4.3 External Network Modelling

4.3 External network modelling

The interconnected system of Europe in 1995

4.3.1

4.3 External network modelling

Line outage External network

4.3.2 Observable network Internal network

External network

External network

4.3 External network modelling

Generator outage

4.3.3 Observable network

4.3 External network modelling

4.3.4

Frequency and power deviation in the UCPTE network Deviation of frequency a) Normal operation

b) Outage of a generation unit

Net tie line interchanges

4.3 External network modelling

4.3.5

Real-time security analysis Network topology Measurements Variance of measurement errors Network parameter

Network equivalent

Contingency set • lines • transformers • generation units

State Estimation

State vector J(x), E{J(x)} Location of bad data

Model of external system

Security analysis

Security constraints: • met • not met Exeeding limits:

• Ith • Vmin, Vmax • Imax

4.3 External network modelling

Alternatives to consider the external network

•

1:1 representation of the external network with complete data transmission of measurements and on-off status quantities

•

Application of an equivalent for the external network

4.3.6

4.3 External network modelling

4.3.7

Available information Observable network External network

Internal network

External network

External network Network region Observable network: - Internal nodes - Boundery nodes External network: - External nodes

Available information Measurements, on-off status quantities, transformator tap-settings, network parameter, Vi, Θi (via state estimation) Network topology (standard topology), network parameter

4.3 External network modelling

4.3.8

diag[V ][Y ] [V ] = [S] ∗

E: external;

VE  YEE diag VB  YBE  VI   0

∗

B: boundary;

YEB YBB YIB

0  YBI  YII 

diag[VE ][YEE VE + YEB VB

∗

I: internal ∗

VE  SE  V  = S   B  B  VI   SI 

]∗ = [SE ]

(1.1)

diag[VB ][YBE VE + YBB VB + YBI VI ] = [SB ] (1.2) ∗

diag[VI ][

YIB VB + YII VI ] = [SI ] ∗

(1.3)

4.3 External network modelling

4.3.9

Elimination of [VE] using eq. (1.1)

[YEE VE + YEBVB ]∗ = diag −1 [VE ]⋅ [SE ] [YEE VE ]∗ = diag −1 [VE ]⋅ [SE ] − [YEBVB ]∗ [VE ]∗ = [YEE∗ ]−1 {diag −1 [VE ]⋅ [SE ] − [YEBVB ]∗ } [VE ] = [YEE ]−1 {diag −1 [VE ]* ⋅ [SE ]* − [YEBVB ]} Introducing [VE] in eq. (1.2)

[ ]{[ ] {[Y ]

diag VB YBE

∗ −1 EE

*

{diag [V ][S ]− [Y ] [V ] }} E

E

[ ] [ ] [ ] [ ] }= [S ] *

*

*

+ YBB VB + YBI VI

∗

*

−1

*

B

EB

B

4.3 External network modelling

4.3.10

{[ ] [ ] [ ] [ ] [Y ] [V ] + [Y ] [V ] } *

*

*

∗ diag[VB ] YBB VB − YBE YEE

[ ]

[ ][ ] [ ] *

∗ = SB − diag VB YBE YEE

−1

−1

*

*

EB

B

*

BI

[ ][ ]

diag −1 VE SE

With the definitions

[Y ] = [Y ] − [Y ][Y ] [Y ] −1

eq BB

BB

BE

EE

EB

[S ] = [S ] − diag[V ][Y ] [Y ] eq B

∗ −1 EE

*

B

B

BE

diag −1 [VE ][SE ]

we can write:

{[ ] [ ] [ ] [ ] }= [S ] ∗

∗

∗

eq diag[VB ] YBB VB + YBI VI

∗

eq B

*

I

4.3 External network modelling

4.3.11

Power flow equations after transformation: eq VB  YBB diag     VI   YIB

YBI   YII 

∗

∗

 VB  Seq B V  =    I   SI 

( 2.1) ( 2 .2 )

With definitions:

[Y ] = [Y ] − [Y ][Y ] [Y ] −1

eq BB

BB

BE

EE

[S ] = [S ] − diag[V ][Y ] [Y ] eq B

∗

B

B

BE

( 2.3)

EB

* −1 EE

[ ][ ]

⋅ diag −1 VE SE

( 2 .4 )

4.3 External network modelling

4.3.12

On-line application: • Topology and network parameters of external, boundary and eq internal network are known; so YBB can be determined

[ ]

using Eq. (2.3):

[Y ] = [Y ] − [Y ][Y ] [Y ] −1

eq BB

•

BB

BE

EE

EB

[S ] , [V ] of external network are unknown; so the calculation of [S ] with the use of Eq. (2.4) is not possible. E

E

eq B

[S ] = [S ] − diag[V ][Y ] [Y ] ∗

eq B

•

B

B

BE

* −1 EE

[ ][ ]

⋅ diag −1 VE SE

[V ],[V ] of boundary and internal network are known as a result of state estimation; therefore [S ] can be determined using Eq. (2.1): [S ] = diag[V ]{[Y ] [V ] + [Y ] [V ] } B

I

eq B

eq B

B

eq * BB

*

B

*

BI

*

I

4.3 External network modelling

4.3.13

Equivalent representation of the external network Pi eq + jQieq

Internal network y ieq 0 y ikeq

passive part: equivalent branches equivaltent shunts active part: equivalent injections

y ikeq y ieq 0 Pi eq + jQieq

4.3 External network modelling

German system with external network

4.3.14

4.3 External network modelling

German system with equivalent for external network

Equivalent branches Equivalent injections

4.3.15

4.3 External network modelling

Variation in time of Peq in boundary nodes (January 9th 1979)

4.3.16

4.3 External network modelling

4.3.17

Reduction of the external network Bus admittance matrix:

E: External B: Boundery I: Internal Elimination of external nodes:

4.3 External network modelling

4.3.18

Example of external network reduction Z 27 = 0 . 0973

+ j 0 . 2691 Z 17 = 0 . 0890

Z 25 = 0 . 0387

+ j 0 . 2359

+ j 0 . 1847 Z 18 = 0 . 1068

+ j 0 . 2807

Z 58 = 0 . 0497

+ j 0 . 2372

Internal network Z 45 = 0 . 0529

+ j 0 . 1465

Z 38 = 0 . 0460

+ j 0 . 2196

Z 43 = 0 . 0364 + j 0 . 1736 Z 46 = 0 . 0511 + j 0 . 2442 Z 56 = 0 . 0579

+ j 0 . 2763

4.3 External network modelling

Example Bus admittance matrix

4.3.19

4.3 External network modelling

Example 1st step

4.3.20

4.3 External network modelling

Example 2nd step

4.3.21

4.3 External network modelling

Example 3rd step

4.3.22

4.3 External network modelling

Example 4th step

4.3.23

4.3 External network modelling

Example 5th step

4.3.24

4.3 External network modelling

4.3.25

Example of external network reduction eq Z 78

Z

eq 67

= j 1 . 0380

= j 0 . 3520

Internal network

eq Z 68 = j 0 . 3213

Equivalent branches (Reactance reduced)