© 2010 D. Kirschen and The University of Manchester1 New Formulations of the Optimal Power Flow...

© 2010 D. Kirschen and The University of Manchester

1

New Formulations of the Optimal Power Flow Problem

Prof. Daniel Kirschen

The University of Manchester


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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


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What is a power system?

Generators

Loads

PowerTransmission Network


5

What is running a power system about?

GreedMinimum cost

Maximum profit

Photo credit: FreeDigitalPhotos.net


6


FearAvoid outages and blackouts



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GreenAccommodate renewables



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Balancing conflicting aspirations

Cost Reliability

Environmental impact


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The Power Flow Problem


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State variables

• Voltage at every node (a.k.a. “bus”) of the network

• Because we are dealing with ac, voltages are represented by phasors, i.e. complex numbers in polar representation: Voltage magnitude at each bus:

Voltage angle at each bus:

Vk

θk


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Other variables

• Active and reactive power consumed at each bus: a.k.a. the load at each bus

• Active and reactive power produced by renewable generators:

• Assumed known in deterministic problems

• In practice, they are stochastic variables

PkW ,Qk

W

PkL ,Qk

L


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What is reactive power?

Active power

Reactive power



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G

€

PkG ,Qk

G

€

PkL ,Qk

L

Pk ,Qk

Injections

W

PkW ,Qk

W

Bus k

Pk =PkG + Pk

W −PkL

Qk =QkG +Qk

W −QkL

There is usually only one P and Q component at each bus


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Pk ,Qk

Injections

Bus k

Two of these four variables are specified at each bus:

• Load bus:• Generator bus:• Reference bus:

Vk∠θk

Pk ,Qk

Pk ,Vk

Vk ,θk


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Pk ,Qk

Line flows

Bus k

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

Vk∠θk

To bus i To bus j

€

Pki ,Qki

€

Pkj ,Qkj

Gki , Bki


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Power flow equations

Pk = VkVi[Gki cosθki + Bki sinθki ]i=1

N

∑

Qk = VkVi[Gki sinθki −Bki cosθki ]i=1

N

∑

with: θki =θk −θi, N : number of nodes in the network

Pk ,Qk

Bus kVk∠θk

To bus i To bus j

€

Pki ,Qki

€

Pkj ,Qkj

Write active and reactive power balance at each bus:

k =1,L N


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The power flow problem


N

∑


N

∑

Given the injections and the generator voltages, Solve the power flow equations to find the voltage magnitude and angle at each bus and hence the flow 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!


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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?


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Linear approximation


N

∑


N

∑Pk = Bkiθki

i=1

N

∑

• Ignores reactive power

• Assumes that all voltage magnitudes are nominal

• Useful when concerned with line flows only


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The Optimal Power Flow Problem(OPF)


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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:

PiG

ViG

tij


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Possible objective functions

• Minimise the cost of producing power with conventional generating units:

• Minimise deviations of the control variables from a given operating point (e.g. the outcome of a market):

min C

i(P

iG )

i=1

g

∑


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Equality constraints

• Power balance at each node bus, i.e. power flow equations


N

∑


N

∑ k =1,L N


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Inequality constraints

• Upper limit on the power flowing though every branch 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


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Formulation of the OPF problem

minu0

f0

x0,u

0( )

g x0 ,u0( ) =0

h x0 ,u0( ) ≤0

x0

u0

: 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.


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The Security Constrained Optimal Power Flow Problem

(SCOPF)


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Bad things happen…



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Sudden changes in the system

• A line is disconnected because of an insulation failure 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


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Security-constrained OPF

• How should the control variables be set to minimise the cost of running the system while ensuring that the operating constraints are satisfied in both the normal and all the contingency states?


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Formulation of the SCOPF problem

minuk

f0

x0,u

0( )

s.t. gk(xk ,uk) =0 k=0,...,Nc

hk(xk ,uk) ≤0 k=0,...,Nc

uk −u0 ≤Δukmax k=1,...,Nc

k =0

k =1,...,Nc

: normal conditions

: contingency conditions

Δukmax

: vector of maximum allowed adjustments after contingency k has occured


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Preventive or corrective SCOPF

minuk

f0

x0,u

0( )

s.t. gk(xk ,uk) =0 k=0,...,Nc

hk(xk ,uk) ≤0 k=0,...,Nc

uk −u0 ≤Δukmax k=1,...,Nc

Preventive SCOPF: no corrective actions are considered

Δukmax =0⇒ uk =u0∀k=1,K Nc

Corrective SCOPF: some corrective actions are allowed

∃ k=1,K Nc Δukmax ≠0


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Size of the SCOPF problem

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

• Pan-European transmission system model contains about 13,000 nodes, 20,000 branches and 2,000 generators

• Based on N-1 criterion, we should consider the outage of 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?)


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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 of corrective actions after a contingency


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The Worst-Case Problems


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Good things happen…



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… but there is no free lunch!

• Wind generation and solar generation can only be predicted with limited accuracy

• When planning the operation of the system a day ahead, some of the injections are thus stochastic variables

• Power system operators do not like probabilistic approaches


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Formulation of the OPF with uncertainty

min cT p0∗−p0

M( )

market-basedgeneration6 74 84+b0

*T c0 +p0nd∗cT( )

additionalgeneration6 74 4 84 4

s.t. g0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) =0

h0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) ≤0

u0 −u0init ≤Δu0

max

p0 −p0M ≤Δp0

max

pminnd b0

T ≤p0ndb0

T ≤pmaxnd b0

T

b0 ∈ 0,1{ }

smin ≤s≤smax

Deviations in cost-free controls

Deviations in market generation

Deviations in extra generation

Decisions about extra generation

Vector of uncertainties


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Worst-case OPF bi-level formulation

maxs

cT p0∗−p0

M( )+b0∗T c0 +p0

nd∗cT( )

s.t. smin ≤s≤smax

p0∗,u0

∗,b0∗,p0

nd∗( ) = arg min cT p0 −p0M( )+b0

T c0 +p0ndcT( )

s.t. g0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) =0

h0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) ≤0

u0 −u0init ≤Δu0

max

p0 −p0M ≤Δp0

max

pminnd b0

T ≤p0ndb0

T ≤pmaxnd b0

T

b0 ∈ 0,1{ }


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Worst-case SCOPF bi-level formulation

maxs

cT p0∗−p0

M( )+b0∗T c0 +p0

nd∗cT( )

s.t. smin ≤s≤smax

p0∗,pk

∗,u0∗,uk

∗,b0∗,p0

nd∗( ) = arg min cT p0 −p0M( )+b0

T c0 +p0ndcT( )

s.t. g0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) =0

h0 (x0 ,u0 ,p0 ,b0 ,p0nd ,s) ≤0

gk(xk ,uk ,pk ,b0 ,p0nd ,s) =0

hk(xk ,uk ,pk ,b0 ,p0nd ,s) ≤0

pk −p0 ≤Δpkmax

uk −u0 ≤Δukmax

pminnd b0

T ≤p0ndb0

T ≤pmaxnd b0

T

b0 ∈ 0,1{ }

Date post:	29-Mar-2015
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© 2010 D. Kirschen and The University of Manchester1 New Formulations of the Optimal Power Flow...

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