DP Ph1 Averaged Voltage Source Inverter

Single-Phase Averaged Voltage Source Inverter with State-Space Nodal Interface (Dynamic Phasor)

This model ports the same grid-following averaged inverter into the dynamic-phasor (DP) domain, as a single positive-sequence complex envelope rather than three abc waveforms. The PLL, power filter, outer power control, and inner current control are baseband and stay real; only the LC filter’s two states are genuine carrier-band envelopes and carry the $-j\omega_n$ shift described in State-Space Nodal.

The terminal input is the PCC voltage envelope

$$u = U ,$$

and the state vector is the mixed real/complex-envelope form

$$\mathbf{x} = \begin{bmatrix} \psi & \phi_{\mathrm{PLL}} & P & Q & \phi_d & \phi_q & \gamma_d & \gamma_q & \operatorname{Re}\{V_c\} & \operatorname{Im}\{V_c\} & \operatorname{Re}\{I_f\} & \operatorname{Im}\{I_f\} \end{bmatrix}^\top ,$$

where $\psi := \theta_{\mathrm{PLL}} - \omega_n t$ is the PLL angle’s deviation from the nominal carrier phase, tracked instead of the raw, unboundedly growing $\theta_{\mathrm{PLL}}$ for relinearization accuracy, and $V_c$, $I_f$ are complex envelopes replacing EMT’s six abc filter states.

The model output is the interface current injected into the MNA system,

$$y = \frac{U - V_c}{R_c}.$$

Model equations

The controller uses the opposite current direction, i.e. positive current denotes inverter injection into the grid,

$$I_{rc} = \frac{V_c - U}{R_c}.$$

Because the DP envelope already demodulates the carrier, the dq quantities are obtained by rotating the envelope by $\psi$ alone, not by the full absolute angle $\theta_{\mathrm{PLL}}$,

$$V_{c,dq} = V_c\, e^{-j\psi}, \qquad I_{rc,dq} = I_{rc}\, e^{-j\psi},$$

with $v_{c,d} = \operatorname{Re}{V_{c,dq}}$, $v_{c,q} = \operatorname{Im}{V_{c,dq}}$, and likewise for $i_{rc,d}$, $i_{rc,q}$.

The instantaneous active and reactive powers are calculated as

$$p = v_{c,d} i_{rc,d} + v_{c,q} i_{rc,q},$$

$$q = -v_{c,d} i_{rc,q} + v_{c,q} i_{rc,d},$$

identical in form to EMT’s; DP::Ph1’s own voltage/current scale already represents total power directly, with no three-phase multiplier.

The PLL and power-filter dynamics are

$$\dot{\psi} = K_{p,\mathrm{PLL}} v_{c,q} + K_{i,\mathrm{PLL}} \phi_{\mathrm{PLL}},$$

$$\dot{\phi}_{\mathrm{PLL}} = v_{c,q},$$

$$\dot{P} = \omega_c(p - P), \qquad \dot{Q} = \omega_c(q - Q).$$

The outer power-control integrators and current references are

$$\dot{\phi}_d = P_{\mathrm{ref}} - P, \qquad \dot{\phi}_q = Q - Q_{\mathrm{ref}},$$

$$i_{d,\mathrm{ref}} = K_{p,P}(P_{\mathrm{ref}} - P) + K_{i,P}\phi_d,$$

$$i_{q,\mathrm{ref}} = K_{p,P}(Q - Q_{\mathrm{ref}}) + K_{i,P}\phi_q.$$

The inner current-control integrators and voltage references are

$$\dot{\gamma}_d = i_{d,\mathrm{ref}} - i_{rc,d}, \qquad \dot{\gamma}_q = i_{q,\mathrm{ref}} - i_{rc,q},$$

$$v_{d,\mathrm{ref}} = K_{p,I}(i_{d,\mathrm{ref}} - i_{rc,d}) + K_{i,I}\gamma_d,$$

$$v_{q,\mathrm{ref}} = K_{p,I}(i_{q,\mathrm{ref}} - i_{rc,q}) + K_{i,I}\gamma_q.$$

The reference voltage is transformed back to a complex envelope, rotating by $\psi$,

$$V_{\mathrm{ref}} = (v_{d,\mathrm{ref}} + j v_{q,\mathrm{ref}})\, e^{j\psi}.$$

The LC filter dynamics carry the envelope’s carrier shift explicitly,

$$\dot{V}_c = \frac{1}{C_f} I_f + \frac{1}{C_f R_c}(U - V_c) - j\omega_n V_c,$$

$$\dot{I}_f = \frac{1}{L_f} \left( V_{\mathrm{ref}} - V_c - R_f I_f \right) - j\omega_n I_f.$$

At each simulation step, the nonlinear model is locally linearized into the affine state-space form, packing the 8 real states and the real/imaginary parts of the 2 complex states into one real 12-vector,

$$\dot{\mathbf{x}} \approx \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{E},$$

$$\mathbf{y} \approx \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u} + \mathbf{F},$$

which is then discretized and stamped into the network equations.

How this is arranged in code, together with the source and the runnable examples, is covered under DP Ph1 averaged VSI implementation.

Last modified 03.08.2026: pybind: restore SystemTopology.add as a deprecated alias (#622) (a0db078)