# Motor Parameter Estimation from Steady-State Models

Estimation of an induction motor's equivalent circuit parameters from steady-state models and manufacturer performance data is the most common approach for offline power system studies, e.g. dynamic motor starting, transient stability, etc [5].

## Parameter Estimation Problem

The characteristics of an induction motor are normally provided by manufacturers in the form of a standard set of performance parameters, with the following parameters being the most common:

• Nominal voltage, $U_n \,$ (V)
• Nominal frequency, $f \,$ (Hz)
• Rated asynchronous speed, $n_{fl} \,$ (rpm)
• Rated (stator) current, $I_{s,fl} \,$ (A)
• Rated mechanical power, $P_{m,fl} \,$ (kW)
• Rated torque, $T_{n} \,$ (Nm)
• Full load power factor, $\cos{\phi_{fl}} \,$ (pu)
• Full load efficiency, $\eta_{fl} \,$ (pu)
• Breakdown torque, $T_{b} / T_{n} \,$ (normalised)
• Locked rotor torque, $T_{lr} / T_{n} \,$ (normalised)
• Locked rotor current, $I_{lr} / I_{s,fl} \,$ (pu)

We know that a set of equivalent circuit parameters can yield specific torque-speed and current-speed curves. So given a set of performance parameters that contain features on the torque-speed and current-speed curves (e.g. breakdown torque, locked-rotor current, etc), is it possible to determine the corresponding equivalent circuit parameters that yield these features? This is the crux of the parameter estimation problem and can be posed as follows - "How can the motor performance parameters be converted into equivalent circuit parameters?".

While all of the performance parameters in the above set can be used in an estimation procedure, there are actually only six indpendent magnitudes that can be formed from them: $P_{m,fl} \,$, $Q_{fl} \,$, $T_{b} \,$, $T_{lr} \,$, $I_{lr} \,$ and $\eta_{fl} \,$ [1]. These independent magnitudes will thus form the basis of the problem formulation, where the independent magnitudes calculated from the equivalent circuit are matched with the performance parameters supplied by the manufacturer.

The basic double cage model is used to illustrate how these six independent magnitudes can be calculated from the equivalent circuit model. Stator and rotor currents at slip "s" can be readily calculated from the equivalent circuit.

Quantities for per-unit active power $P \,$, reactive power $Q \,$ and power factor $\cos{\phi} \,$ at slip "s" can be calculated as follows:

$S(s) = U_{n} I_{s}(s)^{*} \,$
$P(s) = T(s) (1 - s) \,$
$Q(s) = \Im \{ S(s) \} \,$
$\cos{\phi} (s) = \frac{\Re \{ S(s) \}}{|| S(s) ||} \,$

Nominal speed $n_{s} \,$ and full load slip $s_{f} \,$ is calculated as follows:

$n_{s} = \frac{120f}{p} \,$
$s_{f} = 1 - \frac{n_{fl}}{n_s} \,$

where $p \,$ is the number of motor poles

$f \,$ is the nominal frequency (Hz)
$n_{fl} \,$ is the asynchronous speed at full load (rpm)

Calculating the slip at maximum torque $s_{max} \,$ is found by solving the equation:

$\frac{dT}{ds} = 0 \,$

(Under the condition that the second derivative $\frac{d^{2}}{ds^{2}} \lt 0 \,$)

In the double cage model, the solution to this equation is not trivial and it is more convenient to use an estimate, e.g. based on an interval search between s=0 and s=0.5.

## Problem Formulation Ignoring Core Losses

### Single Cage Model (Ignoring Core Losses)

In the single cage model, the locked rotor torque $T_{lr} \,$ and locked rotor current $I_{lr} \,$ are not used because the single cage model does not have enough degrees of freedom to capture both the starting and breakdown torque characteristics without introducing significant errors [1]. As a result, it is more commonplace to only consider the breakdown torque $T_{b} \,$ in the single cage model and simply ignore the torque and current characteristics at locked rotor. For wound-rotor motors, this yields sufficiently accurate results (i.e. in terms of the resulting torque-speed curve). However, a single-cage model is unable to accurately model the torque-speed characteristics of squirrel cage motors, especially those with deep bars.

Without taking into account core losses, the full load motor efficiency $\eta_{fl} \,$ also cannot be used. Therefore, there are only three independent parameters that can be used in the problem formulation: $P_{m,fl} \,$, $Q_{fl} \,$ and $T_{b} \,$.

These independent parameters can be used to formulate the parameter estimation in terms of a non-linear least squares problem, with a set of non-linear equations of the form $\boldsymbol{F}(\boldsymbol{x}) = \boldsymbol{0} \,$:

$f_{1} (\boldsymbol{x}) = P_{m,fl} - P(s_{f}) = 0 \,$
$f_{2} (\boldsymbol{x}) = \sin{\phi} - Q(s_{f}) = 0 \,$
$f_{3} (\boldsymbol{x}) = T_{b} - T(s_{max}) = 0 \,$

where $\boldsymbol{F} = ( f_1, f_2, f_3 ) \,$ $\boldsymbol{x} = ( R_s, X_s, X_m, R_{r}, X_{r} ) \,$ are the equivalent circuit parameters of the single cage model

### Double Cage Model (Ignoring Core Losses)

In the double cage model, the locked rotor torque $T_{lr} \,$ and locked rotor current $I_{lr} \,$ are included as independent parameters. As in the single cage model, the full load motor efficiency $\eta_{fl} \,$ cannot be used without taking into account core losses. Therefore, there are five independent parameters and the following non-linear least squares problem:

$f_{1} (\boldsymbol{x}) = P_{m,fl} - P(s_{f}) = 0 \,$
$f_{2} (\boldsymbol{x}) = \sin{\phi} - Q(s_{f}) = 0 \,$
$f_{3} (\boldsymbol{x}) = T_{b} - T(s_{max}) = 0 \,$
$f_{4} (\boldsymbol{x}) = T_{lr} - T(s=1) = 0 \,$
$f_{5} (\boldsymbol{x}) = I_{lr} - I(s=1) = 0 \,$

where $\boldsymbol{F} = ( f_1, f_2, f_3, f_4, f_5 ) \,$ $\boldsymbol{x} = ( R_s, X_s, X_m, R_{r1}, X_{r1}, R_{r2}, X_{r2} ) \,$ are the equivalent circuit parameters of the double cage model

## Problem Formulation Considering Core Losses

Without taking into account the core (and mechanical) losses, the motor full load efficiency $\eta_{fl} \,$ cannot be used as an independent parameter in the problem formulation. This is because efficiency is calculated based on the ratio of output mechanical power to input electrical power. If the heat losses through the core and rotor frictional losses are not taken into account, then the equivalent circuit is not suitable to accurately estimate motor efficiency. It follows that attempting to use the motor full load efficiency in the estimation of the equivalent circuit without a core loss component would cause errors in the parameter estimates (e.g. the stator resistance would be overestimated).

When core losses are included in the model, then the motor full load efficiency $\eta_{fl} \,$ can also be used as an independent parameter. The problem formulations are restated below for the single cage and double cage models with core losses taken into account.

### Single Cage Model (With Core Losses)

The non-linear least squares problem for the single cage model with core losses is as follows:

$f_{1} (\boldsymbol{x}) = P_{m,fl} - P(s_{f}) = 0 \,$
$f_{2} (\boldsymbol{x}) = \sin{\phi} - Q(s_{f}) = 0 \,$
$f_{3} (\boldsymbol{x}) = T_{b} - T(s_{max}) = 0 \,$
$f_{4} (\boldsymbol{x}) = \eta{fl} - \eta(s_{f}) = 0 \,$

where $\boldsymbol{F} = ( f_1, f_2, f_3 f_4 ) \,$ $\boldsymbol{x} = ( R_s, X_s, X_m, R_{r}, X_{r}, R_{c} ) \,$ are the equivalent circuit parameters of the single cage model (with core losses)

### Double Cage Model (With Core Losses)

The non-linear least squares problem for the double cage model with core losses is as follows:

$f_{1} (\boldsymbol{x}) = P_{m,fl} - P(s_{f}) = 0 \,$
$f_{2} (\boldsymbol{x}) = \sin{\phi} - Q(s_{f}) = 0 \,$
$f_{3} (\boldsymbol{x}) = T_{b} - T(s_{max}) = 0 \,$
$f_{4} (\boldsymbol{x}) = T_{lr} - T(s=1) = 0 \,$
$f_{5} (\boldsymbol{x}) = I_{lr} - I(s=1) = 0 \,$
$f_{6} (\boldsymbol{x}) = \eta{fl} - \eta(s_{f}) = 0 \,$

where $\boldsymbol{F} = ( f_1, f_2, f_3, f_4, f_5 ) \,$ $\boldsymbol{x} = ( R_s, X_s, X_m, R_{r1}, X_{r1}, R_{r2}, X_{r2}, R_{c} ) \,$ are the equivalent circuit parameters of the double cage model (with core losses)

## Parameter Estimation Algorithms

The most common algorithms used to solve the non-linear least squares problems for motor parameter estimation are as follows:

## Software

Many commercial software packages for power systems analysis such as ETAP and DIgSILENT PowerFactory contain integrated routines for estimating motor parameters from manufacturer data. The commercial packages typically use some variant of the Newton-Raphson algorithm.

Moto is a free standalone tool for estimating induction motor parameters based on commonly available manufacturer data (i.e. breakdown torque, locked rotor torque, full load power factor, etc). The program supports a number of estimation algorithms including Newton-Raphson, Levenberg-Marquardt and genetic algorithms.