EMT Ph3 Grid-Forming Inverter

Three-Phase Averaged Grid-Forming Inverter with State-Space Nodal Interface

This model represents a grid-forming averaged voltage source inverter in the EMT domain. The control structure follows the state-space grid-forming converter of Gao2022 (VSG algorithm loop, voltage loop, current loop with active damping), whose grid-following counterpart in the same paper is the basis for the averaged inverter above; the inner voltage/current control and LC filter modeling follow Yazdani2010. Like the grid-following inverter above it is a variable state-space nodal component stamped directly into the MNA system, but instead of a PLL that locks to the grid it carries its own virtual synchronous machine (VSG): the internal angle and voltage magnitude are states driven by active- and reactive-power balance, so the inverter imposes a voltage and can run islanded. The model includes the VSG swing dynamics, a reactive-power/voltage excitation loop, filtered active/reactive power measurement, a cascaded voltage and current controller, a first-order converter/digital-delay approximation, 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 17-element state vector is

$$\mathbf{x} = \begin{bmatrix} P & Q & \omega & \theta & E & \xi_{v,d} & \xi_{v,q} & \xi_{i,d} & \xi_{i,q} & v_{\mathrm{del},d} & v_{\mathrm{del},q} & v_{c,a} & v_{c,b} & v_{c,c} & i_{f,a} & i_{f,b} & i_{f,c} \end{bmatrix}^\top ,$$

where $\theta$ is the VSG angle (there is no PLL), $E$ is the excitation-controlled voltage magnitude, $\xi_{v}$, $\xi_{i}$ are the voltage- and current-loop integrators, and $v_{\mathrm{del}}$ are the two delay states.

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

$$\mathbf{y} = \frac{\mathbf{u} - \mathbf{v}_c}{R_c}.$$

Control structure

graph LR U["PCC voltage u"] --> FILT["LC filter
vc, if"] FILT --> MEAS["Power measurement
p, q"] MEAS --> PF["Measurement filters
P, Q"] PF -->|P| SWING["VSG swing
omega, theta"] PF -->|Q| EXC["Excitation /
Q-V droop -> E"] EXC --> VZ["Virtual impedance
E - Zv*if"] VZ --> VCTRL["Voltage controller
-> i_ref"] VCTRL --> ICTRL["Current controller
-> v_conv"] ICTRL --> DELAY["Converter delay"] DELAY --> FILT SWING -->|theta| VCTRL FILT --> Y["Interface current y"]

The virtual synchronous machine sets the internal angle from the active-power balance and the internal magnitude from the reactive-power/voltage loop; the cascaded voltage and current controllers then track that internal reference through the LC filter. The dashed grid-connected extensions (virtual impedance, feed-forward scaling, Q-V droop) are described below.

Model equations

The physical grid current, positive for injection into the grid, is

$$\mathbf{i}_g = \frac{\mathbf{v}_c - \mathbf{u}}{R_c}.$$

All dq quantities use the VSG angle $\theta$ (amplitude-invariant Park transform $\mathbf{T}(\theta)$),

$$\mathbf{v}_{c,dq} = \mathbf{T}(\theta)\mathbf{v}_c, \qquad \mathbf{i}_{f,dq} = \mathbf{T}(\theta)\mathbf{i}_f, \qquad \mathbf{i}_{g,dq} = \mathbf{T}(\theta)\mathbf{i}_g,$$

and the capacitor current is $\mathbf{i}{\mathrm{cap},dq} = \mathbf{i}{f,dq} - \mathbf{i}_{g,dq}$. Because the Park transform is amplitude invariant, three-phase instantaneous power carries the factor $3/2$,

$$p = \tfrac{3}{2}\,(v_{c,d} i_{g,d} + v_{c,q} i_{g,q}), \qquad q = \tfrac{3}{2}\,(v_{c,q} i_{g,d} - v_{c,d} i_{g,q}),$$

and the PCC voltage magnitude is $U_{\mathrm{pcc}} = \sqrt{v_{c,d}^2 + v_{c,q}^2}$.

The measurement filters are first-order lags,

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

The VSG swing equation sets the angle from the active-power balance,

$$J\dot{\omega} = \frac{P_{\mathrm{ref}} - P}{\omega} - D(\omega - \omega_n), \qquad \dot{\theta} = \omega,$$

with virtual inertia $J$ and damping $D$. The reactive-power/voltage excitation controller sets the internal magnitude,

$$\dot{E} = K_q(Q_{\mathrm{ref}} - Q) + K_u(U_n - U_{\mathrm{pcc}}),$$

an integral law on the reactive error with a voltage-droop term. The excitation defines the dq voltage reference; in the islanded model it is aligned with the d-axis,

$$v_{d,\mathrm{ref}} = E, \qquad v_{q,\mathrm{ref}} = 0 .$$

The voltage controller integrates the voltage error and forms the current reference with the capacitor-current feed-forward and dq decoupling,

$$\dot{\xi}_{v,d} = v_{d,\mathrm{ref}} - v_{c,d}, \qquad \dot{\xi}_{v,q} = v_{q,\mathrm{ref}} - v_{c,q},$$

$$i_{d,\mathrm{ref}} = i_{g,d} - \omega C_f v_{c,q} + K_{p,V}(v_{d,\mathrm{ref}} - v_{c,d}) + K_{i,V}\xi_{v,d},$$

$$i_{q,\mathrm{ref}} = i_{g,q} + \omega C_f v_{c,d} + K_{p,V}(v_{q,\mathrm{ref}} - v_{c,q}) + K_{i,V}\xi_{v,q}.$$

The current controller integrates the current error and forms the converter voltage reference, with inductor decoupling and optional active damping on the capacitor current,

$$\dot{\xi}_{i,d} = i_{d,\mathrm{ref}} - i_{f,d}, \qquad \dot{\xi}_{i,q} = i_{q,\mathrm{ref}} - i_{f,q},$$

$$v_{d,\mathrm{conv}} = v_{c,d} - \omega L_f i_{f,q} + K_{p,I}(i_{d,\mathrm{ref}} - i_{f,d}) + K_{i,I}\xi_{i,d} - K_{ad} i_{\mathrm{cap},d},$$

$$v_{q,\mathrm{conv}} = v_{c,q} + \omega L_f i_{f,d} + K_{p,I}(i_{q,\mathrm{ref}} - i_{f,q}) + K_{i,I}\xi_{i,q} - K_{ad} i_{\mathrm{cap},q}.$$

A first-order lag approximates the converter/digital delay,

$$\dot{v}_{\mathrm{del},d} = \omega_d(v_{d,\mathrm{conv}} - v_{\mathrm{del},d}), \qquad \dot{v}_{\mathrm{del},q} = \omega_d(v_{q,\mathrm{conv}} - v_{\mathrm{del},q}),$$

and its output, transformed back to abc as $\mathbf{v}{\mathrm{inv}} = \mathbf{T}^{-1}(\theta),[v{\mathrm{del},d}\ v_{\mathrm{del},q}]^\top$, drives the LC filter,

$$\dot{\mathbf{v}}_c = \frac{1}{C_f}\left(\mathbf{i}_f + \frac{\mathbf{u} - \mathbf{v}_c}{R_c}\right), \qquad \dot{\mathbf{i}}_f = \frac{1}{L_f}\left(\mathbf{v}_{\mathrm{inv}} - \mathbf{v}_c - R_f\mathbf{i}_f\right).$$

Grid-connected control extensions

The equations above describe the islanded inverter. Three opt-in extensions adapt it to a stiff grid; each defaults to the value that recovers the islanded model exactly, so the eigenstructure is unchanged unless a setter is called.

Virtual output impedance. A virtual impedance $Z_v = R_v + jX_v$ is subtracted from the excitation to form the voltage reference, using the filter current $\mathbf{i}_{f,dq}$,

$$v_{d,\mathrm{ref}} + j v_{q,\mathrm{ref}} = E - Z_v\,(i_{f,d} + j i_{f,q}),$$

i.e.

$$v_{d,\mathrm{ref}} = E - (R_v i_{f,d} - X_v i_{f,q}), \qquad v_{q,\mathrm{ref}} = -(R_v i_{f,q} + X_v i_{f,d}).$$

$Z_v = 0$ recovers $v_{d,\mathrm{ref}} = E,\ v_{q,\mathrm{ref}} = 0$. A finite $R_v$ adds a current-proportional term opposing motion, damping the power-synchronization loop on a stiff grid at the electrical timescale, an alternative to raising the mechanical damping $D$. The drop is taken off the filter-current state $\mathbf{i}_f$ rather than the algebraically reconstructed grid current $\mathbf{i}_g = (\mathbf{v}_c-\mathbf{u})/R_c$; the latter would multiply the reference by a factor $\propto 1/R_c$, amplifying state and linearization error.

Grid-current feed-forward scale. A scalar $\kappa$ scales the grid-current feed-forward in the current reference,

$$i_{d,\mathrm{ref}} = \kappa\, i_{g,d} - \omega C_f v_{c,q} + \dots, \qquad i_{q,\mathrm{ref}} = \kappa\, i_{g,q} + \omega C_f v_{c,d} + \dots,$$

with $\kappa = 1$ the default full feed-forward.

Proportional reactive-power droop. When a cutoff $\omega_q > 0$ is set, the integral excitation is replaced by a proportional Q-V droop,

$$\dot{E} = \omega_q\big(E_{\mathrm{set}} + D_q(Q_{\mathrm{ref}} - Q) - E\big),$$

a first-order lag with a stable fixed point $E^* = E_{\mathrm{set}} + D_q(Q_{\mathrm{ref}} - Q)$ and pole at $-\omega_q$. On a stiff grid the network fixes $U_{\mathrm{pcc}}$, so the reactive error $Q_{\mathrm{ref}} - Q$ cannot be driven to zero and the integral law $\dot E = K_q(Q_{\mathrm{ref}} - Q) + K_u(U_n - U_{\mathrm{pcc}})$ has no reachable equilibrium (reactive windup); the proportional droop always has one. The setpoint $E_{\mathrm{set}}$ is captured at initialization as the operating magnitude, so $\dot{E} = 0$ when $Q = Q_{\mathrm{ref}}$ at $t = 0$.

Linearization and stamping

The model is nonlinear (Park transforms with the moving angle $\theta$, the $1/\omega$ swing term, the power products). It is not linearized by hand; at each simulation step the state and output Jacobians are computed by central finite differences of the nonlinear functions $\mathbf{f}(\mathbf{x},\mathbf{u}) = \dot{\mathbf{x}}$ and $\mathbf{g}(\mathbf{x},\mathbf{u}) = \mathbf{y}$,

$$\mathbf{A} = \frac{\partial \mathbf{f}}{\partial \mathbf{x}},\quad \mathbf{B} = \frac{\partial \mathbf{f}}{\partial \mathbf{u}},\quad \mathbf{C} = \frac{\partial \mathbf{g}}{\partial \mathbf{x}},\quad \mathbf{D} = \frac{\partial \mathbf{g}}{\partial \mathbf{u}},$$

each column $j$ evaluated as $[\mathbf{f}(\mathbf{x}+\delta_j\mathbf{e}_j,\mathbf{u}) - \mathbf{f}(\mathbf{x}-\delta_j\mathbf{e}_j,\mathbf{u})]/(2\delta_j)$ with a mixed relative/absolute step $\delta_j$. Because the grid-connected extensions above all enter through $\mathbf{f}$, they are captured in $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$ automatically. The affine offsets fix the model to the current operating point,

$$\mathbf{E} = \mathbf{f}(\mathbf{x}_0,\mathbf{u}_0) - \mathbf{A}\mathbf{x}_0 - \mathbf{B}\mathbf{u}_0, \qquad \mathbf{F} = \mathbf{g}(\mathbf{x}_0,\mathbf{u}_0) - \mathbf{C}\mathbf{x}_0 - \mathbf{D}\mathbf{u}_0,$$

giving the affine state-space form

$$\dot{\mathbf{x}} \approx \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{E}, \qquad \mathbf{y} \approx \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u} + \mathbf{F},$$

The dq/abc transformations and the nonlinear controls make this local model time varying, so it holds only in a neighbourhood of the operating point it was formed at.

How the linearization is carried out and stamped, together with the source and the runnable examples, is covered under EMT Ph3 grid-forming VSI implementation.

References

[Gao2022] X. Gao, D. Zhou, A. Anvari-Moghaddam, and F. Blaabjerg, “Stability Analysis of Grid-Following and Grid-Forming Converters Based on State-Space Model,” in 2022 International Power Electronics Conference (IPEC-Himeji 2022 - ECCE Asia), 2022, pp. 422-428. Source of both the grid-following and grid-forming state-space control structures. Its eigenvalue analysis finds grid-following control better suited to a stiff grid and grid-forming control to a weak grid; the grid-connected extensions above (virtual impedance, Q-V droop) are what let the grid-forming model stay stable when connected to a stiff grid.
[Yazdani2010] A. Yazdani and R. Iravani, Voltage-Sourced Converters in Power Systems: Modeling, Control, and Applications. Hoboken, NJ: Wiley-IEEE Press, 2010. Basis for the inner voltage/current control and LC-filter modeling of both inverters.

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