Notes on Felix Wu, P. Varaiya, P. Spiller and S. Oren
“Folk Theorems on Transmission Access: Proofs and Counterexamples”
Prepared by A. Kroujiline
____________________________________________________________
Objectives of the paper.
Folk theorems are the implicit assertions concerning the regulation of transmission
access, the determination of power flows, properties of economic dispatch, and the
operations of competitive nodal markets for flow. These theorems are formulated as
explicit mathematical assertions. Some of these assertions are proved to be true, whereas
the counterexamples are provided to negate other assertions, namely
1. uncongested lines do not receive congestion rentss (defined through node price
differences);
2. nodal prices clear markets for power only if allocation is efficient;
3. in an efficient allocation power can only flow from nodes with lower prices to nodes
with higher prices;
4. strengthening transmission lines or building additional lines increases transmission
capacity;
5. transmission capacity rights are compatible with any economically efficient dispatch.
Constraints.1
n
1. Kirchoff Law :
q
2. “Thermal” constraints:
qij  Cij
i 1
i
0
Economic dispatch.
Simultaneous generation and consumption are “netting” by measuring the cost and
benefit of the net injection qi by a single function Ci (qi ).2
1
The third set of constraints is briefly discussed by the authors. These constraints take into account a
possible disconnection of a generator or a transmission line due to the safety switches. Such a fault would
change the power flows in the remaining lines, causing possible “overheating” of some of the lines. The
authors argue that in this case, a proper framework requires differentiation of power by reliability.
2
The per node optimization problems are independent.
Assumptions on Ci (qi ):
1.
2.
3.
4.
increasing function;
convex function;
Ci (0) = 0;
Ci (qi) < 0  qi  0  for the net demander, - Ci (qi ) is the consumer benefit, and
the marginal benefit curve is decreasing;
5. Ci (qi) > 0  qi  0  for the net supplier, Ci (qi ) is the variable cost of
generation, and the marginal cost curve is increasing.
Minimization Problem:
n
 C (q )
Min

i
qi ,
i
(6)
i
i
n
qi  Yij ( i   j ), i  1,..., n
subject to:
(7)
j 1
(Supply equals demand for power at each node.)
qij  Yij ( i   j )  Cij , 1  i, j  n.
(8)
(“Thermal” constraints on transmission of power.)
Lagrangian:


   Ci (qi )   pi Yij (i   j )  qi    ij Yij ( i   j )  Cij
i
i
 j
 i, j

FOC:

0
qi

Ci (qi )
 pi , i  1,..., n
qi

(9)
(10)
(Consumer-supplier equilibrium at a node i : pi equals the marginal cost (benefit) if the
node is a net supplier (consumer). pi  0, according to the Assumption 1.)

0
 i

Y [ p
ij
i
 p j   ij   ji ]  0 , i  1,..., n
(11)
Complementary Slackness Condition:
ij [Yij ( i   j )  Cij ]  0, i, j  1,..., n
(12)
Definitions.
D1. A pair (q,  ) is an (optimal) economic dispatch if it solves (6)-(8)
D2. A triple (q,  , p) is a market equilibrium if (7), (8), (10) holds.
D. Optimal price regulation – making the system operator choose such (q,  ) that it will
fix the nodal prices at the Lagrange multipliers corresponding to the economic dispatch.
This regulation requires the “well behavior” of the agents (truthfully disclosure of the
cost and demand schedules by the consumers and suppliers, unbiased decision makings
by the regulator.)
“Economic Dispatch”  ”Competitive Equilibrium”
If “competitive” means market-clearing prices for power at each node that preclude
arbitrage opportunities from buying power at one node and selling at another, then any
market equilibrium satisfying D2 is competitive.
D3. The merchandizing surplus (MS) at a market equilibrium (q,  , p) is
MS   pi qi 
1
 ( pi  p j )qij
2 i, j
Interpretation: If we imagine a “market maker” who is a party to every purchase and sale
of power, MS is the profit resulting from this merchandizing activity.
Fact 2. At a market equilibrium, MS can be positive or negative.
Fact 3. A market equilibrium need not be an economic dispatch. Hence, a market
equilibrium need not be efficient.
Proof (sketch): Fig. 3 (page 8)
Fact 4. If the market equilibrium is also an economic dispatch, then MS is positive.
D4. Suppose (q,  ) is an economic dispatch. Let pi ,  ij be the associate Lagrange
variables. Then  ij is the (shadow) congestion price for the constraint Yij ( i   j )  Cij
(marginal value of increased thermal capacity,) and
 C
ij
ij
is the (shadow) congestion
i, j
rent.
Fact 5. Suppose (q,  ) is the economic dispatch and pi ,  ij are the associated Lagrange
variables. Then the MS equals the congestion rent
Proof (sketch). (11)*  + (11), then use (7) and (12) (this is the duality result.)
Case of an n-node network with exactly one line being congested.
Assumptions:
1. q1n = C1n (the flow between nodes 1 and n is binding)
2.  1n > 0
3.  ij = 0, for ij  1n (all other flows are not binding)
System (11) contains only n-1 linear independent equations (the matrix form of this
equation is given by formulae (13) on page 10. The sum of elements of each row of the
left-hand side matrix in (13) is zero.) Eliminating one of these equations and defining
i 
pi
, i =1, …, n-1, and
pn
m
1n
pn
,
the following matrix equation is obtained
Y  = h + gm ,
  Y1i
 Y12 

 Y21  Y2i 
Where Y  
 



 Yn 11  Yn 12 
 = (  1,  2, …,  n-1)T,

 Y1n 1 

 Y2 n 1 
 

 Yn 1i 
h = (Y1n, Y2n, …, Yn-1n)T,
 = Y-1 h + Y-1gm = Y-1 h --  m,
where   Y-1g
g = (-Y1n,0, …, 0)T
(15)
Fact 6. Y, h,  depend only on the network and not on the power injections. All entries
of Y-1 and  are positive. All entries of Y-1 h equal 1.
Equation 15 can be rewritten as
p i  p n   i  in,
i =1, …, n-1
(16)
Theorem 1 In any economic dispatch in which line 1n is the only congested line, the
nodal prices are related by (16). Here the  i depend only on line admittances, and
1n  0 is the congestion price for line 1n.
Consequences of Th. 1
1. If 1n = 0 all prices are equal (uncongested case).
2. Price pn is the largest price.
3. If  i   j then pi  p j , even if line ij is not congested. Hence, the nodal price
differences do not equate to congestion prices (  ij =0 ).
4. If  i   j then pi  p j , independently of the power flow direction. So, in any
network (which is not radial) one can construct an economic dispatch involving a line
flow from a higher priced node to a lower priced node.
5. The strengthening of a line (increasing its admittance) may lead to a larger minimum
cost.
30
N1
congested line
25
5
35
5
N2
10
Line 13 is assumed to be congested. By Th 1,

N3
p3 > max {p1,p2}
MS   pi qi = 35p2 – 5p3 – 30p1 < 35(p2 – p1)
According to Fact 4, MS > 0 (economic dispatch)

p1 < p2 < p3
(17)
If line 23 is strengthened, then the flow in line 13 will increase beyond the thermal
capacity of the line (given the same nodal injections.) The economic dispatch will reduce
injection in node 1, leading to a larger minimum cost.
Bilateral Contracts.
Consider the economic dispatch of Fig. 5. Supplier at node 3 will enter the contract with
consumer at node 2 only if the contract price is greater than p3. On the other hand, the
consumer will enter this contract only if the contract price is less than p2. It was shown
before that p3 > p2. Hence, there will be no bilateral contract between node 2 and 3, and
the efficient allocation won’t be achieved.
Two ways of achieving efficiency are considered:
1. Trilateral contracts.
If there is no injection at node 3, the power flow in line 13 will be larger than the thermal
limit of the line. Hence, the injection in node 1 will be reduced by economic dispatch,
reducing the profits at this node. The supplier at node 1 may be willing to purchase some
of the power at node 3 at price p3 and resell it then to node 2 at price p2.
2. System Operator.
The bilateral contracts will be sustained if there is a system operator who imposes a per
MW “transmission charge” of pi  p j (positive or negative) on any bilateral contract to
transfer power from j to i. One of the objections to this method is that an effective
competition in a bilateral transactions market requires knowledge of transmission
charges, obtained a priori or via an adjustment process in which transmission charges
converge to those corresponding to the economic dispatch.
Transmission Capacity Rights. (TCR)
Following Hogan’s paper:
1. “A TCR is defined as the right to put power in one bus and take out the same
amount of power at another bus in the network.”
A TCR is any triple (i, j, Tij) with Tij  0. The matrix of allocated rights is defined as
Ta = { Tija  0, i, j = 1,2, …, n}
2. “We assume that the simultaneous use of all allocated rights is feasible.”
Let
qa = { qia   (Tija  T jia ), ii = 1, …, n}
and
FI = {q: Eq-s (7) and (8) are satisfied}
Then,
qa  FI
(18)
(19)
3. “However, in the contract network, we amend the definition of a capacity right
to allow for either specific performance or receipt of an equivalent rental
payment.”
It is necessary to introduce a system operator, Maxop with the following attributes:
1.
2.
3.
4.
Maxop manages a nodal spot market with the equilibrium prices p = (p1, p2, …, pn)
Maxop select injections so that (q, p) is a ME as in D2.
Maxop collects MS.
Maxop pays to the Tija rights holders a “rent” of (pj – pi) Tija
Two models of TCR:
1. Virtual Capacity Rights (VCR) – weak specification based on 2 and 3 above.
An agent who holds VCR of Tij MW is just indifferent between (i) purchasing Tij MW of
power at i at price pi and having Maxop transfer that power to j, and (ii) collecting a rent
of (pj – pi) Tij and purchasing Tij MW of power at j at price pj.
The VCR may be sold in a secondary market, like any other financial asset. Transmission
rights in the form of VCR means that some agents are initially given claims to portions of
Maxop’s MS. Walsh suggested that initially VCR will be distributed among initial grid
owners to “protect them from economic effects of increased grid congestion in the
future.” Two problems arise within this approach:
a) some VCR-holders may receive a negative income stream (the holder of VCR T32 at
Fig.5 will receive the rent of (p2 – p3)T32 < 0.
b) The MS --pTq will not equal to the rents –pTqa distributed to the holders.
Theorem 2 Suppose q selected by Maxop is the economic dispatch, and suppose the spot
prices equal the dual variables of economic dispatch. Then, Maxop will retain a positive
balance, i.e. --pTq  –pTqa
2. Actual Capacity Rights (ACR) – strong specification based on 1 and 2 above.
A rights holder is allowed to exercise that right or not. There is a need to introduce a
system operator, Minop, who selects a set of nodal prices and a dispatch q0. The Minop
uses the remaining injections to support some or all the transactions requested by other
agents that don’t hold ACR. This will lead to the “second best solution” (see Fig. 6).