Power Electronics

DPsim provides several averaged power-electronic inverter models for simulation using EMT, DP, and SP network modeling domains.

Three-Phase Averaged Voltage Source Inverter with State-Space Nodal Interface

The EMT::Ph3::AvVoltSourceInverterStateSpace model represents a grid-following averaged voltage source inverter in the EMT domain. It is implemented as a variable state-space nodal component and can therefore be directly stamped into the MNA system. 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.

Source code and examples

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

The DP::Ph1::AvVoltSourceInverterStateSpace 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 DP MNA system.

Source code and examples

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

The DP::Ph3::AvVoltSourceInverterStateSpace model extends the grid-following averaged inverter of DP::Ph1 to the three-phase dynamic-phasor (DP) domain. Each phase of the LC filter is represented by an independent complex envelope, $V_{c,a/b/c}$ and $I_{f,a/b/c}$, in contrast to the single positive-sequence envelope employed by DP::Ph1, whereas the controller retains a single positive-sequence $dq$ frame shared by the PLL, the power filter, and the outer and inner control loops. As in DP::Ph1, the control states are baseband quantities and remain real-valued; only the six per-phase filter envelopes are carrier-band quantities, and each carries the $-j\omega_n$ frequency shift introduced in State-Space Nodal.

The terminal input is the PCC voltage envelope of the three phases,

$$u = \begin{bmatrix} U_a & U_b & U_c \end{bmatrix}^\top ,$$

and the state vector concatenates the 8 real control states ahead of the 6 complex per-phase envelopes,

$$\mathbf{x} = \big[\, \psi \;\; \phi_{\mathrm{PLL}} \;\; P \;\; Q \;\; \phi_d \;\; \phi_q \;\; \gamma_d \;\; \gamma_q \;\; V_{c,a} \;\; V_{c,b} \;\; V_{c,c} \;\; I_{f,a} \;\; I_{f,b} \;\; I_{f,c} \,\big]^\top ,$$

where $\psi := \theta_{\mathrm{PLL}} - \omega_n t$ again denotes the deviation of the PLL angle from the nominal carrier phase, retained as a state to preserve relinearization accuracy. Each per-phase envelope contributes its real and imaginary parts to the packed real vector, yielding 20 real states in total.

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

$$y_p = \frac{U_p - V_{c,p}}{R_c}, \qquad p \in \{a, b, c\}.$$

Model equations

The main extension relative to DP::Ph1 is the per-phase projection onto, and redistribution from, the single positive-sequence $dq$ control frame. The three capacitor-voltage envelopes are projected onto a single positive-sequence phasor,

$$\underline{V}_c = V_{c,a} + a\, V_{c,b} + a^2 V_{c,c}, \qquad a = e^{\,j 2\pi/3},$$

and the PCC input $\underline{U}$ is projected identically, so that the coupling-current envelope seen by the controller is $\underline{I}_{rc} = (\underline{V}_c - \underline{U})/R_c$, with positive current again denoting injection from the inverter into the grid. The $dq$ quantities are obtained by rotating the projected envelopes by $\psi$,

$$V_{c,dq} = \tfrac{1}{2}\sqrt{\tfrac{2}{3}}\, e^{-j\psi}\, \underline{V}_c, \qquad I_{rc,dq} = \tfrac{1}{2}\sqrt{\tfrac{2}{3}}\, e^{-j\psi}\, \underline{I}_{rc},$$

with $v_{c,d} = \operatorname{Re}{V_{c,dq}}$, $v_{c,q} = \operatorname{Im}{V_{c,dq}}$, and analogously for $i_{rc,d}$ and $i_{rc,q}$. Taken together, the $1\times 3$ projection, the scalar $dq$ rotation, and the $3\times 1$ redistribution defined below constitute a rank-one $3\times 3$ Park mapping on the envelope triple, which reduces to the single-envelope relation of DP::Ph1 under balanced operation.

The positive-sequence active and reactive power measurements used by the controller are

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

with the projection scaling chosen so that $p$ and $q$ match the total three-phase active and reactive powers under balanced operation; under unbalanced operation they are the positive-sequence components seen by the single-frame controller.

The control chain from the PLL through the inner current loop is identical in form to that of DP::Ph1 and operates on the single positive-sequence $dq$ pair. The PLL and power-filter dynamics read

$$\dot{\psi} = K_{p,\mathrm{PLL}} v_{c,q} + K_{i,\mathrm{PLL}} \phi_{\mathrm{PLL}}, \qquad \dot{\phi}_{\mathrm{PLL}} = v_{c,q},$$
$$\dot{P} = \omega_c(p - P), \qquad \dot{Q} = \omega_c(q - Q).$$

The outer power-control integrators and current references are

$$\dot{\phi}_d = P_{\mathrm{ref}} - P, \qquad \dot{\phi}_q = Q - Q_{\mathrm{ref}},$$
$$i_{d,\mathrm{ref}} = K_{p,P}(P_{\mathrm{ref}} - P) + K_{i,P}\phi_d, \qquad i_{q,\mathrm{ref}} = K_{p,P}(Q - Q_{\mathrm{ref}}) + K_{i,P}\phi_q,$$

and the inner current-control integrators and voltage references are

$$\dot{\gamma}_d = i_{d,\mathrm{ref}} - i_{rc,d}, \qquad \dot{\gamma}_q = i_{q,\mathrm{ref}} - i_{rc,q},$$
$$v_{d,\mathrm{ref}} = K_{p,I}(i_{d,\mathrm{ref}} - i_{rc,d}) + K_{i,I}\gamma_d, \qquad v_{q,\mathrm{ref}} = K_{p,I}(i_{q,\mathrm{ref}} - i_{rc,q}) + K_{i,I}\gamma_q.$$

The single $dq$ voltage reference $V_{\mathrm{ref},dq} = v_{d,\mathrm{ref}} + j v_{q,\mathrm{ref}}$ is redistributed to the per-phase bridge-voltage envelopes through the inverse projection,

$$V_{\mathrm{ref},p} = \bar{a}_p \sqrt{\tfrac{2}{3}}\, V_{\mathrm{ref},dq}\, e^{j\psi}, \qquad \bar{a}_{a/b/c} = \{1,\; a^2,\; a\},$$

so that all three phases are driven by the same positive-sequence command.

The LC-filter dynamics are decoupled per phase within the plant and carry the carrier shift of the envelope explicitly,

$$\dot{V}_{c,p} = \frac{1}{C_f} I_{f,p} + \frac{1}{C_f R_c}(U_p - V_{c,p}) - j\omega_n V_{c,p},$$
$$\dot{I}_{f,p} = \frac{1}{L_f} \left( V_{\mathrm{ref},p} - V_{c,p} - R_f I_{f,p} \right) - j\omega_n I_{f,p},$$

the phases being coupled only through the shared control chain, that is, through $V_{\mathrm{ref},p}$.

At each simulation step the nonlinear model is linearized about the current operating point into the affine state-space form, with the 8 real control states and the real and imaginary parts of the 6 complex per-phase envelopes packed into a single real 20-vector,

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

which is subsequently discretized and stamped into the DP MNA system.

Because the controller operates in a single positive-sequence $dq$ frame, only the positive-sequence component of an unbalanced terminal is regulated. The negative-sequence response is present in the per-phase filter envelopes but is not itself a control state, and the $2\omega_n$ ripple it would otherwise induce in the $dq$ frame is therefore not represented. A dual-sequence controller with a dedicated negative-sequence frame remains the subject of future work.

References

  • M. Mirz, S. Vogel, G. Reinke, and A. Monti, “DPsim: A dynamic phasor real-time simulator for power systems,” SoftwareX, vol. 10, art. 100253, 2019. https://doi.org/10.1016/j.softx.2019.100253
  • A. Yazdani and R. Iravani, Voltage-Sourced Converters in Power Systems: Modeling, Control, and Applications. Hoboken, NJ: Wiley-IEEE Press, 2010. https://ieeexplore.ieee.org/book/5237659
  • 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 Proc. 2022 International Power Electronics Conference (IPEC-Himeji 2022, ECCE Asia), pp. 422–428. https://ieeexplore.ieee.org/document/9806927

Source code and examples

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

The EMT::Ph3::SSN_GFM 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 $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ automatically, with no separate matrix code. 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

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

which is discretized and stamped into the EMT MNA system. The dq/abc transformations and the nonlinear controls make the local model time varying, so the SSN equivalent is recomputed every simulation step.

Source code and examples

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 31.07.2026: chore: strip saved outputs and backfill cell ids across all notebooks (#605) (9e864da)