PMSM Current Controller

Discrete-time permanent magnet synchronous machine current controller

Libraries:
Simscape / Electrical / Control / PMSM Control

Description

The PMSM Current Controller block implements a discrete-time PI-based permanent magnet synchronous machine (PMSM) current controller in the rotor d-q reference frame.

You typically use this block in a series of blocks making up a control structure.

You can generate a current reference in the d-q frame to be used as an input to this block with a PMSM Current Reference Generator.
You can obtain a voltage reference in the abc domain by converting the output of this block using an Inverse Park Transform block.

You can see an example of a full control structure, from machine measurements to machine inputs, in the PMSM Field-Oriented Control block.

Equations

The block is discretized using the backward Euler method due to its first-order simplicity and its stability.

Two PI current controllers implemented in the rotor reference frame produce the reference voltage vector:

$v_{d}^{r e f} = (K_{p_i d} + K_{i_i d} \frac{T_{s} z}{z - 1}) (i_{d}^{r e f} - i_{d}) + v_{d_F F},$

and

$v_{q}^{r e f} = (K_{p_i q} + K_{i_i q} \frac{T_{s} z}{z - 1}) (i_{q}^{r e f} - i_{q}) + v_{q_F F},$

where:

$v_{d}^{r e f}$ and $v_{q}^{r e f}$ are the d-axis and q-axis reference voltages, respectively.
$i_{d}^{r e f}$ and $i_{q}^{r e f}$ are the d-axis and q-axis reference currents, respectively.
$i_{d}$ and $i_{q}$ are the d-axis and q-axis currents, respectively.
K_{p_id} and K_{p_iq} are the proportional gains for the d-axis and q-axis controllers, respectively.
K_{i_id} and K_{i_iq} are the integral gains for the d-axis and q-axis controllers, respectively.
v_{d_FF} and v_{q_FF} are the feedforward voltages for the d-axis and q-axis, respectively, obtained from the machine mathematical equations and provided as inputs.
T_s is the sample time of the discrete controller.

Zero Cancellation

Using PI control results in a zero in the closed-loop transfer function, which can result in undesired overshoot in the closed-loop response. This zero can be canceled by introducing a zero-cancelation block in the feedforward path. The zero cancellation transfer functions in discrete time are:

$G_{Z C_i d} (z) = \frac{\frac{T_{s} K_{i_i d}}{K_{p_i d}}}{z + (\frac{T_{s} - \frac{K_{p_i d}}{K_{i_i d}}}{\frac{K_{p_i d}}{K_{i_i d}}})},$

and

$G_{Z C_i q} (z) = \frac{\frac{T_{s} K_{i_i q}}{K_{p_i q}}}{z + (\frac{T_{s} - \frac{K_{p_i q}}{K_{i_i q}}}{\frac{K_{p_i q}}{K_{i_i q}}})} .$

Voltage Saturation

Saturation must be imposed when the stator voltage vector exceeds the voltage phase limit V_{ph_max}:

$\sqrt{v_{d}^{2} + v_{q}^{2}} \leq V_{p h_m a x},$

where v_d and v_q are the d-axis and q-axis voltages, respectively.

In the case of axis prioritization, the voltages v₁ and v₂ are introduced, where:

v₁ = v_d and v₂ = v_q for d-axis prioritization.
v₁ = v_q and v₂ = v_d for q-axis prioritization.

The constrained (saturated) voltages $v_{1}^{s a t}$ and $v_{2}^{s a t}$ are obtained as follows:

$v_{1}^{s a t} = min (\max (v_{1}^{u n s a t}, - V_{p h_m a x}), V_{p h_m a x})$

and

$v_{2}^{s a t} = min (\max (v_{2}^{u n s a t}, - V_{2_m a x}), V_{2_m a x}),$

where:

$v_{1}^{u n s a t}$ and $v_{2}^{u n s a t}$ are the unconstrained (unsaturated) voltages.
v_{2_max} is the maximum value of v₂ that does not exceed the voltage phase limit, given by $v_{2_m a x} = \sqrt{{(V_{p h_m a x})}^{2} - {(v_{1}^{s a t})}^{2}} .$

In the case that the direct and quadrature axes have the same priority (d-q equivalence) the constrained voltages are obtained as follows:

$v_{d}^{s a t} = min (\max (v_{d}^{u n s a t}, - V_{d_m a x}), V_{d_m a x})$

and

$v_{q}^{s a t} = min (\max (v_{q}^{u n s a t}, - V_{q_m a x}), V_{q_m a x}),$

where

$V_{d_m a x} = \frac{V_{p h_m a x} | v_{d}^{u n s a t} |}{\sqrt{{(v_{d}^{u n s a t})}^{2} + {(v_{q}^{u n s a t})}^{2}}}$

and

$V_{q_m a x} = \frac{V_{p h_m a x} | v_{q}^{u n s a t} |}{\sqrt{{(v_{d}^{u n s a t})}^{2} + {(v_{q}^{u n s a t})}^{2}}} .$

Integral Anti-Windup

An anti-windup mechanism is employed to avoid saturation of integrator output. In such a situation, the integrator gains become:

$K_{i_i d} + K_{a w_i d} (v_{d}^{s a t} - v_{d}^{u n s a t})$

and

$K_{i_i q} + K_{a w_i q} (v_{q}^{s a t} - v_{q}^{u n s a t}),$

where K_{aw_id} and K_{aw_iq} are the anti-windup gains for the d-axis and q-axis, respectively.

Assumptions

The plant model for direct and quadrature axis can be approximated with a first-order system.
This control solution is used only for permanent magnet synchronous motors with sinusoidal flux distribution and field windings.

Examples

Synchronous Reluctance Machine Velocity Control

Control the rotor angular velocity in a synchronous reluctance machine (SynRM) based electrical drive. A high-voltage battery feeds the SynRM through a controlled three-phase converter. An ideal torque source provides the load. The Control subsystem includes a multi-rate PI-based cascade control structure. The control structure has an outer angular-velocity-control loop and two inner current-control loops. The Visualization subsystem contains scopes that allow you to see the simulation results.

Open Model

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