Power system dynamic simulation: an iterative multirate approach

Power system dynamic simulation:

an iterative multirate approach

Frédéric Plumier, Christophe Geuzaine and Thierry Van Cutsem*

Department of Electrical Engineering and Computer Science, University of Liège, Belgium

* Fund for Scientific Research (FNRS)

Objective: Combine accuracy of detailed EMT simulation with efficiency of simplified FF simulation.

1. Power System representation

M M M FF Network EMT Network Interface Injectors

In dynamic simulations, power system com-ponents are modeled by sets of nonlinear stiff hybrid Differential-Algebraic Equations (DAEs).

The

i

-th injector can be described by a DAE

system of the form:

Γ

_i

x

_˙

_i

_{= Φi(xi, V )}

where

x

_i the model’s internal states,

V

¯

_k+1( j)

(Γ

i

)

``

=

(

0

if the

`

-th equation is algebraic

1

if the

`

-th equation is differential

and

V

the vector of network voltages.

Two modeling families are traditionnally used in power system dynamic simulations:

1. Full-Transient or

ElectroMagnetic Transients (EMT)

simulation

2. Fundamental-Frequency (FF)

approxima-tion for which the network can be described by the linear algebraic equations:

0 = DV

_{− I = g(x,V )}

EMT vs FF: Three-phase fault

-10 -8 -6 -4 -2 0 2 4 6 8 0 0.02 0.04 0.06 0.08 0.1 Current magnitude t (s) Full-transient representation ia(t) ib(t) ic(t) 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 0.1 t (s) Phasor representation I(t) Iphi(t)

3. Test system: Nordic-32

g11 g16 g17 g18 g19 g20 g2 g9 g1 g13 g8 g12 g4 g5 g10 g3 g7 g15 g14 g6 4011 4012 1011 1012 1014 1013 1022 1021 2031 cs 4046 4043 4044 4032 4031 4022 4021 4071 4072 4041 1042 1045 1041 4063 4061 ₁₀₄₃ 1044 4047 4051 4045 4062 400 kV 220 kV 130 kV synchronous condenser CS NORTH CENTRAL EQUIV. SOUTH 4042 2032 FF subdomain EMT subdomain

2. Iterative multirate approach

Gauss-Seidel relaxation until convergence

¯

V

_k+1(j)

Interpolation

Extraction

¯

I

_k+1(j+1)

FF

simulation

EMT

simulation

Interpolation

t t + H Vk6 φk Interpolation single-phase to three-phase time k _{k + 1 discretized time} V_k+16 φ_k+1

v

_a

(t), v

_b

(t), v

_c

(t)

+

⇓

FF simulation

EMT simulation

H h

Considering phase

a

, for instance, the voltage evolution is taken as (

n

= 0, 1, 2, . . .

):

v

_a

(t + nh) =

√

2V

a(t + nh) cos (ω(t + nh) + φa(t + nh))

where

V

a(t + nh) = V_a

(t) + (V

a(t + H)

−V

_a

(t)) 

nh

H

and

φ

_a

(t + nh) = φ

_a

(t) + (φ

_a

(t + H)

− φ

_a

(t)) 

nh

H

where

ω

is the nominal angular speed of the system.

Extraction: projection on a rotating frame

+

filtering

I

_0qd

_{= T Iabc}

₌

√

2

3





1

/

√₂ 1

/

√₂ 1

/

√₂

cos

(θ)

cos θ

−

2π₃

cos θ

−

4π₃

−sin(θ) −sin θ −

2π₃

−sin θ −

4π₃









i

_a

(t)

i

_b

(t)

i

c(t)





≈





0

I

_a

cos ψ

_a

I

_a

sin ψ

_a





=

_{⇒ I}

a

=

q

I

_d2

+ I

_q2

ψ

a

= arctan

 I

_q

I

_d



¯

I

_a

(t) = I

_a

e

jψa

θ = ω t

d

q

ψ

_a

¯

V

_a

(t) = V

_a

e

jφa

φ

_a

4. Preliminary results for a three-phase fault

Rotor speed of machine g15

0.996 0.998 1 1.002 1.004 1.006 1.008 1.01 0 1 2 3 4 5

machine speed (pu)

t (s) EMTP-RV FF-EMT Ramses

Voltage at bus 4043

-1.5 -1 -0.5 0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 va (pu) t (s) EMTP-RV FF-EMT