The motivation for the doctoral work arose from testing power electronic devices using digital simulations. Modern power electronic devices incorporate high-frequency switches that are controlled by a control algorithm implemented on an industrial computer. Before connecting such a device to the grid, it is necessary to verify the operation of the entire loop, from the control algorithm software to hardware communication, to ensure desired and safe operation. The most effective way to do this is through real-time digital simulations, where only the operation of power switches and the grid is simulated, while the hardware and software of the tested device are identical to those that will be used in the actual grid. This validation method for the device and its control algorithm is called Hardware-In-the-Loop (HIL) testing.
In this method, delays occur in the signal transmission between the hardware (the tested device) and the simulator, specifically in the computational part of the simulation, which is performed by software. These delays can cause instabilities when using the ideal transformer interface between the device and the simulator, under certain circuit configurations on both sides of this connection. The aim of this doctoral work is to investigate the possibilities of stabilizing this connection between the tested device and the simulator.
To address the issues related to delays in hardware-in-the-loop digital simulations, mathematical methods for stability analysis and delay compensation were utilized. The first step involved defining the interface between the hardware and the simulator using a delay differential equation. The obtained equation was analyzed for the influence of its coefficients on stability. To compensate for the delays, the method of spectrum assignment was employed by adding a system state described by a delay differential equation to the feedback loop at the input of the interface.
The initial chapters are dedicated to the theoretical foundations of digital simulations. The historical development of digital simulations, following the advancement of microelectronic elements, is presented. Microprocessors were the foundation that enabled simulation calculations without the need for specialized scaled test equipment, relying solely on software. However, the breakthrough occurred with the emergence of FPGA technology, which significantly reduced the simulation step to the nanosecond range. This allowed for the simulation of fast switching transients in power electronics.
The second chapter discusses the structure of digital simulations. Electrical networks are modeled using circuits described by concentrated elements, leading to equations in continuous time. To calculate the time variations and transient phenomena of interest in simulations, time must be discretized, enabling computations in discrete steps on a digital computer. The methods of segmentation and parallelization of simulations and interface circuits used for connecting different segments of the network are described.
The subsequent chapter covers dynamic systems with delays and their stability. The focus is on time-invariant systems with constant delays. An overview is provided of stability determination methods based on calculating the rightmost roots of the system's characteristic equation. The fourth chapter presents the theoretical foundations of stabilizing unstable systems with delays using the method of shifting characteristic roots in the complex plane.
In the fifth chapter, the stability of the ideal transformer interface circuit is examined. This is based on the representation of the differential equation of the interface circuit and its spectral analysis. Stability regions for various impedance configurations on both sides of the interface are calculated. The spectrum assignment method is used in an attempt to stabilize these interfaces by moving their eigenvalues to the left half of the complex plane. Limiting parameters of the interfaces that allow for their stabilization are determined.
There were three cases of circuits simulated, divided by the interface of an ideal transformer. The simulations were performed in a fully computational environment using the SIMULINK tool and on a co-simulation system of two real-time digital simulators from RTDS Technologies and Typhoon HIL. The co-simulation system was used to test the interface of the ideal transformer without a load, with a parallel resonant circuit, and with a load represented by an inverter. The closed-loop delay of the co-simulation circuit between the two simulators was measured, and the instability due to delays occurring in the fully computational simulation and the co-simulation of two digital simulators in real-time was compared.
It was found that the instability in co-simulation is mainly expressed by the appearance of unreal harmonics, whose amplitude increases as we approach the stability limit. The method of introducing the system state into the feedback loop is successful in damping these harmonics caused by unfavorable impedance conditions at the circuit division point.
A comparison of the time response deviations of the stabilized co-simulation with the undelayed simulation on just one of the simulators was performed, and an analysis of the harmonics of both simulations was carried out.
The concluding part of the dissertation provides an assessment of the proposed method's applicability in terms of accuracy and implementation complexity. The advantage of spectrum assignment by introducing the state into the feedback loop is the simplicity of implementation, as the circuit of the ideal transformer is defined as the division of the circuit through two impedances composed of resistance and inductance. The analysis of the system's differential equation shows that the presence of inductance allows for the introduction of the derivative of the system state into the feedback loop, which is a condition for assigning the final spectrum. For successful stabilization of the interface, only the knowledge of the inductance ratio on both sides of the interface circuit is sufficient. The accuracy of the simulation depends on the magnitude of the load impedance compared to the interface circuit impedance and the frequency of the observed signal. The larger the period of the observed signal compared to the delay duration, the smaller the errors in the phase angle and amplitude.
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