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 $-\psi$,

$$I_{rc,dq}^- = \tfrac{1}{2}\sqrt{\tfrac{2}{3}}\, e^{\,j\psi}\, \overline{\underline{I}_{rc}^-} .$$

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.

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