes [1,7,8,13]. SSA defines the different states of a system and averages them with respect to some variable, usually time. A "state" of the system is defined as a permutation of the system for a given operating condition, such as when the switch in a boost converter is open, or closed. SSA offers a way to develop a linearized model of the system using small signal analysis. This allows for the use of control design methods based on linear systems and the Laplace transform [1, 10, 15].

-General Procedure
The general procedure for developing an SSA model of a system involves three main parts. First, the different states, or operating conditions, for the system are defined and the variable for how each state is related is determined. Next, important state-space variables are identified and state-space models for these variables are developed. Finally, these state-space models are averaged together via the averaging variable to find the SSA model. The procedure is shown for an ideal boost converter below.

Definition of the System's States
In an ideal boost converter operating in CCM, there are two states for the converter. The overall topology and two states can be seen in Fig. 6. The "ON" state of the converter is defined for when the switch is "ON", or closed. The "OFF" state of the converter is defined for when the switch is "OFF", or open. The constraint that connects these two states is the amount of time that the switch is either in the open or closed position, which is related to the duty cycle of the system.

State Variables and Their Models
In the ideal boost converter, there are two important state variables that define the system. The current through the inductor and the voltage across the capacitor.
Using Kirchhoff's Voltage and Current Laws (KVL and KCL respectively), mathematical models for these variables can be found.
In the "ON" state, the switch is closed and current through the inductor (iL) is In the form of ̇= + , the model for the "OFF" state becomes

Averaging the Models
To derive the final SSA model, these two states are averaged over the constraint that relates them, which is the duty cycle of the system. This converter is operating in CCM, thus there are only two duty ratios of a switching period to consider: the ratio for when the system is in the "ON" state (dcl) and when the system is in the "OFF" state (dop). Averaged matrices ( ̅ , ̅ ) can be written Using the relationship 1 = + and converting to duty cycle ( = ) these matrices can be rewritten in terms of the duty cycle of the system The averaged matrices then become Putting these in the form ̇= ̅ + ̅ the final SSA model is obtained  The hardware is a bidirectional DC-DC boost converter that has been developed to supply power to a hospital during a power outage. This requires the boost converter to quickly provide stable power over a wide range of outputs. For this, the converter uses a secondary inductor after the switches, a nonlinear voltage source, multiple switching devices, and a control system.

-Requirements for the Hardware
Due to the environment that the hardware is designed for, strict requirements on its operation have been implemented. The converter must be able to supply the necessary power output at the desired voltage level in under 50 ms and have the ability to be recharged many times before requiring replacement components. For these reasons, large banks of lithium carbide (LiC) supercapacitors are used as the voltage supply for the hardware. The maximum potential that a bank of these supercapacitors can maintain is 420V, and the minimum potential is 250V without risking damage.
The converter must also be able to supply a wide range of power outputs, from 10kW to 250kW. The hardware needs to be designed to prevent high levels of current from damaging components. The converter is designed to have a voltage output of 520V, and the desired power outputs can require currents of greater than 1000A to be necessary.

-Topology of the Hardware
The converter relies on many devices and systems; the main circuitry for the hardware's boost converter functionality can be seen in Fig. 7.
The supercapacitors can be seen in the top left, feeding directly into the six inductors and switching devices. The currents from the switches then travel to the capacitor and load, then a secondary inductor. The secondary inductor acts to reduce the amount of ripple effect that occurs due to the switching behavior of the boost converter. This is to protect both the load and capacitor from damage.

-Non-ideal Voltage Source
The voltage source for the hardware relies on large banks of lithium carbide (LiC) supercapacitors. Each bank contains 108 of these supercapacitors connected in series. For an increase of 60kW in the power output, another bank of supercapacitors is added in parallel with the other banks. For example, for power outputs ranging from 10kW to 50kW, one bank of supercapacitors is connected to the system; for 60kW to 110kW, two banks of supercapacitors are connected. While the system is on, these supercapacitors drop in potential and act as a non-ideal voltage source. In designing models for this system, this drop in potential must be accounted for.

-Six Inductors and Switching Devices
The hardware for the converter must be able to deliver high levels of current to This means that the cascading nature of the switches is not exact. When the converter first turns on, all the switches are initialized with the same duty cycle and they cascade perfectly, but when the duty cycles begin to differ, they cycle independently of each other.

-Hardware Control System
The voltage source and load of the converter are non-ideal and can change in time. To ensure the proper output of the converter, a control system is implemented to regulate the switching action of the IGBTs. A voltage-mode with current-mode controller is used for each switch. There is one voltage-mode PI controller that works in series with six current-mode PI controllers in the system. A general block diagram of the system can be seen in Fig. 8.  was further developed to include the secondary inductor and parasitic resistances found in the hardware of the converter that operates in CCM. The secondary inductor acts to reduce the amount of ripple in the current that is delivered to the load. This directly affects the power output of the converter and thus the required voltage from the voltage source and the capacitor that is in parallel with the load.
Parasitic resistances in the system play an important role in the converter's ability to provide the required voltage for the load due to the nature of how a boost converter functions [4][5][6]. Since the hardware uses real-world components, the capacitors, switches, and inductors all have non-negligible resistances. This second model accounts for these parasitic resistances.

-Topology and States of Second Model
The overall topology and the possible states for the second model can be seen in Fig. 9. Note that in Fig. 9   Similar to the ideal boost converter, the "ON" state of the converter is defined for when the switch is "ON", or closed, and the "OFF" state of the converter is defined for when the switch is "OFF", or open. Once again, the constraint that connects these two states is the amount of time that the switch is either open or closed, which is related to the duty cycle of the system

-Define State-Variables and Their Models
The key state variables in this model are the current through L1 (iL) and the voltage across the capacitor (vc). Using Kirchhoff's Voltage and Current Laws (KVL and KCL, respectively), equations for these state variables can be found.
In the "ON" state the switch is closed and the current that passes through the first inductor is dependent on the input voltage (Vin) and the voltage drop for the resistance before the switch (RBS). Using KCL, the current is The averaged matrices become Putting these in the form ̇= ̅ + ̅ the SSA model is obtained [ ]

-Final SSA Model
The voltage source in the hardware for the converter is non-ideal as the voltage in the supercapacitors drops in time proportional to the current required by the load.
To compensate for this, the SSA model that has been developed is of a higher-order than the previous model, with the supercapacitor voltage as a state variable instead of as an input for the system.
The hardware utilizes six switching devices and inductors, and the current through each must be monitored. This requires that the developed SSA model must measure these currents, therefore each inductor's current is a state variable. The switches also receive their own duty cycle independently of each other. This means that there can be many states for the system. However, this will be simplified during the initial development of this model and expanded upon after a simpler model has first been defined.

-Topology and States of the Final SSA Model
In this initial model of the hardware, the six switching devices and their inductors have been simplified to a single switch and inductor. Since the converter operates in CCM, there are only two states for the system. A diagram of this simplified converter and its states is shown in Fig. 10. Note that the six switching devices and their inductors will be represented after the development of this model. The state of the switch defines the two states of the converter. The "ON" state corresponds to when the switch is closed, and the "OFF" state corresponds to when the switch is open. The duty cycle is the constraint over which the two states will be averaged.

-Define State-Variables and Their Models
Important key variables for the system are the current through the inductor (iL), the voltage across the capacitor in parallel with the load (vc), and the voltage supplied by the banks of supercapacitors (vLiC).
In the "ON" state the switch is closed and the current that passes through the inductor is dependent on the inductance of the inductor, the voltage drop from the resistance before the switch, and the voltage supplied by the supercapacitors. Using KCL, the current for the inductor iL is In the form of ̇= + , these equations become Where the AOFF matrix takes the form of

-Averaging the States of the Final SSA Model
To derive the final SSA model of the simplified hardware converter, the two states are averaged over the duty cycle of the system. The two segments of time during one switching period to consider for averaging are: during (the "ON" state), and during (1 − ) (the "OFF" state). The averaged matrices ( ̅ , ̅ ) can be written The SSA model for the simplified system becomes

-SSA Model Accounting for Six Inductors, Switches, and Duty Cycles
Due to the fact that each switch receives its own duty cycle independently of the other switches, there are a total of 2 6 , or 64, possible states that the system could exist in at any given time. This requires a very complex SSA averaged model dependent on future knowledge of the duty cycle for each switch at every moment in time. To simplify the model, a compromise was made in designing a model that incorporates all six inductors and switches. Rather than predetermining how each switch will function in regards to every other switch at every moment in time, the concept of the developed model is an overlay of six of the previous models, each with their own duty cycle. A diagram for the system is shown in Fig. 11.

Figure 11 Diagram of the topology for the hardware converter including all six inductors and switches.
By utilizing the previous model and expanding upon it to include all six inductor currents and duty cycles, the final SSA model for the hardware is obtained.

-Simulation Procedure
The hardware converter was tested for a given power output and measurements of important signals were monitored and recorded. These measurements were taken using current and voltage sensors that recorded the data through a filter. Due to this, the recorded data is sampled and the current through the inductors appears to have no ripple effect.
Values necessary for the simulation, such as the initial voltage of the supercapacitors, the initial duty cycle values, and the gains for the controllers were input into the Matlab/Simulink simulation. Of key importance was the resistance of the load, which changed in time due to its implementation in the hardware. The load resistance was calculated using the recorded data for the current and voltage of the load through the relation = The resistance as a function of time was then fed directly into the simulation while it ran. The simulation was then performed and the important signals were recorded and saved. A comparison of the real-world data and the simulation data were plotted and compared. Note that only two inductor currents (iL3 and iL6) and the duty cycle for their respective switches (dc3 and dc6) are plotted due to the high level of similarity between all inductors and switches.     The load power, current, and voltage for low (10kW), medium (60kW and 120kW), and high (250kW) outputs can be seen in Fig. 12-14. At every power output the simulation matches very closely with the hardware's experimental data for each parameter. There are negligible differences between the simulation and hardware thought to be caused by the fact that the simulation is based on a model for the average behavior of the system, and the experimental data fluctuates rapidly.

-Super Capacitor Voltage
The voltage for the supercapacitors (vLiC) is shown in Fig. 15. At low (10kW) and medium (60kW and 120kW) power outputs, the SSA model was able to closely follow the general behavior of the hardware, with only small deviations. At the high (250kW) power output, there was a large deviation from 4-16 seconds.

-Inductor Currents
The current through two inductors (L3 and L6) are shown in Fig. 15 and Fig.   16, respectively. At the low (10kW) power output, there was a small, constant deviation between the SSA model and the hardware. For the medium (60kW and 120kW) outputs, the SSA model closely follows the hardware. At the high (250kW) output, there is a large deviation in the compared currents, beginning at 4 seconds.

-Duty Cycle
The duty cycle for two switches (dc3 and dc6) are shown in Fig. 16 and Fig. 17, respectively. For the low (10kW) power output, there was a nearly constant difference between the duty cycles of the SSA model and the hardware. However, the constant error in dc3 was less than that for dc6, which was reflected in the results for the inductor currents. The simulation also undershot the hardware's experimental data for dc3 and overshot it for dc6. This is thought to be due to the fact that the duty cycle at this power level is low, where small errors would be more noticeable for the simulation. For variables that come before the switch: the supercapacitor voltage, inductor currents, and duty cycles, the simulation was found to be most accurate in the center range of the expected power outputs (60kW and 120kW) with the simulation agreeing very well with the experimental data of the hardware. At low power outputs (10kW), the simulation was able to capture the general behavior of the hardware, however there was a near constant difference in both the inductor currents and the duty cycles of the system. For high power outputs (250kW), the simulation was initially able to follow the general behavior of the hardware converter, but was unable to match the hardware system near the end of the trial. There was an increasing difference between the simulation and hardware for the supercapacitors' voltage, inductor currents, and duty cycles.
Despite these discrepancies, the simulation was determined to be successfully accurate and will be used in optimizing the hardware converter.

-Future Work
This work will be used to predict dangerous levels of operation for the hardware, to optimize the control parameters in its control system, and to determine how to expand the system for a wider range of applications.
Motivation for the creation of the state-space averaged based simulation was the fact that the hardware converter had caught on fire twice during its design. This simulation will be used as a means to predict how the system could react without having to run a trial on the hardware that could damage it. This is necessary when components are exchanged, system parameters are altered, or when testing new power outputs.
The SSA model of the hardware will be used to optimize the control parameters of the control system. Previously, the control parameters for the converter were determined using an informed "guess and check" method due to the converters non-linearity. However, the SSA model offers the means to provide an approximate linear description of a non-linear system (see Ch. 2.2.2). This allows for the use of control optimization techniques based on linear systems.
The hardware converter is designed with the ability to expand the system to meet a wider range of applications. This model will be used to ensure that when the converter is expanded, it will be done in an efficient manner. The SSA model and simulation easily allows for the addition of more inductors and switches, banks of supercapacitors, and changes to the control system. The model will allow for the prediction of how the converter will behave before having to build the full system with the actual hardware.

A.1.2 Topology of Simulation
The hardware converter simulation implemented in Matlab/Simulink can be seen in Fig. 20. Figure 20 Overall topology of the SSA model simulation.
On the far left of the simulation is the "Supercapacitor" subsystem. This calculates the voltage level of the supercapacitors and inputs this into the subsystems for the inductors and switches. The inductors and switches subsystems calculate the current for each inductor. These currents are sent to both the subsystem for the capacitor and load, and the controls subsystem.
The "Capacitor and Load" subsystem determines the voltage and current of the load. The voltage of the load is sent to the "Control System" subsystem. The current of the load is used with the voltage to determine the power output of the converter.
The "Control System" subsystem then calculates the appropriate duty cycle for each switch using the capacitor and load voltage, and the current through each inductor. These duty cycles are then sent to the "Inductors and Switches" subsystem and the "Capacitor and Load" subsystem.
The use of a "From Workspace" block can be seen for inputting the resistance of the load into the simulation. This block takes data from the Matlab workspace and feeds it directly into the simulation. In the hardware experimental tests, the load is simulated using a variable resistor. However, the resistance of the load has been known to fluctuate. To recreate the trials as accurately as possible, the resistance of the load was calculated using the data from the trials then saved in the Matlab workspace and loaded directly into Simulink during the simulation.
To record the values of each of these subsystems, "Scope" blocks are placed on their output. These "Scope" blocks saved the data to the Matlab workspace for use in analysis. The input "Inductor Currents" carries all six inductor currents. These are then summed, divided by the capacitance of the all the combined capacitors, then integrated using the 1 block.  In Fig. 23, the calculation of a single inductor current (iL6) is shown. This is calculated using the equation The inputs iL6, dc6, vc, and vLiC are multiplied by the appropriate values to create each term in the above equation. These terms are then summed and integrated to create the correct current for the sixth inductor.

Inductors and Switches Subsystem
The output from each inductor and switch pair is then combined into the single output named "Inductor Currents". The output "Load Current" is calculated from the load voltage divided by the resistance of the load and the ESR of the capacitor. Figure 25 Diagram for the Control System subsystem.

Control Subsystem
In Fig. 25, a block diagram of the entire control system can be seen. The load voltage is compared to the reference voltage (Vref) and the difference is sent to the voltage-mode controller. The voltage-mode controller determines the appropriate current for the inductors and this commanded current is sent to each current-mode controller where the appropriate duty cycle for each switch is calculated.  Since there are six switches in the system, this commanded current is then divided by six to evenly distribute the current.