EMT Ph3 Averaged Voltage Source Inverter
Three-Phase Averaged Voltage Source Inverter with State-Space Nodal Interface
This model represents a grid-following averaged voltage source inverter in the EMT domain.
Because its state-space form is recomputed as the operating point moves, it is solved simultaneously with the network rather than through a delayed injection.
The model includes a PLL, filtered active/reactive power measurement, outer power control, inner current control, and an LC filter with coupling resistance to the grid node.
The terminal input is the PCC voltage vector
$$\mathbf{u} =
\begin{bmatrix}
u_a & u_b & u_c
\end{bmatrix}^\top ,$$
and the state vector is
$$\mathbf{x} =
\begin{bmatrix}
\theta_{\mathrm{PLL}} &
\phi_{\mathrm{PLL}} &
P &
Q &
\phi_d &
\phi_q &
\gamma_d &
\gamma_q &
v_{c,a} &
v_{c,b} &
v_{c,c} &
i_{f,a} &
i_{f,b} &
i_{f,c}
\end{bmatrix}^\top .$$
The model output is the interface current injected into the MNA system,
$$\mathbf{y} =
\frac{\mathbf{u} - \mathbf{v}_c}{R_c}.$$
Model equations
The controller uses the opposite current direction, i.e. positive current denotes inverter injection into the grid,
$$\mathbf{i}_{rc} =
\frac{\mathbf{v}_c - \mathbf{u}}{R_c}.$$
The Park transformation with PLL angle $\theta_{\mathrm{PLL}}$ is used to obtain dq quantities,
$$\begin{bmatrix}
v_{c,d} \\
v_{c,q}
\end{bmatrix}
=
\mathbf{T}(\theta_{\mathrm{PLL}})\mathbf{v}_c,
\qquad
\begin{bmatrix}
i_{rc,d} \\
i_{rc,q}
\end{bmatrix}
=
\mathbf{T}(\theta_{\mathrm{PLL}})\mathbf{i}_{rc}.$$
The instantaneous active and reactive powers are calculated as
$$p = v_{c,d} i_{rc,d} + v_{c,q} i_{rc,q},$$
$$q = -v_{c,d} i_{rc,q} + v_{c,q} i_{rc,d}.$$
The PLL and power-filter dynamics are
$$\dot{\theta}_{\mathrm{PLL}}
=
\omega_n + K_{p,\mathrm{PLL}} v_{c,q} +
K_{i,\mathrm{PLL}} \phi_{\mathrm{PLL}},$$
$$\dot{\phi}_{\mathrm{PLL}} = v_{c,q},$$
$$\dot{P} = \omega_c(p - P),
\qquad
\dot{Q} = \omega_c(q - Q).$$
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 abc coordinates,
$$\mathbf{v}_{\mathrm{ref}}
=
\mathbf{T}^{-1}(\theta_{\mathrm{PLL}})
\begin{bmatrix}
v_{d,\mathrm{ref}} \\
v_{q,\mathrm{ref}}
\end{bmatrix}.$$
The LC filter dynamics are
$$\dot{\mathbf{v}}_c
=
\frac{1}{C_f}\mathbf{i}_f
+
\frac{1}{C_f R_c}(\mathbf{u} - \mathbf{v}_c),$$
$$\dot{\mathbf{i}}_f
=
\frac{1}{L_f}
\left(
\mathbf{v}_{\mathrm{ref}}
-
\mathbf{v}_c
-
R_f \mathbf{i}_f
\right).$$
At each simulation step, the nonlinear model is locally linearized into the affine state-space form
$$\dot{\mathbf{x}}
\approx
\mathbf{A}\mathbf{x}
+
\mathbf{B}\mathbf{u}
+
\mathbf{E},$$
$$\mathbf{y}
\approx
\mathbf{C}\mathbf{x}
+
\mathbf{D}\mathbf{u}
+
\mathbf{F},$$
which is then discretized and stamped into the EMT MNA system.
How this is arranged in code, together with the source and the runnable examples, is covered under
EMT Ph3 averaged VSI implementation.