DPsim – Power Electronics

Docs: EMT Ph3 Averaged Voltage Source Inverter

Mon, 01 Jan 0001 00:00:00 +0000

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.

Docs: DP Ph1 Averaged Voltage Source Inverter

Mon, 01 Jan 0001 00:00:00 +0000

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

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