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

• Source code: header, implementation
• C++ example
• Python notebook