EXHIBIT 99.487
TRANSMISSION ACCESS AND PRICING WITH MULTIPLE SEPARATE ENERGY FORWARD MARKETS
<Table>
Paul R. Gribik George A. Angelidis Ross R. Kovacs
Perot Systems Corporation Pacific Gas and Electric Company Southern California Edison
Los Angeles, CA San Francisco, CA Rosemead, CA
</Table>
ABSTRACT: California's congestion management protocols provide comparable access
and prices to all users of the transmission system (power exchange and bilateral
contract parties). The users implicitly bid for capacity on major transmission
paths between zones. The Independent System Operator (ISO) allocates the
available transmission capacity on these paths so that it maximizes the value of
this capacity as measured by the users' bids. Everyone scheduling flow on a
congested path is charged the marginal-cost-based price for using the path. The
ISO keeps each party's portfolio of generation and load individually in balance
when adjusting schedules to relieve congestion on interzonal paths. By keeping
the portfolios of the different parties separate, the ISO clears its
transmission market without arranging energy trades between parties. Parties are
responsible for arranging their own trades. The ISO does not become involved in
the energy forward markets.
KEYWORDS: Transmission Congestion, Transmission Access, Transmission Pricing,
Independent System Operator, Power Exchange, Bilateral Transaction, Optimal
Power Flow, Utility Industry Restructuring, Power System Scheduling
I. INTRODUCTION
California's utilities, power producers, consumers and the California Public
Utilities Commission (CPUC) considered two different models for restructuring
California's electric power industry: a power pool model and a bilateral
contracts model. Several parties reached a compromise that combined the two
approaches. In addition, the CPUC's restructuring decision [1] put forth an
industry structure that combined the two approaches. Pacific Gas and Electric,
Southern California Edison, and San Diego Gas and Electric further developed the
industry structure and rules in their Phase I Filing [2] and Phase II Filings
[3, 4] to the Federal Energy Regulatory Commission (FERC).
A Power Exchange (PX) establishes a market in which generators and consumers bid
to sell and buy energy. Bilateral contract parties can operate their own
separate energy markets and schedule their energy transactions outside the PX's
market. An Independent System Operator (ISO), which is separate from the other
market participants, operates the grid.
According to the Phase I Filing, the ISO is to follow certain directives in
managing the transmission system. The ISO would be required to:
i. Provide comparable transmission service (prices and access) to the PX and
bilateral contract parties
ii. Price the use of major transmission paths at the marginal cost to send
appropriate economic signals with respect to limited transmission resources
iii. Restrict adjustments to the schedules submitted by the PX and bilateral
contract parties to adjustments that relieve violations of transmission
limits. (The ISO does not completely reschedule the pooled resources of all
parties to achieve a minimum cost schedule overall. The participants are
expected to reach an efficient outcome without ISO's intervention.)
These directives may conflict with each other [5].
Of particular concern, directive (ii) may not be compatible with directive
(iii). To develop marginal costs that reflect the economic value of transmission
capacity, the ISO must minimize the costs subject to appropriate constraints.
Suppose that the ISO places the resources of the bilateral contract parties and
the PX in a common pool when it reschedules to minimize cost and relieve
violations of transmission limits. (Hogan [6] presents the details of this type
of approach.) The ISO would arrange all trades between market participants that
serve to lower costs. This includes trades that reduce costs but that do not
alleviate violations of transmission limits. This violates directive (iii).
We refined the definition of the congestion management protocols to eliminate
ambiguities and the possibility of such conflicts. Completing their definition
while maintaining the compromise that led to the Phase I Filing required an
understanding of the rationale for adopting (i), (ii) and (iii).
Issues of equity and economics lie behind (i) and (ii). Directive (iii) tends to
reduce the influence of the ISO in the energy futures markets; it prevents the
ISO from compelling different parties to engage in energy trades. It is
necessary to understand some of the business reasons for this, e.g.:
I. The PX and the bilateral contract parties voluntarily provide the ISO with
prices for adjusting their generation and loads to relieve congestion. These
prices may reflect the incremental costs that an energy trader would incur as it
shifts
1
generation from its resources at one location to its resources at another
location. They may not be the prices at which the trader is willing to sell
energy to another energy trader. Consequently, giving the ISO the ability to
make all schedule changes needed to reach a least-cost schedule for a combined
pool would allow the ISO to force parties to engage in energy trades without
their expressed consent.
II. The different energy markets operate under their individual rules.
Differences in those rules could give rise to different strategies for
participants in the different markets. In particular, they may affect the energy
adjustment bids that are sent to the ISO for congestion management. For example,
in one market, a generator's fixed costs may be covered by a payment that is
separate from the payments it receives for energy. In another market, a
generator may only receive payments for energy in which case its fixed costs
must be recovered through its energy payments. In the first case, a generator
may find it advantageous to bid energy prices equal to its incremental energy
costs. In the second, a generator may find it advantageous to bid energy prices
equal to its incremental energy costs plus an estimate of the $/MWh it will need
to cover its fixed costs. If the different markets pass these bids to the ISO
for purposes of congestion management, the adjustment price curves from the
different markets could be shifted relative to one another. Placing all of the
resources in a common pool could result in mixing resources that include fixed
costs in their adjustment bids with those that do not include fixed costs in
their adjustment bids.
The first step in developing the congestion management protocols for the Phase
II Filings is to clearly delineate the markets treated:
o the energy market of the PX
o the energy markets of the bilateral contract parties
o the ancillary services markets
o the transmission market.
Under the congestion management procedures put forth in the Phase II Filings,
the ISO keeps the different energy and ancillary service markets separate as it
relieves congestion. These markets interact through the ISO's transmission
market.
The ISO uses the adjustment prices voluntarily provided by the PX and bilateral
contract parties to determine the value of the available transmission capacity
to each. The ISO adjusts the schedules to maximize the value that transmission
provides to the users. It prices transmission capacity at its marginal value to
the users.
To simplify the exposition, we only treat the mathematical derivation of nodal
and path-based marginal prices. We will not give the details of defining zonal
prices. In addition, we only treat the energy and transmission markets. The
results can be extended to cover ancillary services.
II. PARTICIPANTS IN ENERGY AND TRANSMISSION FORWARD MARKETS
Several entities are involved in the proposed industry structure:
1. Scheduling Coordinators (SCs)
- Manage their own energy forward markets
- Set the rules for their own energy forward markets
- Submit preferred schedules and adjustment bids to the ISO and work
with the ISO to adjust their schedules
2. California Power Exchange (PX) -- a specific scheduling coordinator
- Runs a forward market in which parties can bid to buy and sell energy
- Develops a preferred schedule for its forward market
- Develops marginal costs for its energy transactions
- Submits its preferred schedule and adjustment bids to the ISO and
works with the ISO to adjust its schedule
3. California Independent System Operator (ISO)
- Schedules the use of the transmission system in the forward market
- Develops marginal costs for using transmission
- Controls real-time operation after the forward markets.
These parties schedule their resources in forward markets and they operate in
real time. Scheduling takes place in two forward markets:
o Day-ahead (scheduling resources for each hour of the following day)
o Hour-ahead (scheduling resources for an hour -- a market for deviations
from the day-ahead schedule).
The PX and the other SCs specify their preferred schedules of generation and
loads in each hour of the relevant forward market (day-ahead or hour-ahead).
Each SC, including the PX, must balance its generation and its load (plus its
allocated share of losses) in its preferred schedule.
The PX and the other SCs may voluntarily provide adjustment bids that the ISO
will use for congestion management:
o Adjustment bids for generation consisting of price curves for generation
with generation limits
o Adjustment bids for load reduction consisting of price curves for reducing
load along with limits on reduction.
We only describe congestion management in the forward markets in this paper.
III. INTERZONAL AND INTRAZONAL TRANSMISSION CONGESTION MANAGEMENT
The ISO deals with two types of transmission constraints. Following the
framework of [2], the transmission system is divided into zones using experience
and engineering studies.
2
The interzonal transmission constraints consist of limits on transmission
between zones, e.g. power flow limits on interzonal paths. Intrazonal
transmission constraints consist of limits on parts of the transmission system
within a single zone, e.g. power flow limits on a line within a zone, bus
voltage limits, etc.
Within a zone, it is assumed that transmission constraint violations will be
relatively infrequent and of low cost to relieve. Between zones, it is assumed
that transmission constraint violations will be more frequent and cost more to
relieve.
The ISO combines the preferred schedules of the PX and SCs and checks for
violated transmission constraints in each hour. If there are transmission
constraint violations, the ISO adjusts the schedules to relieve the violations.
Fig. 1 shows the congestion management process in the day-ahead market. The
hour-ahead market is similar except that the ISO neither produces advisory
schedules nor does it give the PX and the other SCs an opportunity to revise
their schedules.
When adjusting schedules to relieve interzonal constraint violations, the ISO
maximizes the value of the available interzonal transmission capacity to the SCs
using it. The PX and other SCs using interzonal transmission pay congestion fees
based on differences in locational marginal costs.
We incorporate constraints in the interzonal congestion management procedures
that keep the energy forward markets of the PX and the other SCs separate. When
the ISO adjusts their schedules to eliminate interzonal transmission constraint
violations, it keeps each party's total schedule in balance. That is, the
adjusted aggregate generation of an SC (or the PX) must equal the adjusted
aggregate loads and allocated share of the losses for that SC (or the PX). These
constraints prevent the ISO from imposing trades between different SCs. For
interzonal transmission, the ISO only manages the transmission forward market.
It does not participate in the energy forward markets by arranging trades
between SCs.
[GRAPHIC]
Fig. 1: Day-Ahead Congestion Management
When adjusting schedules to relieve intrazonal transmission constraint
violations, the ISO adjusts the schedules of the PX and the other SCs as little
as possible. The ISO pays generators whose schedules are increased their bid
prices for the increased output. The ISO charges generators whose schedules are
reduced for replacing the reduced output at their bid prices. Changes to
scheduled load reductions are treated similarly. Any difference between the
total amounts paid and collected by the ISO is charged to all consumers in a
zone as an uplift. Marginal cost pricing is not used for intrazonal congestion
management.
In this paper, we focus on interzonal congestion management and pricing in a
single hour.
IV. TRANSMISSION MODEL
In the ISO protocols, losses are modeled by using generation meter multipliers
(GMMs). The scheduled generation at a location is multiplied by the specific GMM
for that location to provide power to cover losses. Since losses are treated by
GMMs, the interzonal congestion management process uses a lossless network model
to avoid double counting losses. Consequently, the interzonal congestion
management model treats effective generation. To account for the effect of
multiplying generation by the GMMs, we scale the adjustment price curves and the
ranges over which generation may be adjusted by the appropriate GMM for the
location.
The interzonal congestion management protocol uses a DC power flow model for
simplicity and for reliable, robust marginal cost calculation. The DC power flow
equations in matrix notation are written as:
[FORMULA]
(1)
where
3
[FORMULA] is the voltage angle at bus i
[FORMULA] is the real power injection at bus i
[FORMULA] is the real power load at bus i
The DC power flow equations are reviewed in Appendix I.
The interzonal transmission constraints on the California grid are limits on
real power flows on transmission paths between zones. These constraints are
linearized similarly as:
[FORMULA]
(2)
where
H is the linearized interface flow matrix, and
F MAX is the vector of interface flow limits.
V. INTERZONAL CONGESTION MANAGEMENT AND PRICING
For scheduling coordinator k (SC k):
P k is the vector of effective real power generation at buses i
[FORMULA]
D k is the vector of real power loads at buses i [FORMULA OMITTED]
[FORMULA] is the convex adjustment price function for effective generation at
bus i.
[FORMULA] is the convex price for reducing load at bus i.
The ISO solves a linearly-constrained optimal power flow problem to allocate and
price interzonal transmission:
[FORMULA]
(3a)
Subject to
[FORMULA]
(3b)
[FORMULA]
(3c)
[FORMULA]
(3d)
[FORMULA]
(3e)
[FORMULA]
(3f)
Constraints (3b) are the DC power flow equations, including the power balance
for the reference bus. Constraints (3c) are the flow limits on the interzonal
paths, and (3f) are the market separation constraints. The vector of voltage
angles at buses i [FORMULA] with bus N as reference so [FORMULA].
In solving (3), the ISO achieves several goals:
o The ISO schedules the use of interzonal transmission capacity to maximize
its value to the PX and other SCs.
o The ISO clears its forward market for interzonal transmission.
o The energy forward markets of the PX and other SCs are cleared
individually.
The Kuhn-Tucker conditions for the interzonal congestion management optimization
problem (3) provide the marginal prices that the ISO charges for the use of
congested interzonal transmission paths. The Lagrangian for (3) is
[FORMULA]
(4)
Let [FORMULA] for [FORMULA] and [FORMULA] be optimal for (3).
Furthermore, let [FORMULA] satisfy the Kuhn-Tucker conditions at the optimum
point.
The marginal cost of scheduling injection at bus j and equal withdrawal at bus i
does not depend upon the SC and is equal to [FORMULA]. The congestion charges to
SC k are:
[FORMULA]
(5)
The marginal value of capacity on the various interzonal paths are the elements
of [FORMULA]. The congestion charges to SC k can also be calculated by summing
the flow that SC k schedules on each path times the marginal value of capacity
on the path:
[FORMULA]
(6)
The superscript (CARET) indicates that the reference bus is removed.
The congestion charges as calculated by (5) and (6) are equal. Details of the
derivations are in Appendix II.
VI. EXAMPLE
Consider the example problem in Fig. 2 to illustrate the interzonal congestion
management procedure. We have two scheduling coordinators (SC1 and SC2), one of
which may be the PX, and three zones consisting of one bus each. We only treat a
single hour.
The preferred schedules, adjustment price bids, and flow limits on paths between
zones are given in Fig. 2. The adjustment range bid for each generator is 0 MW
to 200 MW. All loads are fixed and constant for the hour. No load reduction
adjustment bids are submitted by the SCs.
4
[GRAPHIC]
Fig. 2: Preferred Schedules and Adjustment Bids
The preferred schedules violate the flow limit on the line between buses 1 and
3. Solving the interzonal congestion management optimization problem (3), the
ISO determines the schedules and locational marginal costs given in Fig. 3.
The ISO determines the following congestion charges:
Transmission congestion charge to SC1
[FORMULA]
Transmission congestion charge to SC2
[FORMULA]
[GRAPHIC]
Fig. 3: Solution and Marginal Costs
To better understand the marginal costs, look at SC1's marginal cost of serving
load at bus 1. The cost of SC1's generator at bus 1 is $5/MWh while the marginal
cost for SC1 at bus 1 is $4/MWh. To understand this difference, look at SC1's
response to a 1 MWh increase in its load at bus 1.
SC1's optimal response is to increase [FORMULA] by 1 MWh at a generation cost of
$10. Because SC1 increased [FORMULA] it frees transmission that SC2 can use to
reduce its generation costs: SC2 increases [FORMULA] by 1 MWh at a generation
cost of $6 and reduces [FORMULA] by 1 MWh with a savings of $12. The total cost
to all SCs increases by ($10+$6-$12) = $4. This is the marginal cost to SC1 at
bus 1.
SC1 is not being altruistic. To see this, calculate SC1's costs before the load
at bus 1 is increased by 1 MWh: Generation Costs = $1300, Congestion Costs =
$300, Total = $1600. Now, calculate SC1's costs after the changes: Generation
Costs = $1310, Congestion Costs = $294, Total = $1604. SC1's total costs
(generation plus congestion) increase by $4. This is SC1's marginal cost at bus
1.
This example shows that an SC may increase its use of more expensive generation
if it can reduce its congestion charges by doing so. A scheduling coordinator
can even receive congestion payments by scheduling reverse flows on congested
transmission lines. Such reverse flows serve to increase the flows that other
scheduling coordinators may schedule on a congested path. In essence, a reverse
flow serves to increase the capacity of the path. A scheduling coordinator will
be paid the marginal value of capacity on the path for its reverse flow. All
SC's are shown the same marginal value for transmission capacity on a given
path.
VII. IMPLICIT VS. EXPLICIT BIDDING FOR INTERZONAL TRANSMISSION
In the congestion management protocols, an SC provides price information that
the ISO uses to allocate capacity on interzonal paths, namely adjustment bids
for its generators and adjustment bids for its loads. The value of interzonal
transmission capacity to the SC is implicit in these bids. The difference
between an SC's generation or load reduction prices at two locations gives the
value to the SC of transmission capacity connecting the two locations. In
solving (3), the ISO allocates the interzonal transmission capacity to the SCs
so that it maximizes the implicit value of the available capacity to the SCs.
We return to the example of the previous section and calculate the amount of
capacity that an SC would use on each path to send 1 MW from one bus to another.
For example, for SC1 to send 1 MW from bus 1 to bus 3, it would use 0.8 MW on
path 1-->3, 0.2 MW on path 1-->2, and 0.2 MW on path 2-->3. Based on the
difference between SC1's generation prices at buses 1 and 3, SC1 implicitly
values this "package" of transmission at $20/MW - 5/MW = $15/MW. Table 1 gives
the transmission capacity on each path that an SC would need to send 1 MW of
power from one bus to another. It also gives the value of the package of
transmission capacity to the SC.
SC1 wants transmission capacity to deliver 80 MW to bus 3. SC2 wants capacity to
deliver 120 MW to bus 3. An SC must also specify the amount of capacity that it
can use to move
5
power out of a location. In this case, each SC can generate up to 200 MW at each
bus, so each can export at most 200 MW from a location. The ISO can allocate up
to 100 MW on path 1-->3, 50 MW on path 1-->2, and 50 MW on path 2-->3.
Table 1: Transmission Capacity Needed to Send 1 MW and
its Value to a Scheduling Coordinator
<Table>
<Caption>
SC Buses MW MW MW Imputed
Sending 1 (From, Path Path Path Value
MW To) 1-->3 1-->2 2-->3 $/MW
=========== ======== ====== ======= ====== =========
SC1 (1,3) 0.8 0.2 0.2 15
SC1 (2,3) 0.4 -0.4 0.6 10
SC2 (1,3) 0.8 0.2 0.2 24
SC2 (2,3) 0.4 -0.4 0.6 18
</Table>
Let [FORMULA] be the amount of network capacity that the ISO allocates to SC k
to move power from bus i to bus j. To maximize the value of transmission, the
ISO would solve:
[FORMULA]
subject to
[FORMULA]
The solution is [FORMULA] [FORMULA] The corresponding allocation of capacity on
each path and the marginal value of transmission capacity on each path are given
in Table 2. These are identical to those determined by the congestion management
optimization (3).
Table 2: Transmission Allocation that Maximizes Value and
the Marginal Value of Capacity on Each Path
<Table>
<Caption>
Path 1-->3 Path 1-->2 Path 2-->3
================ ============== ============== ==============
Cap. to SC1 12 MW -12 MW 18 MW
- ---------------- -------------- -------------- --------------
Cap. to SC2 88 MW 12 MW 32 MW
================ ============== ============== ==============
Marg. Val. $19/MW $0/MW $4/MW
- ---------------- -------------- -------------- --------------
</Table>
The preceding analysis could be used as the foundation on which to develop an
alternate approach for the ISO's interzonal congestion management protocols.
Each SC would be required to bid explicitly the price that it is willing to pay
for a package of transmission capacity from one point on the network to another.
That is, each SC would explicitly bid the prices in Table 1. (The DC power flow
would again be used to determine the capacity required on each path to support
the transport of a unit of power from the source bus to the load bus.) Each SC
would also be required to specify the total amount of transmission capacity that
it wishes to acquire. The ISO would formulate an optimization problem as
outlined in this section to allocate the available capacity on the paths.
We briefly compare and contrast the two alternate methods by which the ISO can
have SCs bid for transmission capacity:
i. the implicit approach based on separation of markets leading to the
interzonal congestion management optimization problem (3)
ii. the explicit approach as outlined in this section.
Under the implicit approach, an SC's bid formulation problem is simpler and the
ISO's capacity allocation problem is smaller than under the explicit approach.
Under the explicit approach, SCs bid directly for transmission packages
connecting generation and load points. In a network with N buses, each SC may
conceivably bid on up to N2 transmission packages. Consequently, there may be up
to N2 variables per SC in the resulting optimization problem. Using the implicit
approach, each SC bids adjustments to generation and load at up to N locations.
Consequently, there are only up to 2N variables per SC in the optimization
problem (3).
VIII. CONCLUSION
Transmission capacity can be efficiently allocated in forward markets when the
energy forward markets of the various scheduling coordinators are kept separate
by the ISO. Each scheduling coordinator can devise strategies and take voluntary
actions to control its costs and gain profits while all participants face the
same transparent marginal price for using a congested transmission path. This
transparent marginal price for transmission supports efficient allocation of
transmission to the highest valued users. It also leads to a stream of hourly
short-run marginal price signals that can indicate the need for investment in
transmission facilities to avoid the stream of hourly congestion costs.
IX. ACKNOWLEDGMENTS
The authors would like to thank the many people who participated in the WEPEX
Congestion Management Subteam. Their open and frank discussion of the issues and
review of the methodologies and models were instrumental in the development of
this approach to interzonal congestion management. In particular, we would like
to acknowledge the work of Ashish Bhaumik, Bill Engelbrecht, Shangyou Hao,
Dianne Hawk, Carl Imparato, and Alex Papalexopoulos. We would also like to thank
Dariush Shirmohammadi for his comments and insight.
6
X. REFERENCES
[1] California Public Utilities Commission Decision 95-12-063 (December 20,
1995), as modified by Decision 96-10-009 (January 10, 1996).
[2] Joint Application of Pacific Gas and Electric Company, San Diego Gas &
Electric Company, and Southern California Edison for Authorization to
Convey Operational Control of Designated Jurisdictional Facilities to an
Independent System Operator, Federal Energy Regulatory Commission Docket
No. EC96-19000, April 29, 1996.
[3] The Phase II Filing of the California Power Exchange Corporation, Federal
Energy Regulatory Commission Docket Nos. EC96-19-001 and ER96-1663-001,
March 31, 1997.
[4] The Phase II Filing of the California Independent System Operator
Corporation, Federal Energy Regulatory Commission Docket Nos. EC96-19-001
and ER96-1663-001, March 31, 1997.
[5] P.R. Gribik, "Transmission Congestion Management and Pricing in Forward
Markets," Report to the WEPEX Congestion Management Subteam, September
1996.
[6] W.W. Hogan, "Contract Networks for Electric Power Transmission: Technical
Reference," John F. Kennedy School of Government, Harvard University, Dec.
1991.
[7] M.C. Caramanis, R.E. Bohn, and F.C. Schweppe, "Optimal Spot Pricing:
Practice and Theory," IEEE Transactions on Power Systems, Volume PAS-101,
No. 9, September 1982.
[8] A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in
Nonlinear Programming, Academic Press, New York, 1983.
APPENDIX I: DC POWER FLOW MODEL
Since interest in congestion management is not limited to power engineers, we
give a brief review of the development of the DC power flow equations.
Start with the real power balance equations at all buses:
[FORMULA] for all buses i (A1)
where
[FORMULA] is the voltage magnitude at bus i
[FORMULA] is the voltage angle at bus i
[FORMULA] is the real power injection at bus i
[FORMULA] is the real power load at bus i
and [FORMULA] is an element of the nodal admittance matrix.
The DC equations are derived from (A1) by assuming that:
o line resistances are negligible so [FORMULA]
o voltage angle differences will be small so [FORMULA]
o the voltage magnitudes will lie within a narrow range about a specified
voltage profile; for simplicity of exposition we assume a flat voltage
profile of 1 per unit.
Under these assumptions, we can write (A1) as:
[FORMULA] for all buses i
APPENDIX II: PRICING INTERZONAL CONGESTION
Calculating the locational marginal cost of serving load is well understood
(e.g. Caramanis et. al. [7]). Our problem involves a slightly more complicated
sensitivity analysis due to the market separation constraints, but the
mathematics is still well defined (e.g. Fiacco [8]).
To calculate an SC's marginal cost of serving a load at a location, we must
determine the rate at which total cost changes as that load varies in
optimization problem (3). To accomplish this, we perturb each SC's locational
loads, which appear in constraints (3b) and (3f), and treat the total cost as a
function of these perturbations:
[FORMULA]
(A2)
where the [FORMULA] element of the vector [FORMULA] is the perturbation of load
at bus i for SC k.
The marginal cost of serving load at bus i for SC k is
[FORMULA]
(A3)
If Total_Cost(o,...,o) is differentiable at (0,...,0), then scheduling
coordinator k's marginal cost of serving load at bus i in its energy forward
market is given by the Lagrange multipliers:
[FORMULA]
(A4)
7
If Total_Cost(o,...,o) is not differentiable at (0,...,0), then the marginal
costs for SC k depend upon the directions in which SC k's loads are moved. In
this case there are multiple sets of Lagrange multipliers that satisfy the
Kuhn-Tucker conditions, and (A4) gives directional derivatives for SC k.
PROPOSITION 1: The congestion price in a forward market for scheduling the use
of interzonal transmission between two locations is the same for all scheduling
coordinators.
PROOF: The congestion prices that the ISO charges scheduling coordinator k for
using interzonal transmission are based on the differences in SC k's locational
marginal costs in SC k's energy forward market. The ISO would charge SC k
[FORMULA]
(A5)
to inject one unit of energy at bus j and withdraw it at bus i. The right-hand
side of (A5) is independent of the scheduling coordinator selected. [ ]
PROPOSITION 2: The following ways of calculating an SC's transmission congestion
charge are equivalent:
o Sum over all buses: {the SC's scheduled withdrawal minus injection at the
bus times the SC's locational marginal cost at that bus}.
o Sum over all interzonal paths: {the SC's scheduled flow on a path times
marginal value of capacity on the path}.
PROOF: Using Proposition 1, the interzonal transmission charges to SC k under
the first approach are given by:
[FORMULA]
(A6)
Partitioning the matrix B and the vectors P* and D* by separating out the
reference bus N, we write the DC power flow equations (1) as:
[FORMULA]
(A7)
The vector of flows on the interzonal paths due to SC k is given by
[FORMULA]
(A8)
The Kuhn-Tucker conditions include
[FORMULA]
(A9)
Partitioning (A9) to separate the reference bus, rearranging terms, and
exploiting the special structure of B:
[FORMULA]
(A10)
where [FORMULA]
Using (3f), (A8) and (A10), we can write (A6) as:
[FORMULA]
(A11)
The final term in (A11) is simply the sum over all interzonal paths of the
product of the flow on an interzonal path due to SC k and the marginal value of
capacity on that path. [ ]
The congestion charges collected from the PX and other SCs will be distributed
to others, e.g. the transmission owners or the owners of financial rights. The
owners of a path (or the rights on a path) would receive the congestion charges
paid by the users of that path. The total due the owners could be defined as the
marginal value of capacity on the path times the available capacity on the path.
If we define the payments in this way, we have:
PROPOSITION 3: The congestion charges billed to the PX and other SCs equal the
payments due to the rights owners.
PROOF: The sum of interzonal congestion charges to the PX and other SCs is:
[FORMULA]
(A12)
The vector of flows on the interzonal paths is given by
[FORMULA] (A13)
Using (A10) and (A13), we can write (A12) as:
[FORMULA]
[GRAPHIC] by complimentary slackness.
The preceding results also hold for more complex problem formulations that treat
multiple hours and that incorporate inter-temporal constraints on generation and
load reduction such as ramping limits.
8
XII. BIOGRAPHIES
PAUL R. GRIBIK is an Associate with Perot Systems Corporation where he assists
utilities in developing methodologies and systems to improve their operations in
the face of changing regulations and increasing competition. He has provided
consulting services to the electric and gas industries for nine years through
Arthur D. Little, Mykytyn Consulting Group, and Perot Systems. He also worked
for the Pacific Gas and Electric Company for ten years. He received a B.S. in
Electrical Engineering in 1971, an M.S. in Industrial Administration in 1973 and
a Ph.D. in Industrial Administration and Operations Research in 1976 from
Carnegie-Mellon University.
GEORGE A. ANGELIDIS was born in Athens, Greece, in 1962. He received his
Ptychion degree from the Aristotle University of Thessaloniki in 1984, and his
M.A.Sc. and Ph.D. degrees from the University of Toronto in 1988 and 1992,
respectively, all in Electrical Engineering. He is currently with the Pacific
Gas and Electric Company where he works on issues related to the California
electricity industry restructuring. His research interests and expertise are in
advanced computer applications in large-scale electric power systems, with
emphasis on steady-state and dynamic analysis and optimization. Dr. Angelidis is
a member of IEEE and the Technical Chamber of Greece.
ROSS R. KOVACS (SM '87) received a BEE from Georgia Institute of Technology and
an MBA in Finance from Georgia State University. From 1977 to 1993, he worked
for the Southern Company in various areas including system planning,
transmission planning, financial analysis and electrical engineering. Since
1993, he has worked for Southern California Edison in the Grid Planning and
Strategy division where his major responsibilities have been power industry
restructuring, leading development of open access tariffs, and developing new
methods for a restructured power industry. Ross is a registered Professional
Engineer in the state of Georgia.
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