Motor#

Induction Motor group

Common Parameters: u, name

Available models: Motor3, Motor5

Motor3#

Third-order induction motor model.

See "Power System Modelling and Scripting" by F. Milano.

To simulate motor startup, set the motor status u to 0 and use a Toggle to control the model.

Parameters#

Name

Symbol

Description

Default

Unit

Properties

idx

unique device idx

u

\(u\)

connection status

1

bool

name

device name

bus

interface bus id

mandatory

Sn

\(S_n\)

Power rating

100

Vn

\(V_n\)

AC voltage rating

110

fn

\(f\)

rated frequency

60

rs

\(r_s\)

rotor resistance

0.010

non_zero,z

xs

\(x_s\)

rotor reactance

0.150

non_zero,z

rr1

\(r_{R1}\)

1st cage rotor resistance

0.050

non_zero,z

xr1

\(x_{R1}\)

1st cage rotor reactance

0.150

non_zero,z

rr2

\(r_{R2}\)

2st cage rotor resistance

0.001

non_zero,z

xr2

\(x_{R2}\)

2st cage rotor reactance

0.040

non_zero,z

xm

\(x_m\)

magnetization reactance

5

non_zero,z

Hm

\(H_m\)

Inertia constant

3

kWs/KVA

power

c1

\(c_1\)

1st coeff. of Tm(w)

0.100

c2

\(c_2\)

2nd coeff. of Tm(w)

0.020

c3

\(c_3\)

3rd coeff. of Tm(w)

0.020

zb

\(z_b\)

Allow working as brake

1

Variables#

Name

Symbol

Type

Description

Unit

Properties

slip

\(\sigma\)

State

v_str

e1d

\(e'_{d}\)

State

real part of 1st cage voltage

v_str

e1q

\(e'_{q}\)

State

imaginary part of 1st cage voltage

v_str

vd

\(V_{d}\)

Algeb

d-axis voltage

vq

\(V_{q}\)

Algeb

q-axis voltage

p

\(P\)

Algeb

v_str

q

\(Q\)

Algeb

v_str

Id

\(I_{d}\)

Algeb

v_str

Iq

\(I_{q}\)

Algeb

te

\(\tau_{e}\)

Algeb

v_str

tm

\(\tau_{m}\)

Algeb

v_str

a

\(\theta\)

ExtAlgeb

Bus voltage phase angle

v

\(V\)

ExtAlgeb

Bus voltage magnitude

Initialization Equations#

Name

Symbol

Type

Initial Value

slip

\(\sigma\)

State

\(1.0 u\)

e1d

\(e'_{d}\)

State

\(0.05 u\)

e1q

\(e'_{q}\)

State

\(0.9 u\)

vd

\(V_{d}\)

Algeb

vq

\(V_{q}\)

Algeb

p

\(P\)

Algeb

\(u \left(I_{d} V_{d} + I_{q} V_{q}\right)\)

q

\(Q\)

Algeb

\(u \left(I_{d} V_{q} - I_{q} V_{d}\right)\)

Id

\(I_{d}\)

Algeb

\(1\)

Iq

\(I_{q}\)

Algeb

te

\(\tau_{e}\)

Algeb

\(u \left(I_{d} e'_{d} + I_{q} e'_{q}\right)\)

tm

\(\tau_{m}\)

Algeb

\(u \left(\alpha + \beta \sigma + \sigma^{2} c_{2}\right)\)

a

\(\theta\)

ExtAlgeb

v

\(V\)

ExtAlgeb

Differential Equations#

Name

Symbol

Type

RHS of Equation "T x' = f(x, y)"

T (LHS)

slip

\(\sigma\)

State

\(u \left(- \tau_{e} + \tau_{m}\right)\)

\(M\)

e1d

\(e'_{d}\)

State

\(u \left(\omega_{b} \sigma e'_{q} - \frac{I_{q} \left(- x' + x_{0}\right) + e'_{d}}{T'_{0}}\right)\)

e1q

\(e'_{q}\)

State

\(u \left(- \omega_{b} \sigma e'_{d} - \frac{- I_{d} \left(- x' + x_{0}\right) + e'_{q}}{T'_{0}}\right)\)

Algebraic Equations#

Name

Symbol

Type

RHS of Equation "0 = g(x, y)"

vd

\(V_{d}\)

Algeb

\(- V u \sin{\left(\theta \right)} - V_{d}\)

vq

\(V_{q}\)

Algeb

\(V u \cos{\left(\theta \right)} - V_{q}\)

p

\(P\)

Algeb

\(- P + u \left(I_{d} V_{d} + I_{q} V_{q}\right)\)

q

\(Q\)

Algeb

\(- Q + u \left(I_{d} V_{q} - I_{q} V_{d}\right)\)

Id

\(I_{d}\)

Algeb

\(u \left(- I_{d} r_{s} + I_{q} x' + V_{d} - e'_{d}\right)\)

Iq

\(I_{q}\)

Algeb

\(u \left(- I_{d} x' - I_{q} r_{s} + V_{q} - e'_{q}\right)\)

te

\(\tau_{e}\)

Algeb

\(- \tau_{e} + u \left(I_{d} e'_{d} + I_{q} e'_{q}\right)\)

tm

\(\tau_{m}\)

Algeb

\(- \tau_{m} + u \left(\alpha + \beta \sigma + \sigma^{2} c_{2}\right)\)

a

\(\theta\)

ExtAlgeb

\(P\)

v

\(V\)

ExtAlgeb

\(Q\)

Services#

Name

Symbol

Equation

Type

wb

\(\omega_b\)

\(2 \pi f\)

ConstService

x0

\(x_0\)

\(x_{m} + x_{s}\)

ConstService

x1

\(x'\)

\(\frac{x_{m} x_{R1}}{x_{m} + x_{R1}} + x_{s}\)

ConstService

T10

\(T'_0\)

\(\frac{x_{m} + x_{R1}}{\omega_{b} r_{R1}}\)

ConstService

M

\(M\)

\(2 H_{m}\)

ConstService

aa

\(\alpha\)

\(c_{1} + c_{2} + c_{3}\)

ConstService

bb

\(\beta\)

\(- c_{2} - 2 c_{3}\)

ConstService

Config Fields in [Motor3]

Option

Symbol

Value

Info

Accepted values

allow_adjust

1

allow adjusting upper or lower limits

(0, 1)

adjust_lower

0

adjust lower limit

(0, 1)

adjust_upper

1

adjust upper limit

(0, 1)

Motor5#

Fifth-order induction motor model.

See "Power System Modelling and Scripting" by F. Milano.

To simulate motor startup, set the motor status u to 0 and use a Toggle to control the model.

Parameters#

Name

Symbol

Description

Default

Unit

Properties

idx

unique device idx

u

\(u\)

connection status

1

bool

name

device name

bus

interface bus id

mandatory

Sn

\(S_n\)

Power rating

100

Vn

\(V_n\)

AC voltage rating

110

fn

\(f\)

rated frequency

60

rs

\(r_s\)

rotor resistance

0.010

non_zero,z

xs

\(x_s\)

rotor reactance

0.150

non_zero,z

rr1

\(r_{R1}\)

1st cage rotor resistance

0.050

non_zero,z

xr1

\(x_{R1}\)

1st cage rotor reactance

0.150

non_zero,z

rr2

\(r_{R2}\)

2st cage rotor resistance

0.001

non_zero,z

xr2

\(x_{R2}\)

2st cage rotor reactance

0.040

non_zero,z

xm

\(x_m\)

magnetization reactance

5

non_zero,z

Hm

\(H_m\)

Inertia constant

3

kWs/KVA

power

c1

\(c_1\)

1st coeff. of Tm(w)

0.100

c2

\(c_2\)

2nd coeff. of Tm(w)

0.020

c3

\(c_3\)

3rd coeff. of Tm(w)

0.020

zb

\(z_b\)

Allow working as brake

1

Variables#

Name

Symbol

Type

Description

Unit

Properties

slip

\(\sigma\)

State

v_str

e1d

\(e'_{d}\)

State

real part of 1st cage voltage

v_str

e1q

\(e'_{q}\)

State

imaginary part of 1st cage voltage

v_str

e2d

\(e''_{d}\)

State

real part of 2nd cage voltage

v_str

e2q

\(e''_{q}\)

State

imag part of 2nd cage voltage

v_str

vd

\(V_{d}\)

Algeb

d-axis voltage

vq

\(V_{q}\)

Algeb

q-axis voltage

p

\(P\)

Algeb

v_str

q

\(Q\)

Algeb

v_str

Id

\(I_{d}\)

Algeb

v_str

Iq

\(I_{q}\)

Algeb

v_str

te

\(\tau_{e}\)

Algeb

v_str

tm

\(\tau_{m}\)

Algeb

v_str

a

\(\theta\)

ExtAlgeb

Bus voltage phase angle

v

\(V\)

ExtAlgeb

Bus voltage magnitude

Initialization Equations#

Name

Symbol

Type

Initial Value

slip

\(\sigma\)

State

\(1.0 u\)

e1d

\(e'_{d}\)

State

\(0.05 u\)

e1q

\(e'_{q}\)

State

\(0.9 u\)

e2d

\(e''_{d}\)

State

\(0.05 u\)

e2q

\(e''_{q}\)

State

\(0.9 u\)

vd

\(V_{d}\)

Algeb

vq

\(V_{q}\)

Algeb

p

\(P\)

Algeb

\(u \left(I_{d} V_{d} + I_{q} V_{q}\right)\)

q

\(Q\)

Algeb

\(u \left(I_{d} V_{q} - I_{q} V_{d}\right)\)

Id

\(I_{d}\)

Algeb

\(0.9 u\)

Iq

\(I_{q}\)

Algeb

\(0.1 u\)

te

\(\tau_{e}\)

Algeb

\(u \left(I_{d} e''_{d} + I_{q} e''_{q}\right)\)

tm

\(\tau_{m}\)

Algeb

\(u \left(\alpha + \beta \sigma + \sigma^{2} c_{2}\right)\)

a

\(\theta\)

ExtAlgeb

v

\(V\)

ExtAlgeb

Differential Equations#

Name

Symbol

Type

RHS of Equation "T x' = f(x, y)"

T (LHS)

slip

\(\sigma\)

State

\(u \left(- \tau_{e} + \tau_{m}\right)\)

\(M\)

e1d

\(e'_{d}\)

State

\(u \left(\omega_{b} \sigma e'_{q} - \frac{I_{q} \left(- x' + x_{0}\right) + e'_{d}}{T'_{0}}\right)\)

e1q

\(e'_{q}\)

State

\(u \left(- \omega_{b} \sigma e'_{d} - \frac{- I_{d} \left(- x' + x_{0}\right) + e'_{q}}{T'_{0}}\right)\)

e2d

\(e''_{d}\)

State

\(u \left(\omega_{b} \sigma e'_{q} - \omega_{b} \sigma \left(- e''_{q} + e'_{q}\right) - \frac{I_{q} \left(- x' + x_{0}\right) + e'_{d}}{T'_{0}} + \frac{- I_{q} \left(x' - x''\right) - e''_{d} + e'_{d}}{T''_{0}}\right)\)

e2q

\(e''_{q}\)

State

\(u \left(- \omega_{b} \sigma e'_{d} + \omega_{b} \sigma \left(- e''_{d} + e'_{d}\right) - \frac{- I_{d} \left(- x' + x_{0}\right) + e'_{q}}{T'_{0}} + \frac{I_{d} \left(x' - x''\right) - e''_{q} + e'_{q}}{T''_{0}}\right)\)

Algebraic Equations#

Name

Symbol

Type

RHS of Equation "0 = g(x, y)"

vd

\(V_{d}\)

Algeb

\(- V u \sin{\left(\theta \right)} - V_{d}\)

vq

\(V_{q}\)

Algeb

\(V u \cos{\left(\theta \right)} - V_{q}\)

p

\(P\)

Algeb

\(- P + u \left(I_{d} V_{d} + I_{q} V_{q}\right)\)

q

\(Q\)

Algeb

\(- Q + u \left(I_{d} V_{q} - I_{q} V_{d}\right)\)

Id

\(I_{d}\)

Algeb

\(u \left(- I_{d} r_{s} + I_{q} x'' + V_{d} - e''_{d}\right)\)

Iq

\(I_{q}\)

Algeb

\(u \left(- I_{d} x'' - I_{q} r_{s} + V_{q} - e''_{q}\right)\)

te

\(\tau_{e}\)

Algeb

\(- \tau_{e} + u \left(I_{d} e''_{d} + I_{q} e''_{q}\right)\)

tm

\(\tau_{m}\)

Algeb

\(- \tau_{m} + u \left(\alpha + \beta \sigma + \sigma^{2} c_{2}\right)\)

a

\(\theta\)

ExtAlgeb

\(P\)

v

\(V\)

ExtAlgeb

\(Q\)

Services#

Name

Symbol

Equation

Type

wb

\(\omega_b\)

\(2 \pi f\)

ConstService

x0

\(x_0\)

\(x_{m} + x_{s}\)

ConstService

x1

\(x'\)

\(\frac{x_{m} x_{R1}}{x_{m} + x_{R1}} + x_{s}\)

ConstService

T10

\(T'_0\)

\(\frac{x_{m} + x_{R1}}{\omega_{b} r_{R1}}\)

ConstService

M

\(M\)

\(2 H_{m}\)

ConstService

aa

\(\alpha\)

\(c_{1} + c_{2} + c_{3}\)

ConstService

bb

\(\beta\)

\(- c_{2} - 2 c_{3}\)

ConstService

x2

\(x''\)

\(\frac{x_{m} x_{R1} x_{R2}}{x_{m} x_{R1} + x_{m} x_{R2} + x_{R1} x_{R2}} + x_{s}\)

ConstService

T20

\(T''_0\)

\(\frac{\frac{x_{m} x_{R1}}{x_{m} + x_{R1}} + x_{R2}}{\omega_{b} r_{R2}}\)

ConstService

Config Fields in [Motor5]

Option

Symbol

Value

Info

Accepted values

allow_adjust

1

allow adjusting upper or lower limits

(0, 1)

adjust_lower

0

adjust lower limit

(0, 1)

adjust_upper

1

adjust upper limit

(0, 1)