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

$$\dot{\psi} = K_{p,\mathrm{PLL}} v_{c,q} + K_{i,\mathrm{PLL}} \phi_{\mathrm{PLL}},$$