State-Space Nodal

The state-space nodal (SSN) method represents a component by its own continuous state-space model and couples it to the network through the nodal admittance matrix. A component is described by

$$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}, \qquad \mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u},$$

where $\mathbf{x}$ is the internal state, and the input $\mathbf{u}$ and output $\mathbf{y}$ are the terminal quantities exchanged with the network (a voltage and the corresponding current). Trapezoidal discretisation of $(\mathbf{A}, \mathbf{B})$ yields a discrete model $(\mathbf{A}_d, \mathbf{B}_d)$ and a Norton equivalent: a constant conductance $\mathbf{W}$ stamped into the system matrix plus a history current source recomputed each step from the previous state and input. Because the component is solved simultaneously with the network in the same nodal system, SSN is numerically robust without the parasitic snubbers that delayed current-injection schemes require.

This builds directly on Nodal Analysis and is the companion of State-Space Extraction, which recovers a state-space model from an MNA simulation rather than starting from one.

Shift to the Dynamic-Phasor Envelope

In the Dynamic Phasors (shifted-frequency) domain a physical quantity is written as a slowly varying complex envelope on a carrier $\omega_s$,

$$x(t) = \operatorname{Re}\{\, X(t)\, e^{j \omega_s t} \,\},$$

so that differentiation maps to $\frac{dx}{dt} \rightarrow \frac{dX}{dt} + j \omega_s X$. The physical state-space model therefore becomes, for the envelope,

$$\frac{d\mathbf{X}}{dt} = (\mathbf{A} - j \omega_s \mathbf{I})\,\mathbf{X} + \mathbf{B}\,\mathbf{U}.$$

DPsim does not solve the dynamic-phasor system as a complex matrix: every complex admittance $g = g_r + j g_i$ is stamped as a real $2 \times 2$ block $\left[\begin{smallmatrix} g_r & -g_i \ g_i & g_r \end{smallmatrix}\right]$, with the real and imaginary node parts living in separate halves of a real-valued system. The DP-SSN model matches this by carrying the envelope in real/imaginary-split form $[\mathbf{X}_r; \mathbf{X}_i]$ and writing the carrier shift explicitly as a real augmented model,

$$\mathbf{A}_\text{aug} = \begin{bmatrix} \mathbf{A} & \omega_s \mathbf{I} \\ -\omega_s \mathbf{I} & \mathbf{A} \end{bmatrix}, \qquad \mathbf{B}_\text{aug} = \begin{bmatrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B} \end{bmatrix}.$$

The same real trapezoidal discretisation used by the EMT SSN models then applies unchanged, the Norton conductance comes out as a real block that stamps directly into the augmented network, and the interface reads and writes the real and imaginary node halves directly with no complex assembly. The $\pm \omega_s$ coupling block is algebraically exact; placing a rotating-frame term inside the real state matrix follows the established SSN practice for rotating-machine models.

Components

The single-phase dynamic-phasor SSN models are:

Full_Serial_RLC, a series resistor-inductor-capacitor one-port with a hand-derived state-space model, used as the reference component.
GenericTwoTerminalVTypeSSN and GenericTwoTerminalITypeSSN, which accept a user-supplied $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$ and build the V-type (voltage input, current output) or I-type (current input, voltage output) stamping accordingly.

All three reproduce the classical dynamic-phasor stamping of the same circuit, and the reconstructed time-domain waveform matches the EMT and EMT-SSN results within discretisation error.

Three-Phase Components

The same real-augmented model extends per phase to DP::Ph3. The $3 \times 3$ $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ matrices are general, so off-diagonal entries can couple the phases together:

Full_Serial_RLC, the three-phase series RLC one-port.
GenericTwoTerminalVTypeSSN and GenericTwoTerminalITypeSSN, the three-phase generic V-type and I-type components.

As in the single-phase case, all three reproduce the classical three-phase dynamic-phasor stamping exactly, and the reconstructed time-domain waveform matches the EMT and EMT-SSN results within discretisation error once corrected for the RMS-to-peak scaling that EMT::Ph3 sources apply and DP::Ph3 sources do not, since the DP envelope already carries the complex amplitude directly. The notebooks below only exercise the symmetrical, diagonal case; coupling between phases is not covered by existing tests.

Variable Components

The components above are fixed-parameter LTI systems: $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$ are built once from the component’s parameters and never change. Some SSN components instead depend on the operating point and are relinearized and re-stamped every step (via MNAVariableCompInterface), the same simultaneous nodal solve but around a Jacobian frozen at the previous step’s converged state rather than a constant matrix. DP::Ph1::AvVoltSourceInverterStateSpace, an averaged grid-following inverter, is one such component; see Power Electronics for its state vector and model equations.

Validation and Examples

Two notebooks accompany the single-phase models, both on a single-carrier series RLC one-port. examples/Notebooks/Circuits/DP_generalizedSSN_RLC.ipynb validates the DP-SSN models against the classical dynamic-phasor stamping and the EMT and EMT-SSN waveforms. examples/Notebooks/Circuits/DP_SSN_RLC_accuracy.ipynb studies the time-step and frequency-dependent accuracy against a small-step EMT reference.

The three-phase analogues, examples/Notebooks/Circuits/DP_Ph3_generalizedSSN_RLC.ipynb and examples/Notebooks/Circuits/DP_Ph3_SSN_RLC_accuracy.ipynb, repeat both studies on DP::Ph3 circuits, including a current-driven network with generic V-type and I-type components and a three-phase fault transient.