Reference Frames & Initialization

Phasor Approximation

Under the phasor approximation, the network equations are:

$$\bar{\mathbf{I}} = \mathbf{Y} \bar{\mathbf{V}}$$

where $\bar{\mathbf{I}}$ is the vector of complex currents, $\bar{\mathbf{V}}$ is the vector of complex voltages, and $\mathbf{Y}$ is the bus admittance matrix.

The voltage at bus $i$ takes the form:

$$v_i(t) = \sqrt{2} \, \text{Re}\left[(v_{xi}(t) + j\,v_{yi}(t)) \, e^{j\omega_N t}\right]$$

where $v_{xi} + j\,v_{yi}$ is the voltage phasor in rectangular coordinates on $(x,y)$ axes rotating at $\omega_N = 2\pi f_N$.

Synchronous Reference Frame

The $(x,y)$ axes rotating at $\omega_N$ form a synchronous reference. After a disturbance, the system may settle at frequency $f \ne f_N$, causing phasor components to oscillate at $|f - f_N|$.

This reference is suited for short-term simulations or when the model includes an infinite bus driving the frequency back to $f_N$.

Center of Inertia (COI) Reference

In the COI reference, the axes rotate at:

$$\omega_{coi} = \frac{\sum_{i=1}^m M_i \omega_i}{\sum_{i=1}^m M_i}$$

where $M_i = 2H_i \frac{S_{Ni}}{S_B}$ is the inertia coefficient of the $i$-th machine.

When the system reaches equilibrium at frequency $f$, all phasor components tend to constant values, enabling larger time steps. The COI reference is well suited for long-term simulations.

Specifying the Reference Frame

The reference is specified in Solver Settings via:

$OMEGA_REF SYN ;    # Synchronous reference
$OMEGA_REF COI ;    # Center of inertia reference

The presence of a Thévenin equivalent (infinite bus) forces the synchronous reference.

Network Equations

With all phasors referred to the $(x,y)$ axes, the network equations decompose into:

$$\begin{bmatrix} \mathbf{i}_x \\ \mathbf{i}_y \end{bmatrix} = \begin{bmatrix} \mathbf{G} & -\mathbf{B} \\ \mathbf{B} & \mathbf{G} \end{bmatrix} \begin{bmatrix} \mathbf{v}_x \\ \mathbf{v}_y \end{bmatrix}$$

For a network with $N$ buses, there are $2N$ equations involving $4N$ variables.

Initialization Procedure

The dynamic simulation is initialized as follows:

1. Start from initial bus voltages (from the power flow solution)

2. Compute power flows in network branches and shunts

3. Determine bus power injections by summing flows at incident branches

4. Share bus injection among components using one of two methods:

   • (i) Explicit powers: $P^c_j = P^{c0}_j$, $Q^c_j = Q^{c0}_j$
   • (ii) Fractions: $P^c_j = f_{Pj} \cdot P^{inj}_i$, $Q^c_j = f_{Qj} \cdot Q^{inj}_i$

   These methods are mutually exclusive: $f_{Pj} \times P^{c0}_j = 0$.

5. Assign remaining power to an impedance load: if the unassigned power is above an internal tolerance, an automatic constant-admittance load (named M_bus) is created:

$$(G_i^r - jB_i^r) \cdot V_i(0)^2 = -(P_i^r + jQ_i^r)$$

Tip

For a single component taking the whole bus injection, set $f_P = 1$, $P^{c0} = 0$, $f_Q = 1$, $Q^{c0} = 0$. This allows reusing data at different operating points.

Note

Names starting with M_ are reserved for automatic impedance loads. Check their values through the “Load Initialization” menu in the STEPSS GUI.

Initialization Output Example

NUMBER OF IMPEDANCE LOADS :     3  (M_ type:     3 )

load name              bus name                  P          Q

M_2                    2                       90.002     17.997
M_3                    3                        0.013     -0.011
M_4                    4                       -0.017     -0.022

Here, M_3 and M_4 are negligible (rounding artifacts), while M_2 suggests a missing load specification at bus 2.