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New Formulations of the Optimal Power
Flow Problem
Prof. Daniel Kirschen
The University of Manchester
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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
Power
Transmission Network
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2010 D. Kirschen and The University of Manchester
What is running a power system about?
GreedMinimum cost
Maximum profit
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What is running a power system about?
FearAvoid outages and blackouts
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What is running a power system about?
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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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
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What is reactive power?
Active power
Reactive power
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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
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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:
Vkk
Pk ,QkPk ,VkVk,k
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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
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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
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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!
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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
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
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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:
Pi
G
Vi
G
tij
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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
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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
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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.
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2010 D. Kirschen and The University of Manchester
Bad things happen
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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
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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?
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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
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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
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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?)
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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 ofcorrective 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 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
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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
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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
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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