Chapter model is chosen the mathematical equations


1.1 Simulation of power
electronic circuits

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!

order now

To simulate a power electronic circuit or any
electrical circuit, there are a few procedures that are required to be followed.
At first a mathematical model of the circuit under test is formed, then the
circuit is represented by some mathematical or network equations and finally
some techniques are chosen to solve those equations. After a proper model is
chosen the mathematical equations for the model can be developed by using
Maxwell’s cyclic current, widely known as Kirchhoff’s Voltage Law (KVL), nodal
analysis or Kirchhoff’s Current Law (KCL) and the relatively newer approach
called Modified Nodal Analysis (MNA). Nodal analysis had some advantages over
mesh analysis, one being reduced number of equations compared to mesh analysis.
But there were difficulties in the use of the classical nodal analysis particularly
in computer simulation; certain elements such as voltage sources, dependent
sources, transformers etc. could not be included in the analysis unless some conversion
was done to some extent but on conversion there was always loss of information
about the model 1. MNA was proposed by Chung-Wen Ho, in 1975 2, 3
to resolve the limitations of the classical nodal analysis. MNA can
considerably reduce computation time for solving the network matrices and is
easier to implement on a computer thus making it more suitable for simulation
of electrical circuits. Real Time Simulation (RTS) of power electronic circuits
demands even faster computation times and MNA was further upgraded to a Fixed
Admittance Modified Nodal Analysis Method (FAMNM) 4, 5. This method allowed
the system equations to be solved in very small time steps as required by RTS
of fast switching power electronic converters.

1.2 Real Time Simulation

Whenever a computer simulation of a computer model
of any physical process or a system is done in parallel to its physical
counterpart can be referred to as the Real Time Simulation (RTS). The virtual representation of physical system i.e. a
virtual model runs simultaneously and also for the same time as the physical
system. They may share common input variables and come out with comparable
output. One good example for a RTS can be operation of the Fuel
Injection System of a modern day computer controlled car engine, the onboard
computer (Engine Control Unit) calculates the duration of operation and the
interval between each operation based upon the throttle input, camshaft
position, inputs from Oxygen Sensors, inputs from NOx sensors etc. all of which
are measured in real time.

A computer does all the computations using an
operating system which eventually does all of its calculations in the form of
0’s and 1’s. All the differential equations, state equations or any
mathematical functions representing a physical system will converted to a
discrete system of 0’s and 1’s, these will be solved by the computer simulation
software using their own solvers. The solvers use different numerical methods
to do the computations and each may take different amount of time and produce
results with different accuracy.

To work with RTS the simulation associated would
be for discrete time with a constant step size. Variable time-steps simulation
is not suitable for RTS and hence the time is incremented in equal step sizes
called Simulation Time Steps and the simulation itself is often called Fixed
Time Simulation.

As mentioned earlier the differential equations
and the mathematical functions representing the model are solved to perform the
I/O operations and to obtain the output of the model. However during a
‘discrete non real time simulation’ the actual time required to solve the
aforementioned equations and functions may be more or less than the simulation
time step. But in case of real time simulation it is necessary that (apart from
the precise modeling of the physical system) all the computations are done
within the simulation time step so that the model under test can accurately
represent the functioning and perform all the I/O operations of its equivalent
real or physical system.

If the computations are not complete within the
simulation time step the real time simulation results are not accurate, which
is also referred to as an ‘overrun’. Moreover, if the computations are done
before the simulation time step is complete then the remaining time, called the
‘idle-time’ 7 is simply lost, which is in contrast to the accelerated
simulations where the remaining time would be utilized to perform the
computations of the subsequent time step. Fig. 1.1. to Fig. 1.3. distinguishes
between the characteristics of a real time and non-real time simulations.

Fig. 1. 1. Timing
diagram when Computation Time is less than Simulation Time Step (non-real time)

Fig. 1. 2. Timing
diagram when Computation Time is greater than Simulation Time Step (non-real time)

Fig. 1. 3. Timing
diagram during Real Time Simulation

1.3 Selection of the real time simulator

As suggested in 7-12 selection of real time
simulators can be done based upon the applications to which they are intended for
and can be categorized as:

Rapid Control Prototyping

In a Rapid Control Prototyping a physical setup
is always used and the controller is implemented in a real time simulator. The
presence of a virtual controller enables it to be configured with more
flexibility and be debugged easily. Since the modelling and testing of the
controller becomes easy and fast, the prototype model of the controller can be
developed sooner to a final robust product.

Hardware in the Loop (HIL):

In Hardware in the Loop a virtual model of the
physical system is run on a real time simulator this virtual model emulates and
behaves like a physical test bench; this virtual model is then controlled by a
physical controller. In a variation of HIL, another real time simulator can
function as a physical controller and feed the virtual model in a separate
simulator. This configuration is very beneficial because the physical
controller can be tested even without a physical setup, the tests are very
repeatable and can be done without any fear of damage as in case of a real
hardware based setup. The work in this literature uses Hardware in the Loop

Software in the Loop (SIL):

Software in the Loop is possible as the
simulators grow more and more powerful. This allows the controller and the
virtual plant model to be implemented in the same real time simulator. Here no
physical input/outputs (I/O) are used and that ensures that the fidelity of the
signals are maintained. Moreover the simulations can now run only in the
virtual mode and there are no constraints in following the time clock of the
real world. Simulation can now take their own time and if resources are
available simulations can run faster than the real world time while maintaining
the integrity of the results.

1.4 CPU and FPGA Based Simulation

The computer architecture has changed a lot in the last two
decades, the advent of multiple cores, increase in parallel processing
capabilities, decrease in the I/O latencies, faster working memories and
improved hardware interfaces 13 have made the modern computer quite suitable
for real time simulations. Fig. 1.4. shows an Intel chipset architecture widely
used until 2011, Fig. 1.5. shows an architecture used currently in most modern

Fig. 1. 4. Computer
chipset architecture prevalent before Sandy Bridge microarchitecture introduced
in 2011

Fig. 1. 5. Modern
computer chipset using Nehalem microarchitecture

However, even though the CPU of a modern
computer has a very high clock frequency the sequential nature of the operating
system and the latencies still present at the i/o communications buses/ports
allows it to have a minimal sampling time of about 5-10 ?s. This sampling time
is often enough for the real time simulation of systems with slower dynamics
such as a motor but for fast systems like the high frequency switches in the
power electronic systems, this sampling rate is inadequate. So a methodology
was developed wherein the models requiring very low sampling times are
simulated in Field Programmable Logic Gate Arrays (FPGAs) 14 – 16, the
highly parallel structure of the FPGA allowed very high sampling rates with
simulation time steps as low as 250ns. Fig. 1.6. shows a CPU based simulation
and Fig. 1.7. shows a FPGA based simulation.

Fig. 1. 6. CPU
Based Simulation

Fig. 1. 7. FPGA
Based Simulation

1.5 Discrete Time Switch Model in Real Time

There are a number of models for representing a switch
in order to make it suitable for computer simulations. Some simulators model
the switch as a small resistance when it is ON and a large resistance when it
is OFF 5, the value of the ON and OFF resistances are updated at every step
of the iteration in the numerical method used for the simulation 5. The
trouble with this approach is that for a large network with a number of
switches huge amount of computational resources may be consumed for the updating
of the resistance parameters at every step of the simulation. 17 proposes a
model with a RC circuit for an open circuit and an inductor for a short
circuit, this representation removes the need of a system matrix inversion when
the switch state changes from ON to OFF or OFF to ON. Another model proposed in
18 represents the switch as a resistance across a capacitance, when the
switch turns ON this resistance goes low and goes high when it is turned OFF.

In the switch model mentioned in 5 an ideal
switch is modelled as a conductance in parallel with a current source. This
conductance will not change with the iteration steps and the current source
shall provide details of the state of the switch. The work in this thesis shall
consider this model for real time simulation.

Validation of Power Electronic Circuits

Validation of power electronic systems is
necessary to ensure that a virtual model behaves like a real physical system to
the extent possible. Real time simulation of fast switching power electronic
converters using a FPGA based ‘Electric Hardware Solver (eHS) 8 is discussed
in 16. The design flow for the solver as well as validation results of the
RTS of a number of power electronic converters are also covered. The results
were validated by comparing the output of the eHS with those from the solver of
SimPowerSystems (SPS). Some of the shortcomings of this method of simulating
fast switching devices has also been discussed. Notwithstanding the
shortcomings the technique of performing real time simulation of power
electronic systems in FPGAs has a lot of potential.

Multilevel Inverters

An introduction to the different topologies and
working principles of multilevel converters are covered in 20. A more
detailed overview about NPC inverters along with the modulation techniques is
provided in 21. Different modulation schemes for a three level inverter are
introduced in 22. A comprehensive working of the three level NPC inverter is
discussed in 23, 24, this will also be covered in detail in the Chapter 3
along with a step by step procedure in the hardware validation of the same. The
results obtained from the simulation of a three phase three level inverter in
SimScape Power Systems (SPS), a FPGA based electrical Hardware Solver (eHS) and
the physical converter shall be compared. Furthermore few techniques for
optimizing the switch conductance for the Pejovic discrete time switch model
shall be discussed in more details in Chapter 4.


A brief introduction to circuit simulation and
real time simulation of power electronic converters is provided in this
chapter. The chapter introduces some of the current trends in real time
simulation and also familiarizes the reader with the basics of RTS. The chapter
also introduces some switch models which are often used in the RTS of
electrical systems. Certain constraints in the RTS of power electronic systems
are also discussed, this chapter also reckons with related work done by other
researchers and gives a glimpse of the work that shall be done to complete this