dtu cee summer school 2020 - optimal power flow (dc and ac...
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Optimal Power Flow (DC-OPF and AC-OPF)
DTU Summer School 2017
Spyros Chatzivasileiadis
What is optimal power flow?
2 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Optimal Power Flow (OPF)• In its most realistic form, the OPF is a non-linear, non-convex problem,which includes both binary and continuous variables.
• Goal: minimize an objective function, respecting all physical, operational,and technical constraints, such as:
• Ohm’s law and Kirchhoff laws• Operational limits of generators• Loading limits of transmission lines• Voltage levels• and many, many others
• Disclaimer:
• Realistic OPF implementations include thousands of variables andconstraints
• Here we focus on the most“fundamental” formulations of OPF• These can be extended with several additional constraints toaccurately model the problem and/or system at hand
3 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Use of OPF in the industry
• RTE, France: they invented the OPF! (Carpentier,1962)
• Their focus is mostly on the optimization of thesystem operation and not so much on markets
• They are working on the application of convexrelaxations of OPF on their system (see DanMolzahn’s talk!)
• CAISO, California, USA
• Electricity markets: An OPF runs every day(Day-Ahead), every hour, every 15 minutes, andevery 5 minutes.
• Depending on the problem they run a DC-OPFwith unit commitment, a standard DC-OPF, orjust Economic Dispatch
4 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Use of OPF in the industry
• PJM, East Coast, USA
• Security-Constrained Economic Dispatch every 5 minutes• final test phase of optimal voltage control every 5 minutes (they useAC-OPF as well)
• EUPHEMIA
• one common algorithm to calculate electricity prices across Europe, andallocate cross border capacity on a day-ahead basis
• 19 European countries, over 150 million EUR in matched trades daily
• PLEXOS
• June 2000: PLEXOS was first-to-market with electric power marketsimulation based entirely on mathematical programming
• Features: generation capacity expansion planning, transmission expansionplanning, hydro-thermal coordination, ancillary services
5 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Summarizing• DC-OPF
• market clearing uses DC-OPF (at the moment)• convex• can solve fast; can be applied in very large problems
• AC-OPF
• primarily used for optimization of operation and control actions• non-convex (in its original form)• continuous efforts to decrease computation time and increase systemsize
• Trends
• Incorporating uncertainty• Decomposition (and other) methods to solve very large problems• Guarantees for a global minimum (convex relaxations)• Market design• Coupled energy networks and markets, e.g. gas and electricity
6 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Outline
• Economic Dispatch
• DC-OPF
• AC-OPF
7 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Outline• Economic Dispatch
• used in power exchanges, e.g. EPEX, etc.• Supply must meet demand• Generator limits
• DC-OPF
• extends Economic Dispatch• considers the power flows! (in a linearized form)• includes the power flow limits of the lines• only active power; no losses
• AC-OPF
• full AC power flow equations• active and reactive power flow• current, voltage• losses
8 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Outline• Economic Dispatch
• used in power exchanges, e.g. EPEX, etc.• Supply must meet demand• Generator limits
• DC-OPF
• extends Economic Dispatch• considers the power flows! (in a linearized form)• includes the power flow limits of the lines• only active power; no losses
• AC-OPF
• full AC power flow equations• active and reactive power flow• current, voltage• losses
8 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Outline• Economic Dispatch
• used in power exchanges, e.g. EPEX, etc.• Supply must meet demand• Generator limits
• DC-OPF
• extends Economic Dispatch• considers the power flows! (in a linearized form)• includes the power flow limits of the lines• only active power; no losses
• AC-OPF
• full AC power flow equations• active and reactive power flow• current, voltage• losses
8 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Economic Dispatch
min∑i
ciPGi
subject to:
PminGi
≤ PGi≤ Pmax
Gi∑i
PGi= PD
• The Economic Dispatch does not consider any network flows or networkconstraints!
• We assume a copperplate network, i.e. a lossless and unrestricted flow ofelectricity from A to B.
9 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Economic Dispatch
min∑i
ciPGi
subject to:
PminGi
≤ PGi≤ Pmax
Gi∑i
PGi= PD
• The Economic Dispatch does not consider any network flows or networkconstraints!
• We assume a copperplate network, i.e. a lossless and unrestricted flow ofelectricity from A to B.
9 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Economic Dispatch
min∑i
ciPGi
subject to:
PminGi
≤ PGi≤ Pmax
Gi∑i
PGi= PD
• The Economic Dispatch does not consider any network flows or networkconstraints!
• We assume a copperplate network, i.e. a lossless and unrestricted flow ofelectricity from A to B.
9 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Can we solve the economic dispatch problem without using anoptimization solver?
Yes! With the help of the merit order curve.
10 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Can we solve the economic dispatch problem without using anoptimization solver?
Yes! With the help of the merit order curve.
10 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
The Merit-Order Curve
0 A B C D
cG1
cG2
cG3
cG4
price
power
PD
A = PmaxG1
B = A+ PmaxG2
C = B + PmaxG3
D = C + PmaxG4
11 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
The Merit-Order Curve
0 A B C D
cG1
cG2
cG3
cG4
price
powerPD
A = PmaxG1
B = A+ PmaxG2
C = B + PmaxG3
D = C + PmaxG4
11 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
The Merit-Order Curve
0 A B C D
cG1
cG2
cG3
cG4
price
powerPD
• cG3 is the systemmarginal price
• Gens G1 and G2are fullydispatched
• Gen G4 is notdispatched at all
• Gen G3 is partiallydispatched
• G3 is the“marginalgenerator”
12 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
The Merit-Order Curve: An Example
Merit-Order of the German conventional generation in 2008. Source:Forschungsstelle fur Energiewirtschaft e. V.
13 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Merit-Order Curve, Marginal Generators,and Line Congestion
0 A B C D
cG1
cG2
cG3
cG4
price
powerPD
• Although G3 hasenough capacity,it cannot produceenough to coverthe demand dueto line congestion
• Instead G4, amore expensivegen that does notcontribute to theline congestion,must produce themissing power
• In a DC-OPF context, there is no longer a single system marginal price(we will observe different nodal prices in different nodes)
14 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
DC-OPF
min∑i
ciPGi
subject to:PminGi
≤ PGi≤ Pmax
Gi
B · θ = PG −PD
1
xij
(θi − θj) ≤ Pij,max
• The DC-OPF with the standard power flow equations contains both thepower generation PG and the voltage angles θ in the vector of theoptimization variables.
15 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
DC-OPF
min∑i
ciPGi
subject to:PminGi
≤ PGi≤ Pmax
Gi
B · θ = PG −PD
1
xij
(θi − θj) ≤ Pij,max
• The DC-OPF with the standard power flow equations contains both thepower generation PG and the voltage angles θ in the vector of theoptimization variables.
15 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
DC-OPF
min∑i
ciPGi
subject to:PminGi
≤ PGi≤ Pmax
Gi
B · θ = PG −PD
1
xij
(θi − θj) ≤ Pij,max
• The DC-OPF with the standard power flow equations contains both thepower generation PG and the voltage angles θ in the vector of theoptimization variables.
15 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Exercise
1 2
3
cG1 = 60 $/MWh, cG2 = 120 $/MWh
Pload = 150 MW
PmaxG1 = 100 MW, Pmax
G2 = 200 MW
X12 = 0.1 pu, X13 = 0.3 pu, X23 = 0.1pu, BaseMVA = 100 MVA
Pmax13 = 40 MW (line limit)
1 What are the optimization variables? Formthe optimization vector
2 Formulate the objective function
3 Formulate the constraints
16 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
DC-OPF in Matlab
How would you transfer yourproblem formulation to Matlab?
17 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Discussion Points• sin δ ≈ δ
• δ is in rad!
• B · θ = P
• B is in p.u.• θ is in rad, ⇒ dimensionless• P must be in p.u.
• Bus Admittance Matrix B in DC-OPF
• bij =1xij
⇒ positive
• all off-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This differs from the case where zij = rij + jxij . In thatcase, it is yij = gij + jbij with bij is negative.
• If the DC-OPF does not converge, check that the admittance matrix B iscorrect!
18 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Discussion Points• sin δ ≈ δ
• δ is in rad!
• B · θ = P
• B is in p.u.• θ is in rad, ⇒ dimensionless• P must be in p.u.
• Bus Admittance Matrix B in DC-OPF
• bij =1xij
⇒ positive
• all off-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This differs from the case where zij = rij + jxij . In thatcase, it is yij = gij + jbij with bij is negative.
• If the DC-OPF does not converge, check that the admittance matrix B iscorrect!
18 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Discussion Points• sin δ ≈ δ
• δ is in rad!
• B · θ = P
• B is in p.u.• θ is in rad, ⇒ dimensionless• P must be in p.u.
• Bus Admittance Matrix B in DC-OPF
• bij =1xij
⇒ positive
• all off-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This differs from the case where zij = rij + jxij . In thatcase, it is yij = gij + jbij with bij is negative.
• If the DC-OPF does not converge, check that the admittance matrix B iscorrect!
18 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Discussion Points• sin δ ≈ δ
• δ is in rad!
• B · θ = P
• B is in p.u.• θ is in rad, ⇒ dimensionless• P must be in p.u.
• Bus Admittance Matrix B in DC-OPF
• bij =1xij
⇒ positive
• all off-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This differs from the case where zij = rij + jxij . In thatcase, it is yij = gij + jbij with bij is negative.
• If the DC-OPF does not converge, check that the admittance matrix B iscorrect!
18 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
4-slide “break”
DC-OPF: linear program = convex
AC-OPF: non-linear non-convex problem in its original form⇒ recent efforts to convexify the problem
Why?
19 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Convex vs. Non-convex Problem
Convex Problem Non-convex problem
x
Cost f(x)
x
Costf(x)
One global minimum Several local minima
20 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Several local minima: So what?
Example: Optimal Power Flow Problem• Assume that the difference in thecost function of a local minimumversus a global minimum is 5%
• The total electric energy cost inthe US is ≈ 400 Billion$/year
• 5% amounts to 20 billion US$ ineconomic losses per year
• Even 1% difference is huge
• Convex problems guarantee thatwe find a global minimum ⇒convexify the OPF problem
x
Costf(x)
21 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Convexifying the Optimal Power Flow problem(OPF)
• Convex relaxations transform theOPF to a convex Semi-DefiniteProgram (SDP)
• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1
x
Costf(x)
Convex Relaxation
1Javad Lavaei and Steven H Low. “Zero duality gap in optimal power flow problem”. In:IEEE Transactions on Power Systems 27.1 (2012), pp. 92–10722 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Convexifying the Optimal Power Flow problem(OPF)
• Convex relaxations transform theOPF to a convex Semi-DefiniteProgram (SDP)
• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1
x
Costf(x)
f(x)
Convex Relaxation
1Javad Lavaei and Steven H Low. “Zero duality gap in optimal power flow problem”. In:IEEE Transactions on Power Systems 27.1 (2012), pp. 92–10722 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Convexifying the Optimal Power Flow problem(OPF)
• Convex relaxations transform theOPF to a convex Semi-DefiniteProgram (SDP)
• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1 x
Costf(x)
f(x)
Convex Relaxation
1Javad Lavaei and Steven H Low. “Zero duality gap in optimal power flow problem”. In:IEEE Transactions on Power Systems 27.1 (2012), pp. 92–10722 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Break is over... More in Dan Molzahn’s lecture tomorrow!
23 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF• Minimize
Costs, Line Losses, other?
• subject to:
AC Power Flow equations
Line Flow Constraints
Generator Active Power Limits
Generator Reactive Power Limits
Voltage Magnitude Limits
(Voltage Angle limits to improve solvability)
(maybe other equipment constraints)
Line Current Limits
Apparent Power Flow limits
Active Power Flow limits
• Optimization vector: [P Q V θ]T
24 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF• Minimize
Costs, Line Losses, other?
• subject to:
AC Power Flow equations
Line Flow Constraints
Generator Active Power Limits
Generator Reactive Power Limits
Voltage Magnitude Limits
(Voltage Angle limits to improve solvability)
(maybe other equipment constraints)
Line Current Limits
Apparent Power Flow limits
Active Power Flow limits
• Optimization vector: [P Q V θ]T
24 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF• Minimize
Costs, Line Losses, other?
• subject to:
AC Power Flow equations
Line Flow Constraints
Generator Active Power Limits
Generator Reactive Power Limits
Voltage Magnitude Limits
(Voltage Angle limits to improve solvability)
(maybe other equipment constraints)
Line Current Limits
Apparent Power Flow limits
Active Power Flow limits
• Optimization vector: [P Q V θ]T
24 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF• Minimize
Costs, Line Losses, other?
• subject to:
AC Power Flow equations
Line Flow Constraints
Generator Active Power Limits
Generator Reactive Power Limits
Voltage Magnitude Limits
(Voltage Angle limits to improve solvability)
(maybe other equipment constraints)
Line Current Limits
Apparent Power Flow limits
Active Power Flow limits
• Optimization vector: [P Q V θ]T
24 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF2
obj.function min cTPG
AC flow SG − SL = diag(V )Y∗busV
∗
Line Current |Y line,i→jV | ≤ Iline,max
|Y line,j→iV | ≤ Iline,max
or Apparent Flow |V iY∗line,i→j,i-rowV
∗| ≤ Si→j,max
|V jY∗line,j→i,j-rowV
∗| ≤ Sj→i,max
Gen. Active Power 0 ≤ PG ≤ PG,max
Gen. Reactive Power −QG,max ≤ QG ≤ QG,max
Voltage Magnitude Vmin ≤ V ≤ Vmax
Voltage Magnitude Vmin ≤ V ≤ Vmax
Voltage Angle θmin ≤ θ ≤ θmax
2All shown variables are vectors or matrices. The bar above a variable denotes complexnumbers. (·)∗ denotes the complex conjugate. To simplify notation, the bar denoting a complexnumber is dropped in the following slides. Attention! The current flow constraints are defined asvectors, i.e. for all lines. The apparent power line constraints are defined per line.25 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Current flow along a line
Vi Vj
Rij + jXij
jBij
2
jBij
2
π-model of the line
It is:
yij =1
Rij + jXij
ysh,i = jBij
2+ other shunt
elements connected to that bus
i → j : Ii→j = ysh,iVi + yij(Vi − Vj) ⇒ Ii→j =[ysh,i + yij −yij
] [Vi
Vj
]
j → i : Ij→i = ysh,jVj + yij(Vj − Vi) ⇒ Ij→i =[−yij ysh,j + yij
] [Vi
Vj
]
26 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Current flow along a line
Vi Vj
Rij + jXij
jBij
2
jBij
2
π-model of the line
It is:
yij =1
Rij + jXij
ysh,i = jBij
2+ other shunt
elements connected to that bus
i → j : Ii→j = ysh,iVi + yij(Vi − Vj) ⇒ Ii→j =[ysh,i + yij −yij
] [Vi
Vj
]
j → i : Ij→i = ysh,jVj + yij(Vj − Vi) ⇒ Ij→i =[−yij ysh,j + yij
] [Vi
Vj
]
26 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Current flow along a line
Vi Vj
Rij + jXij
jBij
2
jBij
2
π-model of the line
It is:
yij =1
Rij + jXij
ysh,i = jBij
2+ other shunt
elements connected to that bus
i → j : Ii→j = ysh,iVi + yij(Vi − Vj) ⇒ Ii→j =[ysh,i + yij −yij
] [Vi
Vj
]
j → i : Ij→i = ysh,jVj + yij(Vj − Vi) ⇒ Ij→i =[−yij ysh,j + yij
] [Vi
Vj
]26 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Line Admittance Matrix Yline
• Yline is an L×N matrix, where L is the number of lines and N is thenumber of nodes
• if row k corresponds to line i− j:
• Yline,ki = ysh,i + yij• Yline,kj = −yij
• yij =1
Rij + jXijis the admittance of line ij
• ysh,i is the shunt capacitance jBij/2 of the π-model of the line
• We must create two Yline matrices. One for i → j and one for j → i
27 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Bus Admittance Matrix Ybus
Si = ViI∗i
Ii =∑k
Iik,where k are all the buses connected to bus i
Example: Assume there is a line between nodes i−m, and i− n. It is:
Ii = Iim + Iin
= (yi→msh,i + yim)Vi − yimVm + (yi→n
sh,i + yin)Vi − yinVn
= (yi→msh,i + yim + yi→n
sh,i + yin)Vi − yimVm − yinVn
Ii = [ysh,im + yim + ysh,in + yin︸ ︷︷ ︸Ybus,ii
−yim︸ ︷︷ ︸Ybus,im
−yin︸︷︷︸Ybus,in
][Vi Vm Vn]T
28 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Bus Admittance Matrix Ybus
• Ybus is an N ×N matrix, where N is the number of nodes
• diagonal elements: Ybus,ii = ysh,i +∑
k yik, where k are all the busesconnected to bus i
• off-diagonal elements:
• Ybus,ij = −yij if nodes i and j are connected by a line3
• Ybus,ij = 0 if nodes i and j are not connected
• yij =1
Rij + jXijis the admittance of line ij
• ysh,i are all shunt elements connected to bus i, including the shuntcapacitance of the π-model of the line
3If there are more than one lines connecting the same nodes, then they must all be added toYbus,ij , Ybus,ii, Ybus,jj .29 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC Power Flow Equations
Si = ViI∗i
= ViY∗busV
∗
For all buses S = [S1 . . . SN ]T :
Sgen − Sload = diag(V )Y ∗busV
∗
30 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
From AC to DC Power Flow Equations• The power flow along a line is:
Sij = ViI∗ij = Vi(y
∗sh,iV
∗i + y∗ij(V
∗i − V ∗
j ))
• Assume a negligible shunt conductance: gsh,ij = 0 ⇒ ysh,i = jbsh,i.
• Given that R << X in transmission systems, for the DC power flow we
assume that zij = rij + jxij ≈ jxij . Then yij = −j1
xij.
• Assume: Vi = Vi∠0 and Vj = Vj∠δ, with δ = θj − θi.
I∗ij =− jbsh,iVi + j1
xij
(Vi − (Vj cos δ − jVj sin δ))
=− jbsh,iVi + j1
xij
Vi − j1
xij
Vj cos δ −1
xij
Vj sin δ
31 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
From AC to DC Power Flow Equations (cont.)
• Since Vi is a real number, it is:
Pij =ℜ{Sij} = Viℜ{I∗ij} = − 1
xij
ViVj sin δ
• With δ = θj − θi, it is:
Pij =1
xij
ViVj sin(θi − θj)
• We further make the assumptions that:
• Vi, Vj are constant and equal to 1 p.u.• sin θ ≈ θ, θ must be in rad
Then
Pij =1
xij
(θi − θj)
32 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Some additional points...
• Nodal prices
In a market context, the nodal prices are:• the lagrangian multipliers of the equality constraints Bθ = P• of a DC-OPF (at the moment)• with objective function the minimization of costs
• Power Transfer Distribution Factors (PTDFs)
• PTDFs are linear sensitivies that relate the line flows to the powerinjections
• the DC-OPF can be formulated with respect to PTDFs• PTDFs eliminate the need of θ as optimization variable• In the zonal pricing in Europe PTDFs are used to model the flowsbetween the zones
33 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Some additional points...
• Nodal prices
In a market context, the nodal prices are:• the lagrangian multipliers of the equality constraints Bθ = P• of a DC-OPF (at the moment)• with objective function the minimization of costs
• Power Transfer Distribution Factors (PTDFs)
• PTDFs are linear sensitivies that relate the line flows to the powerinjections
• the DC-OPF can be formulated with respect to PTDFs• PTDFs eliminate the need of θ as optimization variable• In the zonal pricing in Europe PTDFs are used to model the flowsbetween the zones
33 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
AC-OPF
• Resources about AC-OPF from the US Federal Energy RegulatoryCommission (FERC)
https://www.ferc.gov/industries/electric/indus-act/
market-planning/opf-papers.asp
• Overview paper on Economic Dispatch and DC-OPF:
R.D. Christie, B. F. Wollenberg, I. Wangesteen, TransmissionManagement in the Deregulated Environment, Proceedings of the IEEE,vol. 88, no. 2, February 2000
34 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Wrap-up
DC-OPF AC-OPF
• market clearing uses DC-OPF (atthe moment)
• convex
• can solve fast; can be applied invery large problems
but
• only active power flow
• no losses and no voltage levels
DC approximations more suitablefor transmission systems
• primarily used for optimization ofoperation and control actions
• full AC power flow equations
but
• non-convex (in its original form) →no guarantee that we find theglobal optimum
• computationally expensive andpossibly intractable for very largesystems
efforts to decrease computationtime and increase system size
35 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017
Thank you!
36 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 12, 2017