Network model

Here we discuss the models used to describe the network in PowerSimulationsDynamics.jl. This is based on a standard current injection model as defined in Power System Modelling and Scripting. The numerical advantages of current injection models outweigh the complexities of implementing constant power loads for longer-term transient stability analysis. The network is defined in a synchronous reference frame (SRF), named the RI (real-imaginary) reference frame, rotating at the constant base frequency $\Omega_b$.

In simple terms, PowerSimulationsDynamics.jl internally tracks the current-injection balances at the nodal level from all the devices on the system. Based on the buses and branches information, the system constructor computes the admittance matrix $\boldsymbol{Y}$ assuming nominal frequency and this is used for static branch modeling. The algebraic equations for the static portions of the network are as follows:

\[ \begin{align} 0 = \boldsymbol{i}(\boldsymbol{x}, \boldsymbol{v}) - \boldsymbol{Y}\boldsymbol{v} \end{align}\]

where $\boldsymbol{i} = i_r + ji_i$ is the vector of the sum of complex current injections from devices , $\boldsymbol{x}$ is the vector of states and $\boldsymbol{v} = v_r + jv_i$ is the vector of complex bus voltages. Equations (1) connect all the port variables, i.e., currents, defined for each injection device. Components that contribute to (1) by modifying the current $\boldsymbol{i}$ are (i) static injection devices, (ii) dynamic injection devices, and (iii) dynamic network branches. Components that contribute to modify the admittance matrix $\boldsymbol{Y}$ are static branches.

Static Branches (or Algebraic Branches)

Lines

Each line is defined using a $\pi$ model connecting two buses $(n,m)$, with a series resistance $r$ and reactance $x$, and a shunt capacitance at both ends $(c_n, c_m)$. The values are already in system per unit. Then each branch contributes to the admittance matrix as follows:

\[\begin{align} Y_{nn} &+\!= \frac{1}{r+jx} + jc_n \\ Y_{nm} &+\!= \frac{-1}{r+jx} \\ Y_{mm} &+\!= \frac{1}{r+jx} + jc_m \\ Y_{mn} &+\!= \frac{-1}{r+jx} \\ \end{align}\]

Two-Windings Transformers

Similarly to lines these are defined by a series reactance and impedance. The equations are equivalently of the lines without the shunt capacitance.

Dynamic Branches

Dynamic network branches contribute directly to (1) by modifying the vector of complex currents. Their parameters are also the series resistance $r$ and reactance $x$, and a shunt capacitance at both ends $(c_n, c_m)$ for a line $\ell$. In addition, they define 3 new additional differential equations per line (6 in total for real and imaginary part):

\[\begin{align} \frac{l}{\Omega_b} \frac{d\boldsymbol{i}_{\ell}}{dt} &= (\boldsymbol{v}_n - \boldsymbol{v}_m) - (r+jl) \boldsymbol{i}_{\ell} \\ \frac{c_n}{\Omega_b} \frac{d\boldsymbol{v}_n}{dt} &= \boldsymbol{i}_n^{\text{cap}} - jc_n\boldsymbol{v}_n \\ \frac{c_m}{\Omega_b} \frac{d\boldsymbol{v}_m}{dt} &= \boldsymbol{i}_m^{\text{cap}} - jc_m\boldsymbol{v}_m \end{align}\]

Since all the values are in per unit, the reactance is equal to the inductance.

A detail discussion about the effects of different line models in the modeling of inverters is presented in Grid Forming Inverter Small Signal Stability: Examining Role of Line and Voltage Dynamics