pf opf scopf

8/2/2019 PF OPF SCOPF

1/39

1

New Formulations of the Optimal Power

Flow Problem

Prof. Daniel Kirschen

The University of Manchester


2/39

2


3/39

3

Outline

A bit of background

The power flow problem

The optimal power flow problem (OPF)

The security-constrained OPF (SCOPF)

The worst-case problem


4/39

4

What is a power system?

Generators

Loads

Power

Transmission Network


5/39

2010 D. Kirschen and The University of Manchester

What is running a power system about?

GreedMinimum cost

Maximum profit

Photo credit: FreeDigitalPhotos.net


6/39



FearAvoid outages and blackouts



7/39 2010 D. Kirschen and The University of Manchester


GreenAccommodate renewables



8/398

Balancing conflicting aspirations

Cost Reliability

Environmental impact


9/399

The Power Flow Problem


10/39


11/3911

Other variables

Active and reactive power consumed at eachbus:

a.k.a. the load at each bus

Active and reactive power produced byrenewable generators:

Assumed known in deterministic problems

In practice, they are stochastic variables

PkW,Qk

W

PkL

,QkL


12/3912

What is reactive power?

Active power

Reactive power



13/3913

G

Pk

G,Q

k

G Pk

L,Q

k

L

Pk ,Qk

Injections

W

Pk

W,Q

k

W

Bus k

Pk PkG

PkW

PkL

Qk QkG Qk

W QkL

There is usually only one Pand Qcomponent at each bus


14/3914

Pk ,Qk

Injections

Bus k

Two of these four variables are specified at each bus:

Load bus: Generator bus: Reference bus:

Vkk

Pk ,QkPk ,VkVk,k


15/3915

Pk ,Qk

Line flows

Bus k

The line flows depend on the bus voltage magnitudeand angle as well as the network parameters(real and imaginary part of the network admittance matrix)

Vkk

To bus i To bus jPki ,Qki Pkj ,Qkj

Gki ,Bki


16/3916

Power flow equations

Pk

Vk

Vi

[Gki

coski

Bki

sinki

]i1

N

Qk VkVi[Gki sinki Bki coski ]

i1

N

with:ki

k

i, N: number of nodes in the network

Pk

,Qk

Bus kVkk

To bus i To bus jPki ,Qki Pkj ,Qkj

Write active and reactive power balance at each bus:

k 1,L N


17/3917

The power flow problem

Pk VkVi[Gki coski Bki sinki ]i1

N

Qk VkVi[Gki sinki Bki coski ]i1

N

Given the injections and the generator voltages,Solve the power flow equations to find the voltagemagnitude and angle at each bus and hence theflow in each branch

k 1,L N

Typical values of N:GB transmission network: N~1,500Continental European network (UCTE): N~13,000

However, the equations are highly sparse!


18/3918

Applications of the power flow problem

Check the state of the network

for an actual or postulated set of injections

for an actual or postulated network configuration

Are all the line flows within limits?

Are all the voltage magnitudes within limits?


19/3919

Linear approximation

Pk VkVi[Gki coski Bki sinki ]i1

N

Qk VkVi[Gki sinki Bki coski ]i1

N

Pk Bkiki

i1

N

Ignores reactive power

Assumes that all voltage magnitudes are nominal

Useful when concerned with line flows only


20/39 2010 D. Kirschen and The University of Manchester

The Optimal Power Flow Problem(OPF)


21/3921

Control variables

Control variables which have a cost:

Active power production of thermal generating units:

Control variables that do not have a cost:

Magnitude of voltage at the generating units:

Tap ratio of the transformers:

Pi

G

Vi

G

tij


22/3922

Possible objective functions

Minimise the cost of producing power withconventional generating units:

Minimise deviations of the control variables froma given operating point (e.g. the outcome of amarket):

min Ci(P

i

G)

i1

g

min c

i

Pi

G ci

Pi

Gi1

g


23/39


24/3924

Inequality constraints

Upper limit on the power flowing though everybranch of the network

Upper and lower limit on the voltage at every

node of the network Upper and lower limits on the control variables

Active and reactive power output of the generators

Voltage settings of the generators Position of the transformer taps and other control

devices


25/39

25

Formulation of the OPF problem

minu0

f0x

0,u

0g x

0,u

0 0h x

0,u

0 0

x

0

u

0

: vector of dependent (or state) variables

: vector of independent (or control) variables

Nothing extraordinary, except that we are dealingwith a fairly large (but sparse) non-linear problem.


26/39


27/39


Bad things happen



28/39

28

Sudden changes in the system

A line is disconnected because of an insulationfailure or a lightning strike

A generator is disconnected because of a

mechanical problem A transformer blows up

The system must keep going despite such events

N-1 security criterion


29/39

29

Security-constrained OPF

How should the control variables be set tominimise the cost of running the system whileensuring that the operating constraints aresatisfied in both the normal and all thecontingency states?


30/39

30

Formulation of the SCOPF problem

minuk

f0x

0,u

0 s.t. g

k(x

k,u

k) 0 k 0,...,N

c

hk(x

k,u

k) 0 k 0,...,N

c

uk u0 ukmax

k 1,...,Nc

k 0

k 1,...,Nc

: normal conditions

: contingency conditions

uk

max

: vector of maximum allowed adjustments aftercontingency khas occured


31/39

31

Preventive or corrective SCOPF

minuk f0 x0 ,u0 s.t. g

k(x

k,u

k) 0 k 0,...,N

c

hk(x

k,u

k) 0 k 0,...,N

c

uk

u0

uk

maxk 1,...,N

c

Preventive SCOPF: no corrective actions are considered

uk

max 0 uk

u0k 1,K N

c

Corrective SCOPF: some corrective actions are allowed

k 1,K Ncu

k

max 0


32/39

32

Size of the SCOPF problem

SCOPF is (Nc+1) times larger than the OPF

Pan-European transmission system model containsabout 13,000 nodes, 20,000 branches and 2,000generators

Based on N-1 criterion, we should consider the outageof each branch and each generator as a contingency

However:

Not all contingencies are critical (but which ones?) Most contingencies affect only a part of the network (but what

part of the network do we need to consider?)


33/39

33

A few additional complications

Some of the control variables are discrete:

Transformer and phase shifter taps

Capacitor and reactor banks

Starting up of generating units

There is only time for a limited number ofcorrective actions after a contingency


34/39

34

The Worst-Case Problems


35/39


Good things happen



36/39

36

but there is no free lunch!

Wind generation and solar generation can onlybe predicted with limited accuracy

When planning the operation of the system a

day ahead, some of the injections are thusstochastic variables

Power system operators do not like probabilisticapproaches


37/39

37

Formulation of the OPF with uncertainty

min cTp

0

p0

M market-basedgeneration

6 74 84

b0

*Tc

0p

0

ndcT

additionalgeneration6 74 4 84 4

s.t. g0(x

0,u

0,p

0,b

0,p

0

nd,s) 0

h0

(x0

,u0

,p0

,b0

,p0

nd,s) 0

u0

u0

init u0

max

p0

p0

M p0

max

pmin

ndb

0

T p0

ndb

0

T pmax

ndb

0

T

b0

0,1 s

min s s

max

Deviations in cost-free controls

Deviations in market generation

Deviations in extra generation

Decisions about extra generation

Vector of uncertainties


38/39

38

Worst-case OPF bi-level formulation

maxs

cTp

0

p0

M b0T c0 p0ndcT s.t. s

min s s

max

p0

,u

0

,b

0

,p

0

nd arg min cT p0 p0M b0T c0 p0ndcT

s.t. g0 (x0 ,u0 ,p0 ,b0 ,p0nd

,s) 0

h0(x

0,u

0,p

0,b

0,p

0

nd,s) 0

u0

u0

init u0

max

p0

p0

M p0

max

pmin

ndb

0

T p0

ndb

0

T pmax

ndb

0

T

b0

0,1


39/39

Worst-case SCOPF bi-level formulation

maxs c

T

p0

p0M

b0

T

c0 p0nd

c

T

s.t. s

min s s

max

p0

,p

k

,u

0

,u

k

,b

0

,p

0

nd arg min cT p0 p0M b0T c0 p0ndcT s.t. g

0(x

0,u

0,p

0,b

0,p

0

nd,s) 0

h0(x

0,u

0,p

0,b

0,p

0

nd,s) 0

gk(x

k,u

k,p

k,b

0,p

0

nd,s) 0

hk(x

k,u

k,p

k,b

0,p

0

nd,s) 0

pk p0 pkmax

uk

u0

uk

max

pmin

ndb

0

T p0

ndb

0

T pmax

ndb

0

T

b0

0,1

pf opf scopf

Documents