Tech:DeltaSPIM

=Overview=

This page describes the implementation of a single phase induction motor model using dynamic phasors. The model was first presented in [1]. The model is created to represents the impact of residential single phase induction motor, (air-conditioners, compressors, etc.) on power system dynamic performance. Our particular interest is in modeling the impact on dynamic voltage stability and oscillation damping. The model is expected to:


 * Accurately capture the sensitivities of motor real and reactive power requirements as a function of its voltage and frequency
 * Reasonably predict the stalling phenomenon, as well as accurately represent motor current, real and reactive power during the stalled state

=Dynamic Phasor Model=

$$ |V_{s}| = ( r_{ds} + j \frac{\omega_{s}}{\omega_{b}} X_{ds}^{'} ) (I_{ds}^{R} + jI_{ds}^{I}) + j ( \frac{\omega_{s}}{\omega_{b}} ) \frac{X_{m}}{X_{r}} (\psi_{dr}^{R} + j\psi _{dr}^{I}) $$

$$ |V_{s}| = ( r_{qs} + j \frac{\omega_{s}}{\omega_{b}} X_{qs}^{'} + j \frac{\omega_{b}}{\omega_{s}} X_{c} ) (I_{qs}^{R} + jI_{qs}^{I}) + j ( \frac{\omega_{s}}{\omega_{b}} ) \frac{n X_{m}}{X_{r}} (\psi_{qr}^{R} + j\psi _{qr}^{I}) $$

$$ T_{0}^{'} \frac{d}{dt} (\psi _{f}^{R} + j\psi _{f}^{I}) = X_{m} (I _{f}^{R}+ jI _{f}^{I} ) - ( sat(\psi_{f},\psi_{b}) + j (\omega_{s} - \omega_{r}) T_{0}^{'} ) (\psi _{f}^{R} + j\psi _{f}^{I}) $$

$$ (\psi_{b}^{R} + j \psi_{b}^{I}) = \frac{X_{m} (I_{b}^{R} + jI_{b}^{I}) }{(sat(\psi_{f},\psi_{b}) + j (\omega_{s} + \omega_{r}) T_{0}^{'} )}$$

$$ \frac{2H}{\omega_{b}} \frac{d}{dt} \omega_{r} = \frac{X_{m}}{X_{r}} ({I_{f}^{I} \psi_{f}^{R} - I_{f}^{R} \psi_{f}^{I} - I_{b}^{I} \psi_{b}^{R} + I_{b}^{R} \psi_{b}^{I}} ) - T_{mech} $$

$$ I_{S} = ( (I_{ds}^{R} + jI_{ds}^{I}) + (I_{qs}^{R} + jI_{qs}^{I}) ) e^{j \phi } $$

$$ \begin{bmatrix} \psi_{f}^{R} + j \psi_{f}^{I} \\\ psi_{b}^{R} + j \psi_{b}^{I} \end{bmatrix} = \frac{1}{2} \begin{bmatrix}1 & -j \\ 1 & j \end{bmatrix} \begin{bmatrix}\psi_{dr}^{R} + j \psi_{dr}^{I} \\ \psi_{qr}^{R} + j \psi _{qr}^{I} \\\end{bmatrix} $$

$$ \begin{bmatrix} \psi_{dr}^{R} + j \psi_{dr}^{I} \\\ psi_{qr}^{R} + j \psi_{qr}^{I} \end{bmatrix} = \begin{bmatrix}1 & 1 \\ j & -j \end{bmatrix} \begin{bmatrix}\psi_{f}^{R} + j \psi_{f}^{I} \\ \psi_{b}^{R} + j \psi _{b}^{I} \\\end{bmatrix} $$

$$ \begin{bmatrix}I_{f}^{R} + j I_{f}^{I} \\ I_{b}^{R} + j I_{b}^{I} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & -jn \\ 1 & jn \end{bmatrix} \begin{bmatrix} I_{ds}^{R} + j I_{ds}^{I} \\ I_{qs}^{R} + j I_{qs}^{I} \\\end{bmatrix} $$

$$ \begin{bmatrix}I_{ds}^{R} + j I_{ds}^{I} \\ I_{qs}^{R} + j I_{qs}^{I} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ \frac{j}{n} & -\frac{j}{n} \end{bmatrix} \begin{bmatrix} I_{f}^{R} + j I_{f}^{I} \\ I_{f}^{R} + j I_{f}^{I} \\\end{bmatrix} $$

=GridLAB-D Implementation=

motor object
The motor model is divided into two portions, a steady state model and a delta mode model. Both models are obtained from the above mentioned equations. The steady state model is an iterative model solving the dynamic phasor equation at each time step. This model is valid for time steps greater than 0.3 milliseconds. The delta mode model also uses the dynamic phasor equations and is valid for time steps below 0.3 milliseconds. Switching between the model is handled by user specified settings for the speed and voltage of the motor. These properties can be found below. A minimalist motor could be created with object motor { name MotorOne; phases AN; nominal_voltage 240.0; }

which is the same as specifying

object motor { name MotorOne; phases AN; nominal_voltage 240.0; base_power 3500; n 1.22; Rds 0.0365; Rqs 0.0729; Rr 0.0486; Xm 2.28; Xr 2.33; Xc_run -2.779; Xc_start -0.7; Xd_prime 0.1033; Xq_prime 0.1489; A_sat 0.7212; b_sat 5.6; H 0.04; To_prime 0.1212; capacitor_speed 50; trip_time 10; reconnect_time 300; mechanical_torque 1.0448; iteration_count 1000; delta_mode_voltage_trigger 80; delta_mode_rotor_speed_trigger 80; delta_mode_voltage_exit 95; delta_mode_rotor_speed_exit 95; maximum_speed_error 1e-10; motor_status RUNNING; motor_override ON; }

Motor Parameters
As with all powerflow objects, phases and nominal_voltage are inherently part of node.

Motor Model Verification
In order to verify that the motor model works as intended it was subjected to a voltage ramp signal as seen in Fig. 1. It is expected, as the voltage magnitude drops below ~0.6 pu, that the motor will stall, as it does in this simulation. From the data it can also be verified that the object switches into delta mode in order to capture the stall and out again when the motor is running with nominal voltage and rotor speed. After the motor is stalled it trips of at ~38 seconds due to thermal overload. The motor tries to reconnect again at ~58 seconds. The motor remains stalled until ~77 seconds where the voltage has recovered enough for the motor to start. The full simulation can be seen in Fig. 1 and 2.



Motor State of Development
Motor is considered a experimental model.

= References =
 * 1) Bernard Lesieutre, Dmitry Kosterev, John Undrill, "Phasor Modeling Approach for Single Phase A/C Motors," IEEE Power and Energy Society General Meeting, 2008.