Application of the four-pole Poincaré-Steklov in interface construction for HIL simulation

INTRODUCTION

The article considers the possibility of using the Poincaré–Steklov filter to build an interface for hardware in the loop simulation (HIL) of the system.
Hardware in the loop simulation assumes that the system is divided into parts and one part of the system is modeled numerically, and the other part of the system is represented by a real physical object. At the same time, at each step, parts of the system exchange data (calculated and measured values) with a certain delay τ through the software-hardware interface.
The literature describes various options for constructing hardware-software interfaces ITM, TLM, TFA, PCD, DIM, GCS, Poincaré-Steklov filter, which should ensure the stability and convergence of the results of semi-natural simulation to the simulation results of the original system.
The article shows that the Poincaré–Steklov filter, with the right choice of parameters, can ensure the stability and convergence of the results of semi-natural modeling of systems.

TASK SETTING

In the article, it is necessary to consider the possibility of using the Poincaré-Steklov quadripole to build an interface in semi-natural modeling of systems, as well as to investigate the stability and convergence of the results of modeling the system in parts when using this interface.
The Poincaré-Steklov filter in its Z and Y-forms is shown in fig. 1 and 2.

Fig. 1. Z-form of the Poincaré-Steklov quadripole.

The Z-parameters of the Poincaré-Steklov quadripole for this representation are given below:

Fig. 2. Y-form of the Poincaré-Steklov quadripole.

The Y-parameters of the Poincaré-Steklov quadripole for this representation are given below:

As can be seen from fig. 1 and 2, the Poincaré-Steklov filter (D) consists of controlled current and voltage sources, the values of which are equal to the values of the corresponding currents and voltages at the inputs of the quadripole, delayed by τ. Conductivity/resistance y1/z1 and y2/z2 are called stabilizing elements (parameters). The values of the stabilizing elements are always real numbers (there is no reactive component) and significantly affect the stability and convergence of the HIL simulation results.
Suppose that the original system is represented by two parts A and B, the behavior of each of which is described by matrices of the 2nd order (the order of the matrix does not matter, the main thing is that they have one “common” node, i.e. they “intersect” with one coefficient)

In this case, we assume that the behavior of the original (not divided into parts) system is described by the system of equations given below:

We assume that this system of equations was compiled using a modified method of nodal potentials.
Let's break this system into parts using the Y-shape of the Poincaré-Steklov filter, and this can be done in two ways: directly according to the circuit diagram of the quadripole or using the Y-parameters of this quadripole.
Let's demonstrate the first way of splitting the system into parts (directly according to the four-pole circuit diagram):

after transformations we get:

where, using the negative superscript -1 and -2, it is shown that the value is taken with a delay of τ and 2τ, respectively;

Thus, the use of the Y-parameters of the quadripole made it possible to get rid of the controlled voltage sources e1 and e2 in the quadripole but complicated the control of the current sources j1 and j2.
A similar (dual) result can be obtained using the loop current method to compose a system of equations describing the original system and the Z-shape of the Poincaré-Steklov filter in order to break it into parts.

DIVISION OF THE ORIGINAL SYSTEM INTO PARTS

Consider the source system shown in Fig. 3. A voltage in the form of a meander is applied to the input of the system, inductance L1 = L2 = 1mH, Y1 = Y2 =Y4 = 1000Sm, Y3 = 0.5Sm, C1 = 3e-3.

Fig. 3. Source system.

Interest in the study of such systems arises when using HIL technology to simulate the operation of three-phase inverters.
Let's split the source system into parts using the Poincaré-Steklov filter as shown in Fig. 4.

Fig. 4. Partitioned system.

Similarly to the one described above, using the modified method of nodal potentials and the Y-form of the Poincaré-Steklov filter, we will compose a system of equations describing the behavior of the original system and divided into parts. Moreover, when compiling the equations of a system divided into parts, we apply both methods described above to make sure that they give equivalent results.
In addition, we model and compare the solutions of the original system and the broken one using two different methods for numerically solving the system of differential equations: the inverse Euler’s formula and the trapezoid method.

RESULTS OF SIMULATION OF THE SYSTEM BY PARTS

Let us simulate with a step h = 1 ms the original and divided system in MATLAB, while the parts of the system will exchange data with each other at each simulation step only once, i.e. delay τ = h = 1 ms. This method of numerical simulation of a system divided into parts is as close as possible to the processes occurring in the semi-natural simulation of systems.
The result of the simulation will be a comparison of the current through L1 and the voltage on C1 for the original system and the system divided into parts.
Below in fig. 5 shows the results of modeling the original system and the system divided into parts using the inverse Euler’s formula and the first method of partitioning the system into parts. Before modeling, using the Schur complement, the values of the stabilizing elements y2 = 0.999, y1 = 3.4789 were calculated.

Fig. 5 Graphs of the change in current through L1 and voltage on C1 for the original system (CurL1, UC1) and divided into parts (CurPL1, UPC1) at y2 = 0.999, y1 = 3.4789.

As seen in fig. 5, the HIL process is stable but converges slowly to the simulation results of the original system.
Below in fig. 6 shows the results of modeling the original system and the system divided into parts using the inverse Euler’s formula but the values of the stabilizing elements were chosen equal, i.e. y1 = y; y2 = y; where y = (0.999+3.4789)/2 = 2.2389 is the average value.

Fig. 6. Graphs of changes in current through L1 and voltage on C1 for the original system (CurL1, UC1) and divided into parts (CurPL1, UPC1) at y1 = y2 = 2.2389.

As seen in fig. 6, the HIL process is stable and compared to the experimental results shown in fig. 5 converges faster.


Michael Maksimov, Ph.D., Associate Professor.
Brown Grape Technologies

If you like our scientific work or have questions related to HIL modeling, please email us.
info@browngrape.com
We will be glad to participate in joint projects and share our experience.