Power Electronics
Averaged voltage source inverter models and their control.
Every inverter model here is averaged: the switching is not represented, and the converter is
treated as a controllable voltage behind its filter. Averaging removes the switching frequency from
the problem, which is what allows a step size set by the control bandwidth rather than by the
carrier. It also means these models say nothing about switching losses, harmonic injection or any
behaviour that depends on the modulation itself.
The control that sits on top of each is derived separately under
converter control, because the same cascade appears in more
than one of these models.
Choosing among them
The models differ along two axes: which domain they are written in, and whether the converter
follows the grid or forms it.
EMT Ph3 averaged VSI is the reference formulation. All
fourteen states are real, the three filter phases are represented individually, and there is no
carrier, so nothing is assumed about the bandwidth of what it carries.
DP Ph1 averaged VSI is the same converter as a single
positive-sequence envelope. Its six real filter states become two complex envelopes, which is the
saving the envelope description buys, at the cost of being unable to represent an unbalance.
DP Ph3 averaged VSI restores per-phase representation in the
envelope domain, with one complex envelope per phase and a controller that keeps a single
positive-sequence frame. Because three independent phase envelopes admit a negative-sequence
component, it carries negative-sequence current control that the single-phase model has no need for.
EMT Ph3 grid-forming VSI is the one that differs in
kind rather than in representation. It carries its own frequency and angle as states instead of
tracking the grid’s, so it can energise a network with no other source. Its control is nonlinear
enough that the model is linearized numerically at each operating point rather than written in
closed form.
What they share
All four are solved simultaneously with the network rather than through a delayed injection, using
the state-space nodal method described under
SSN components. All four are therefore re-formed as the
operating point moves, and all four make the system matrix change at every step, which is the cost
of the approach.
1 - 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.
2 - 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.
3 - DP Ph3 Averaged Voltage Source Inverter
Three-Phase Averaged Voltage Source Inverter with State-Space Nodal Interface (Dynamic Phasor)
This model extends the single-phase grid-following averaged inverter 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 of the single-phase model, 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 the single-phase case, 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 6 complex per-phase envelopes ahead of the 8 real control states, keeping the carrier-band and baseband blocks separate,
$$\mathbf{x} =
\big[\,
V_{c,a} \;\;
V_{c,b} \;\;
V_{c,c} \;\;
I_{f,a} \;\;
I_{f,b} \;\;
I_{f,c} \;\;
\psi \;\;
\phi_{\mathrm{PLL}} \;\;
P \;\;
Q \;\;
\phi_d \;\;
\phi_q \;\;
\gamma_d \;\;
\gamma_q
\,\big]^\top ,$$
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, or 22 with the optional negative-sequence loop described below.
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 real and imaginary parts of the 6 complex per-phase envelopes and the 8 real control states 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.
In this default configuration the controller operates in a single positive-sequence $dq$ frame, so 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.
Optional negative-sequence current control
A second, negative-sequence current-control loop can be added alongside the positive-sequence one, giving the dual-sequence structure of Yazdani and Iravani, chapter 8. The two configurations answer different questions: without the loop the model has the same 20 states and the same eigenvalue count as its EMT::Ph3 counterpart, which is what a cross-domain comparison requires, while with it the model gains 2 states and can regulate an unbalanced terminal.
The negative-sequence quantities are obtained by projecting the same three envelopes onto the conjugate sequence set,
$$\underline{V}_c^- = V_{c,a} + a^2 V_{c,b} + a\, V_{c,c},
\qquad
\underline{I}_{rc}^- = \frac{\underline{V}_c^- - \underline{U}^-}{R_c}.$$
A negative-sequence component rotates backwards relative to the PLL frame, so in envelope terms its $dq$ image follows from conjugating the projected phasor and rotating by $+\psi$ rather than $-\psi$,
$$I_{rc,dq}^- = \tfrac{1}{2}\sqrt{\tfrac{2}{3}}\, e^{\,j\psi}\, \overline{\underline{I}_{rc}^-} .$$
Both sequence images are baseband
This is what keeps the extension cheap. The negative-sequence loop costs only the two real integrator states $\gamma_{nd}$ and $\gamma_{nq}$, with no second carrier and no $2\omega_n$ term anywhere in the model.The loop itself is the same PI structure as the positive-sequence inner loop,
$$\dot{\gamma}_{nd} = i_{nd,\mathrm{ref}} - i_{rc,nd},
\qquad
\dot{\gamma}_{nq} = i_{nq,\mathrm{ref}} - i_{rc,nq},$$
$$v_{nd,\mathrm{ref}}
=
K_{p,I}(i_{nd,\mathrm{ref}} - i_{rc,nd}) +
K_{i,I}\gamma_{nd},
\qquad
v_{nq,\mathrm{ref}}
=
K_{p,I}(i_{nq,\mathrm{ref}} - i_{rc,nq}) +
K_{i,I}\gamma_{nq},$$
reusing the inner-loop gains $K_{p,I}$ and $K_{i,I}$. Its output is redistributed to the per-phase bridge voltages through the sequence-orthogonal set, and adds to the positive-sequence command of the previous section,
$$V_{\mathrm{ref},p}
=
\bar{a}_p \sqrt{\tfrac{2}{3}}\, V_{\mathrm{ref},dq}\, e^{j\psi}
+
a_p \sqrt{\tfrac{2}{3}}\, \overline{V_{\mathrm{ref},dq}^-}\, e^{j\psi},
\qquad
a_{a/b/c} = \{1,\; a,\; a^2\}.$$
The two references $i_{nd,\mathrm{ref}}$ and $i_{nq,\mathrm{ref}}$ default to zero, which makes the loop a negative-sequence suppressor. A non-zero pair commands a deliberate negative-sequence injection instead, as required by some unbalanced fault ride-through grid codes.
The state vector grows to 22 by appending the two integrators after the control block, so that the envelope and positive-sequence control indices are unaffected. Under a single-line-to-ground fault, enabling the loop suppresses the negative-sequence component of the injected current by about 40 percent while moving the positive-sequence component by less than 0.1 percent.
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
How this is arranged in code, together with the source and the runnable examples, is covered under
DP Ph3 averaged VSI implementation.
4 - 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.