6. The DC Optimal Power Flow
Quantitative Operational Energy Economics
Anthony Papavasiliou
1 / 36
6. The DC Optimal Power Flow
1
Centralized Optimal Power Flow
The OPF Using PTDFs
The OPF Using Phase Angles
2
Decentralized DC Optimal Power Flow
Bilateral Trading and Competitive Prices
Locational Marginal Pricing
2 / 36
Table of Contents
1
Centralized Optimal Power Flow
The OPF Using PTDFs
The OPF Using Phase Angles
2
Decentralized DC Optimal Power Flow
Bilateral Trading and Competitive Prices
Locational Marginal Pricing
3 / 36
Equivalent Models of DC Power Flow
DC power flow equations: System of linear equalities where
nodal power injections imply unique combination of power
flows:
using nodal power injections, and translating to power
flows through PTDFs (Fkn )
using nodal power injections, and translating to power
flows through reactance (Xk )
Optimal power flow problem (OPF): System operator
regulates injections such that welfare is maximized, while
respecting the transmission constraints and operating
constraints of generators and loads.
4 / 36
The OPF Using PTDFs
max
pg ,dl ≥0
X
X
fl (dl ) −
l∈L
fg (pg )
g∈G
s.t. pg ≤ Pg , (µg ), dl ≤ Dl , (νl )
−
fk ≤ Tk , (λ+
k ), −fk ≤ Tk , (λk )
X
fk −
Fkn rn = 0, (ψk )
n∈N
rn −
X
pg +
g∈Gn
X
X
dl = 0, (ρn )
l∈Ln
rn = 0, (φ)
n∈N
5 / 36
Notation and New Constraints
Notation
N: nodes in the network, including the hub
K : transmission lines
Gn , Ln : set of generators / loads located in bus n
rn : power injected in node n (and shipped to the hub)
Fkn : PTDF of bus n on line k
Constraints
Thermal and stability limits on flow of power over lines
Translation of power injections to flows through PTDFs
(since h ∈ N, we set Fkh = 0)
Power balance constraints
Definition of rn as net amount of power injected in node n
6 / 36
KKT Conditions of the OPF Using Bus Angles
X
rn = 0, fk −
n∈N
X
n∈N
−
X
X
Fkn rn = 0 , rn −
Fkn ψk + ρn + φ = 0 ,
λ+
k
pg +
X
g∈Gn
l∈Ln
λ−
k
+ ψk = 0
−
dl = 0
k ∈K
0 ≤ µg ⊥ Pg − pg ≥ 0 , 0 ≤ νl ⊥ Dl − dl ≥ 0
−
0 ≤ λ+
k ⊥ Tk − fk ≥ 0 , 0 ≤ λk ⊥ Tk + fk ≥ 0
0 ≤ pg ⊥ ∇fg (pg ) + µg − ρn(g) ≥ 0 ,
0 ≤ dl ⊥ −∇fl (dl ) + νl + ρn(l) ≥ 0
n(g) and n(l): node where generator g / load l are located
7 / 36
Analogies to Economic Dispatch KKT Conditions
For 0 < pg < Pg : ρn = ∇fg (pg ). For 0 < dl < Dl :
ρn = ∇fl (dl )
For pg = Pg : ρn ≥ ∇fg (Pg ). For dl = Dl : ρn ≤ ∇fl (Dl )
For pg = 0: ρn ≤ ∇fg (0). For dl = 0, ρn ≥ ∇fl (0)
ρn in OPF resembles role of system lambda in economic
dispatch:
Generators with marginal costs below ρn should produce,
loads with marginal benefit above ρn should consume
Generators with marginal cost above ρn should not
produce, loads with marginal benefit below ρn should not
consume
8 / 36
Sensitivity Interpretation of OPF Dual Variables
φ: marginal change in welfare from a marginal requirement
to produce, or a marginal requirement to decrease
consumption, in the hub node (expected to be negative)
−
λ+
k and λk : marginal benefit of increasing line capacity
ρn : marginal increase in welfare from a requirement to
marginally increase consumption, or a requirement to
marginally decrease generation, in node n
ρn for inelastic demand: marginal cost increase by
marginally increasing required production in a certain
location, or marginal savings from marginally decreasing
required production (since demand and consumer benefit
in perturbed problem remains the same). Very helpful
interpretation in understanding locational marginal prices.
9 / 36
Components of ρn
From KKT conditions of OPF:
X
X
ρn = −φ +
Fkn λ−
−
Fkn λ+
k
k
k ∈K
k ∈K
Since Fkh = 0, ρh = −φ
−
Obviously, λ+
k ⊥ λk . If line is congested in forward
direction, it contributes −Fkn λ+
k . If line is congested in
negative direction, it contributes Fkn λ−
k.
Interpretation: marginal increase in available power at
node n can be consumed there (marginal benefit ρn ).
Alternatively, this marginal production can be shipped to
the hub node (marginal benefit of −Fkn λ+
k if fk = Tk , or
−
marginal benefit of Fkn λk if fk = −Tk ) and consumed at the
hub (marginal benefit −φ).
10 / 36
OPF for a 2-Node Network
A
B
∼
∼
Suppose
CA = 50$/MWh, CB = 60$/MWh
DB = 100MW, high value of lost load
No capacity limit on line
Optimal solution
Only A produces
ρA = 50$/MWh (since unit A strictly within its operating
region)
ρB = 50$/MWh (since buses A and B are indistinguishable)
11 / 36
OPF for a 2-Node Network (Case 2)
Suppose TAB = 100MW
Optimal solution
Same optimal dispatch
ρA = 50$/MWh (since 0 < pA < PA )
Since generator B is not producing, ρB ≤ 60$/MWh
Choose A as hub. FAB,B = −1 (reference direction of line
arbitrarily chosen as A → B).
Marginal increase in capacity of line: no additional benefit
Marginal decrease: generator B produces, generator A
backs off. Marginal cost increase of 10$/MWh.
Therefore, 0 ≤ λ+
AB ≤ 10
ρB = ρA + λ+
AB , therefore 50$/MWh ≤ ρB ≤ 60$/MWh
(same conclusion from sensitivity interpretation of ρB )
12 / 36
OPF for a 2-Node Network (Case 3)
Suppose TAB = 100MW and DB = 110 MW
Solution:
pA = 100MW, pB = 10MW
ρA = 50$/MWh, ρB = 60$/MWh
13 / 36
Motivation
Reasons for examining alternative formulation:
Describes models where topology of network changes
(e.g. transmission line switching or transmission line
outages)
Different KKT conditions suggest different market designs.
Practical viability of alternative market designs is fiercely
debated in the industry, but the blueprint should be inspired
from KKT conditions of OPF.
14 / 36
The OPF Using Phase Angles
max
pg ,dl ≥0
−
X
g∈Gn
X
fl (dl ) −
l∈L
pg −
X
fg (pg )
g∈G
X
k =(·,n)
fk +
X
dl +
l∈L
X
fk = 0, (ρn )
k =(n,·)
fk − Bk (θm − θn ) = 0, k = (m, n), (γk )
−
fk ≤ Tk , (λ+
k ), −fk ≤ Tk , (λk )
pg ≤ Pg , (µg ), dl ≤ Dl , (νl )
Differences:
Power balance constraints (imposed node-by-node,
requires that production plus incoming flows equal
consumption plus outgoing flows)
Power flow equations using Bk (inverse of reactance Xk )
15 / 36
KKT Conditions of the OPF Using Line Flows
New KKT conditions:
X
pg +
g∈Gn
X
k =(·,n)
fk =
X
dl +
l∈L
X
fk
k =(n,·)
fk = Bk (θm − θn ), k = (m, n)
γk + λ+
− λ−
k − ρn + ρm = 0, k = (m, n)
Xk
X
−
Bk γk +
Bk γk = 0, (θn )
k =(n,·)
k =(·,n)
ρn retains same meaning as in previous OPF model
Conclusions regarding how dispatch depends on ρn
remain the same
16 / 36
Relationships Among Dual Multipliers
−
ρn − ρm = λ+
k − λk + γk , k = (m, n)
Congestion in the positive direction (λ+
k > 0) tends to
cause (but does not guarantee) higher marginal value ρn ,
relative to ρm . The opposite true when λ−
k < 0.
Interpretation: if marginal value of power is higher in one
location than in another, power should be flowing from
region of lower valuation to region of higher valuation
This is not always the case. We will see a counterexample.
17 / 36
Table of Contents
1
Centralized Optimal Power Flow
The OPF Using PTDFs
The OPF Using Phase Angles
2
Decentralized DC Optimal Power Flow
Bilateral Trading and Competitive Prices
Locational Marginal Pricing
18 / 36
Loop Flows and Contract Paths
Economic dispatch can be organized using markets. Presence
of a transmission network complicates things.
Heart of the problem: loop flows. A trade between two
market participants in an electric power network causes
flows over all lines of a network.
Contract path: a path over which a traded good will travel.
This does not exist in electricity.
In early days of deregulation loop flows were ignored and
contract paths were used anyways. They failed.
19 / 36
Example: Failure of Contract Paths [Hogan, 1992]
∼
∼
2
1
3
Generators in nodes 1 and 2, load in node 3
Line 1-3 can carry 600MW
Electrical network is symmetric: Every MW from node 1 to
load uses (2/3)MW of line 1-3. Every MW from node 2 to
load uses (1/3)MW of line 1-3.
20 / 36
Why Contract Paths Fail
How many rights should be issued for the contract path from
generator region to load region?
Option 1: 900MW
Advantage: no matter which generators produce, line 1-3
will never be overloaded (since the most stress occurs
when node 1 is shipping power)
Disadvantage: even if generators in node 2 are cheaper,
the market limits their use
Option 2: 1800MW
Advantage: we can fully utilize generators in node 2
Disadvantage: line 1-3 will be overloaded if generators in
node 1 use the rights
Conclusion: by using rights on contract paths we either limit the
amount of trade to inefficient levels, or we violate the power
transfer limits of lines
21 / 36
Externalities and Markets for Transmission
Rerouting of power from bilateral trade over multiple links of the
network causes externalities.
Externalities are consequences from the actions of an
economic agent on the welfare of other agents
Optimization interpretation: externalities are coupling
constraints in economic agents’ optimization problems that
are not accounted for because their violation is not
penalized
In order to deal with externalities, economics recommends the
creation of a market for the scarce resource (transmission).
The exact definition of the product ‘transmission rights’ leads to
different market designs.
22 / 36
Bilateral Trading and Competitive Prices [Stoft, §5-3.1]
A
B
∼
∼
Bus A: remote bus with cheap generation, low demand:
∇fA (pA ) = (20 + pA /50)$/MWh
DA = 100MW
Bus B: city bus with expensive generation, high demand
∇fB (pB ) = (40 + pB /50)$/MWh
DB = 800MW
TAB = 500MW
Obvious definition of transmission rights
Suppliers can sell to consumers located in the same node
Suppliers can sell power to the other bus, but then they (or
the buyers) need to buy transmission rights
23 / 36
Competitive Price of Transmission Rights
Competitive equilibrium requires that no opportunities for
profitable trade exist:
ρB ≤ ρA + λAB , otherwise consumers at B would profitably
change to importing from A
ρB ≥ ρA + λAB , otherwise consumers at B would profitably
change to buying power from B
Therefore, λAB = ρB − ρA
24 / 36
Formal Derivation of λAB
Formally, denote dA and dB as power bought at locations A and
B from loads at B:
max −ρA dA − ρB (DB − dA ) − λAB dA
dA ≥0
dA ≤ DB , (µ)
KKT conditions:
0 ≤ µ ⊥ DB − dA ≥ 0
0 ≤ dA ⊥ µ + ρA − ρB + λAB ≥ 0
Generators in A will export to B, because their marginal
cost after having covered DA is (20 + 100/50) = 22$/MWh
(cheaper than cheapest generators in B)
dA < DB since TAB < DB
From KKT conditions, µ = 0 and λAB = ρB − ρA
25 / 36
Solution of 2-Node Bilateral Trading System
Suppose the line were not congested. Then generators at
A would cover entire demand at B at marginal cost of
(20 + 900/50) = 38$/MWh. Therefore, line must be
congested.
Generators at A produce pA = 600MW, generators at B
produce pB = 300MW
Nodal prices are ρA = 32$/MWh and ρB = 46$/MWh
Transmission rights prices are λAB = 14$/MWh
The same allocation is obtained from the centralized solution of
the OPF problem
26 / 36
Locational Marginal Pricing
Originally proposed by Schweppe, Caramanis, Tabors and
Bonn in 1988
Proposition adopted about 15 years later in most US
electricity markets and elsewhere
Experience suggests that this has been the most
successful framework for organizing electricity markets
with transmission constraints
LMP pricing is considered tabu in Europe
LMP design generalizes uniform pricing auction for
economic dispatch
27 / 36
Rules of the Auction
Generator bids. Suppliers submit incremental
price-quantity bids
Consumer bids. Loads submit decremental price-quantity
bids
Obligations and payoffs. System operator solves OPF
problem and uses ρn as uniform market clearing price for
bus n.
Supply bids that are ‘in the money’ (i.e. bid below ρn ) have
to supply and receive ρn per MWh. Supply bids that are ‘out
of the money’ should not be produced and receive no
payoff.
Demand bids that are ‘in the money’ (i.e. bid above ρn )
need to consume and pay ρn per MWh. Demand bids that
are ‘out of the money’ cannot consume and do not pay
anything.
28 / 36
Justification of LMP Pricing
In order to follow format of auction: constant marginal costs Cg
for generators, constant valuations Vl for loads.
For all g ∈ Gn : {max ρ?n pg − Cg pg |pg ≤ Pg , (µg )} with
pg ≥0
corresponding KKT conditions:
0 ≤ pg ⊥ Cg − ρ?n + µg ≥ 0
0 ≤ µg ⊥ Pg − pg ≥ 0
For all l ∈ Ln : {max Vl dl − ρ?n dl |dl ≤ Dl , (νl )} with corresponding
dl ≥0
KKT conditions:
0 ≤ dl
⊥ ρ?n − Vl + νl ≥ 0
0 ≤ νl
⊥ Dl − dl ≥ 0
29 / 36
Justification of LMP Pricing Continued
System operator solves centralized OPF using market
participant bids. KKT conditions of OPF include:
0 ≤ µg ⊥ Pg − pg ≥ 0
0 ≤ νl
⊥ Dl − dl ≥ 0
0 ≤ pg ⊥ Cg + µg − ρn(g) ≥ 0
0 ≤ dl
⊥ −Vl + νl + ρn(l) ≥ 0
KKT conditions of generators are identical to KKT
conditions of centralized OPF when ρ?n = ρn(g)
KKT conditions of loads are identical to KKT conditions of
centralized OPF when ρ?n = ρn(g)
Conclusion: if ρn(g) is used as a uniform market clearing price,
generators and loads will be dispatched optimally.
30 / 36
Locational Marginal Pricing in PJM
Figure: LMPs in PJM on February 15, 2014. 05:40 (upper left), 08:40
(upper right), 09:20 (lower left), 09:55 (lower right).
31 / 36
Example: LMPs in a 3-Node Network [Hedman]
∼
∼
C
A
B
∼
CA = 50$/MWh, CB = 70$/MWh, CC = 100$/MWh
Symmetrical transmission network
DA = 0MW, DB = 50MW, DC = 100MW
32 / 36
LMPs in a 3-Node Network: Case 1
Suppose all lines have limit of 25MW. Optimal solution:
pA = 25MW, pB = 75MW, pC = 50MW
fAB = 0MW, fBC = fAC = 25MW
In order to confirm, examine cost impact of marginal changes:
Marginal decrease in production of A is costlier
+1MW from A (+50$/h): compensated by +1MW from C
(+100$/h) and -2MW from B (-2 · 70$/h). Net effect: +10$/h.
-1MW from B (-70$/h): compensated by +(1/2)MW from A
(+(1/2) · 50$/h) and +(1/2)MW from C (+(1/2) · 100$/h).
Net effect: +5$/h.
+1MW from B (+70$/h): compensated by +1MW from C
(+100$/h) and -2MW from A (-2 · 50$/h). Net effect: +70$/h.
Marginal decrease in production of C is infeasible
Marginal increase in production of C is costlier
33 / 36
LMPs in a 3-Node Network: Case 1 Continued
ρA = 50$/MWh, ρB = 70$/MWh and ρC = 100$/MWh since
all units are strictly within their operating range
What did we learn? It is possible for two buses that are
connected by a line which is not congested to have
different LMPs. The fact that a line is not congested does
not mean we can transport power over that line from a
cheap bus (bus A) to an expensive bus (bus B).
34 / 36
LMPs in a 3-Node Network: Case 2
Suppose all data as in case 1, except TBC = 50MW and
TAC = 60MW. Solution:
pA = 80MW, pB = 70MW, pC = 0MW
fAB = 20MW, fBC = 40MW and fAC = 60MW
ρC = $90/MWh: +1MW of demand in load C results in
-1MW from A (-50$/MWh), and + 2MW from B
(+140$/MWh). Net effect: 90$/MWh.
35 / 36
LMPs in a 3-Node Network: Case 3
Suppose all data as in case 1, except TAB = 25MW,
TBC = 50MW, TAC = 50MW. Solution:
pA = 50MW, pB = 100MW, pC = 0MW
fAB = 0MW, fBC = fAC = 50MW
If load at node C were to increase, the only way to serve it
would be by an increase in the output of generator C.
Therefore, ρC = 100$/MWh is a valid LMP.
If we reduce consumption in C by 1 MW we get +1MW from
A (+50$/MWh) and -2MW from B (-140$/MWh). Net effect:
-90$/MWh. Therefore, ρC = 90$/MWh is also a valid LMP.
36 / 36
© Copyright 2024 ExpyDoc
