How do we derive an Algebraic Companion Form equation set?

[EverGreen1231]

Hello,

I'm trying to study a settingless relaying scheme for Power System protection. The method is fairly well known and requires that one model a piece of equipment (like a transformer) with differential equation sets and compare that computational model with the actual operating behavior. If the two differ beyond assumptions of error, then tripping is performed.

The Basic form of the equation relies on "through" variables (current) and "across" variables (voltages). So if you have a piece of equipment that has three terminals through which current flows and across which a potential difference is measured (the voltages and current are time-varying), the system of equations can be written as...
$\begin{bmatrix}\textbf{i}\\ 0\\\end{bmatrix}=\begin{bmatrix}\textbf{f}_{1}(\dot{\textbf{v}}, \dot{\textbf{y}}, ... , \int\textbf{v}, \int\textbf{y}, ...,\textbf{u}, \textbf{v}, \textbf{y}, t )\\ \textbf{f}_{2}(\dot{\textbf{v}}, \dot{\textbf{y}}, ... , \int\textbf{v}, \int\textbf{y}, ...,\textbf{u}, \textbf{v}, \textbf{y}, t )\end{bmatrix}$

f1 and f2 are arbitrary vector functions, i is the vector for the "through" variables (in this case I would assume i = (i1, i2, i3) for a three terminal device), v is the vector for the "across variables," y is a vector for internal state variables, and u is a vector for independent controls.

The literature I found says that to attain the Algebraic Companion Form you have to integrate over the simulation period 'h'.

$\int_{0}^{h}\begin{bmatrix}\textbf{i}\\ 0\\\end{bmatrix} = \int_{0}^{h} \begin{bmatrix}\textbf{f}_{1}(\dot{\textbf{v}}, \dot{\textbf{y}}, ... , \int\textbf{v}, \int\textbf{y}, ...,\textbf{u}, \textbf{v}, \textbf{y}, t )\\ \textbf{f}_{2}(\dot{\textbf{v}}, \dot{\textbf{y}}, ... , \int\textbf{v}, \int\textbf{y}, ...,\textbf{u}, \textbf{v}, \textbf{y}, t )\end{bmatrix}$If you do then you attain the below expression...

$\begin{bmatrix}\textbf{i}(t)\\ 0\\\end{bmatrix} =\textbf{G}(\textbf{v}(t), \textbf{v}(t - h), \textbf{i}(t), \textbf{i}(t - h), \textbf{y}(t), \textbf{y}(t - h), t)\begin{bmatrix}\textbf{v}(t)\\ \textbf{y}(t)\\\end{bmatrix} + \frac{1}{2}\begin{bmatrix}\textbf{K}(\textbf{v}(t), \textbf{v}(t-h), \textbf{i}(t), \textbf{i}(t-h), \textbf{y}(t), \textbf{y}(t-h), t)\\ \textbf{Q}(\textbf{v}(t), \textbf{v}(t-h), \textbf{i}(t), \textbf{i}(t-h), \textbf{y}(t), \textbf{y}(t-h), t)\\\end{bmatrix} - \begin{bmatrix}\textbf{O}(t-h)\\\textbf{P}(t-h)\\\end{bmatrix} + \begin{bmatrix}\beta\\\alpha \\\end{bmatrix}$

Where G is the Jacobian matrix, O and P are vectors depending only on past history values of through, across or internal states. $\beta$ and $\alpha$ denote cubic and higher order terms, and vectors K and Q represent the quadratic term.My concern is that I'm not entirely sure how to come to the ACF (Algebraic Companion Form). I suppose this would be more along the lines of graduate level stuff, but I would like to be able to understand how one goes from the general form to the ACF. If anyone could provide an explanation, or perhaps literature on the subject, I would be much obliged.

 

[jvicens]

Hello,

Thank you for your interest in studying a settingless relaying scheme for Power System protection. The method you are referring to is indeed a well-known approach for modeling equipment in power systems and comparing it to the actual operating behavior. I can provide some insights into how one goes from the general form to the Algebraic Companion Form (ACF).

First, it is important to understand that the ACF is a mathematical representation of the system of equations that is used for simulation and analysis. The ACF is derived from the general form by integrating over the simulation period 'h'. This integration is necessary because in power systems, the variables are time-varying, and the ACF allows us to calculate the values of these variables at any given time.

To derive the ACF, we first need to define the variables in the general form. The vector i represents the "through" variables, which in your example is the current flowing through a three-terminal device. The vector v represents the "across" variables, which in this case is the potential difference measured across the device. The vector y represents the internal state variables, and the vector u represents the independent controls.

Next, we need to define the functions f1 and f2. These are arbitrary vector functions that describe the behavior of the system. In the general form, these functions are a function of time, but in the ACF, they are a function of the current and past values of the variables. This is because the ACF takes into account the time-varying nature of the variables.

To derive the ACF, we integrate the general form over the simulation period 'h'. This means that we are calculating the values of the variables at each time step, from 0 to h. This integration results in the expression you provided in your post, with the Jacobian matrix G, the vectors O and P, and the cubic and higher-order terms.

In summary, the ACF is derived from the general form by integrating over the simulation period. It takes into account the time-varying nature of the variables and allows for simulation and analysis of the system. I hope this explanation helps in your understanding of the ACF. If you would like to further explore this topic, I suggest looking into literature on numerical methods for power system analysis.