CLEARING PRICES COMPUTATION FOR INTEGRATED GENERATION,
RESERVE AND TRANSMISSION MARKETS
Shangyou Hao Dariush Shirmohammadi
Perot Systems Corporation
Los Angels, CA USA
Abstract: In a competitive electricity market, trading and scheduling for
electric energy, transmission and reserve related services are often the
centerpieces of market functions. The operation of these three markets may
reside in centralized pools, system operators or power exchanges. However, they
are often conducted separately and sequentially in practice.
This paper sets forth a new formulation to determine schedules and clearing
prices for integrated electric energy, transmission and reserve markets
simultaneously. The formulation yields locational market clearing prices when
transmission congestion is present. Example results based on the new formulation
are presented.
I. INTRODUCTION
Complex interactions between daily generation schedules, transmission network
and reserve requirement need to be accounted for in determining schedules and
clearing prices of a competitive electricity market. The delivery of electric
energy is not possible without the use of transmission system and capacity
reserves. In many established electricity markets, energy scheduling,
transmission congestion and reserve management are decomposed into component
products, allowing these products to be traded in centralized power pools,
system operators or power exchanges.
However, these component products are often traded and scheduled separately and
sequentially in practice. For example, in the California [1], day-ahead energy
schedules are determined by Scheduling Coordinators; transmission congestion is
managed by the Independent System Operator (ISO); and unloaded capacities along
with capacity price bids are then submitted into regulation, spinning,
non-spinning and replacement reserve markets. Each market is independently
priced and scheduled at different time. Similar arrangements exist in the
electricity markets of New York [2], PJM [3] and others around the world.
In the power pool of England and Wales, dispatch orders and prices for energy
are determined using unconstrained system marginal price that is set at the
highest offer price of generating units being dispatched [4]. However, when
transmission constraints are present, constrained dispatch is executed and
schedules are adjusted. The rescheduling costs are then charged to consumers as
transmission service uplift.
Many have studied formulations and algorithms of generation scheduling and
pricing methodologies, Ma [5] reviewed different dispatch algorithms such as
merit order dispatch, sequential and joint dispatch of energy and reserve. He
proposed a joint dispatch algorithm that minimizes costs and incorporates
several zonal interface constraints with heuristics to account for inter-zonal
trading. Hybrid method of solving for multi-commodity schedules is investigated
by Kwok [6] for the interim ISO New England. Post [7] studied how energy
schedules can be determined by a central power pool to minimize total generation
costs while individual bidders construct their bids to maximize profits. Guan
[8] discussed the limitation of using ramp rates in Unit Commitment algorithm
and proposed a concept of realizable schedules. Auction design and schedule
issues related to using different objectives were studied in [9]. Sheble [10]
proposed a framework for coordinating financial and physical electricity
markets.
In [11], Cadwalader recognized the need for security-constrained, economic
dispatch methodologies for computing market equilibrium of these component
markets. He pointed out that prices are the integral part of the market
equilibrium along with the schedules that maximize the bidders' profits or
utility. However, in the proposed formulation, clearing prices are still the
by-products of an OPF cost minimization problem and derived from the dual
variable of the solution. Chattopadhyay [12] discussed the modeling requirement
for price based generation scheduling and examined the importance to add
transmission constraints in scheduling energy and reserves.
The formulations studied so far as based on variations of the unit commitment or
OPF techniques. While traditional unit commitment techniques can be used to
minimize bidding costs, clearing prices for generation capacity and transmission
are difficult to compute. On the other hand, while OPF techniques have been
deployed to compute transmission prices, it has difficulties for scheduling
reserves and enforcing clearing pricing rules.
Our objective is to set forth a new formulation that solves for schedules and
clearing prices for electric energy, transmission and reserve capacity markets
simultaneously. This should lead to better economic efficiency and reduced
transactions costs due to the integration of the three component markets and
simplification of market processes.
The paper has five main sections. After the introduction, Section II lists the
notations used in this paper. The proposed formulation is described in detail in
Section III. We will also discuss the features and properties of the new
formulation. Section IV presents application examples. Section V contains a
summary of the paper.
0-7803-6681-6/01/$10.00(C) 2001 IEEE 262
II. NOTATION
[Formula Omitted]
III. FORMULATION
A. Market Structures
Various pricing and scheduling rules have developed in order to address unique
issues facing each market. However, there are some common attributes for the
established electricity markets. In order to set up the formulation, we make a
few assumptions about the market structures used in this paper. These
assumptions are intended to elaborate the proposed algorithm and solution;
however, the algorithm and solution are not necessarily limited by the
assumptions.
Specifically, we consider the following market rules:
- - One reserve market is assumed and the impact of ramp rate is ignored.
- - The DC transmission system model is used and the B matrix is symmetric (no
phase shifters).
- - No double auction is considered for simplicity. This means that demands
are treated as inelastic.
- - Schedules of different time periods are independently calculated. That
means that hourly generation schedules are determined independently.
- - Both the energy and capacity demands are known.
- - Market pricing mechanism is based on marginal cost and clearing prices
principles. When there is no congestion, uniform prices are used for all
bidders.
We assume that bidders submit to the market operator a monotonically increasing
and non-negative price curve as a function of output quantity. This curve
specifies the minimum price that the bidders will accept for the unit to operate
at that level. The curve will be used by the market operator to schedule reserve
capacity and to compute reserve clearing prices. Further, the energy payment
from the reserved capacity when being called upon during real-time operation,
will be equal to or more than that of the forward energy clearing prices.
As shown in Figure 1, a generator can potentially receive three payments:
forward energy, reserved capacity and real-time energy if being called. For a
marginal unit whose forward energy schedule is x, total schedule is x+y, and
real-time instructed output is z, the three payments are represented by areas
of OECX, ABCD and CGZX, respectively.
[Graphic Omitted]
B. Proposed Formulation
The proposed formulation is a minimization problem as follows:
[Formula Omitted]
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[Formula Omitted]
C. DISCUSSIONS
1. Objective
The objective in (1) represents the total payment by consumers. The sum of the
energy and reserve payments are computed as inner product of the demand vectors
with the clearing price vectors. The clearing prices are used not only to pay
suppliers, but also to charge consumers. As a neutrality condition to a market
operator, the total payment from consumers is equal to the sum of payment
credited to generators and network surplus when transmission congestion is
present. Thus, the use of total consumer payment minimization tends to reduce
the transmission congestion costs as well.
2. Constraints
The DC network model for the transmission system is represented by (2) and (3)
which describe network balance for the energy and reserve capacity schedules,
respectively. Voltage angels of reference bus are set using (4) and (5). Eq.(6)
ensures that all price and schedule variables are positive.
Transmission branch flow limits are enforced by (7) and (8). These limits are
applicable to energy schedules and to reserve usage. For simplicity, we assume
the flow direction is known and flow constraints are unidirectional.
The behaviors of profit maximizing bidders are modeled by (9) and (10). These
constraints ensure that energy and reserve clearing prices are no less than
their bids. Consequently, all generators are sufficiently compensated with the
final clearing prices. Eq.(11) is used to enforce the generation output limit.
We use (12) and (13) as network price constraints. They are critically important
in that they link the clearing prices at different buses to satisfy the network
surplus requirements. Network surplus is defined as the difference of payment
received from demand users and credited for suppliers. When there is no
transmission congestion, constraints (12) and (13) will ensure that the clearing
prices at all buses are identical and network surplus is zero. If B(X)=0, (12)
becomes:
BP(x)=0(1)
The rank of B admittance matrix is n-1. Thus the solution for P(x) must lie in
the null space of B. Using the property of B matrix, we can obtain:
P(x)=P[1,1,1,...1](t) (16)
Where p is a scalar. Hence, all clearing prices are identical. Without these
constraints, the solution can lead to different clearing prices when there is
no transmission congestion. When transmission congestion is present, we will
show later that network surplus is the sum of product of branch congestion
prices and branch flow, and market operator receives adequate revenue from the
resulting clearing prices.
The motivation for introducing the network price constraints stems from the
solution process used in traditional OPF formulation. The gradient vector of
Langrangian with respect the bus angel yields the necessary optimality
condition to the OPF problem. This condition is the basis of locational
marginal prices in the OPF formulation and sets forth many desired properties
between the locational prices and branch congestion costs. In order to inherit
these desired properties, we therefore impose this optimality condition in the
original OPF problem as the new network surplus constraints.
Eqs.(14) and (15) are optional market separation constraints. These
constraints, if enforced, ensures that a set of energy schedules or capacity
reserves are balanced for a set of predefined market participants. These
requirements are often used in bilateral trading arrangement. In California and
Texas, for example, energy schedules for each scheduling coordinator must be
balanced.
The proposed formulation is different from the traditional pricing
methodologies such as used in [5] and [6]. In our formulation, price variables
for energy, reserve and transmission are explicitly used so that payments of
these markets are minimized. This is in contrast to the cost based objectives
in [5] and [6]. Moreover, we introduced the new constraints (12) and (13) and
ensure that the resulting prices across the different locations reflect the
network conditions.
The proposed formulation can be extended to handle practical and
implementational issues. For example, bi-directional flows are modeled by
adding bidirectional flow constraints. Inclusion of demand bids is accommodated
by decomposing the x and y vectors into separated supply and demand components.
Finally, when ramp-rate and dependent auctions are considered, the objective
function needs to be modified to be the total payments for the scheduling
period, and generator outputs of different hours need to be constrained by its
ramp rate. In addition, pricing and bidding rules specific to a market can be
modeled with additional constraints. For instance, when a unit participants
only in the reserve (or energy) market, the energy variable x (or reserve
variable y) can be constrained to be zero value.
D. PROPERTIES OF THE NEW FORMULATION.
Price clarity and simplicity: The explicit use of pricing variables provides
price clarity to markets. With the traditional method of scheduling, prices are
often treated as the by-products of the solution process. In contrast, the new
formulation uses prices as control variables for scheduling and no separate
process is needed for pricing. Each bidder acts as an agent to maximize its own
profit by observing the price signals. In addition, opportunity costs often
need to be computed for compensation for redispatched or constrained generators
by market operators. Use of opportunity costs has been very controversial and
often adds to price ambiguity.
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With the new formulation, there is no need for computing these opportunity
costs.
COMPETITION FOR TRANSMISSION USE: The formulation allocates transmissions to
energy and reserve use according to the economic bid data rather than
heuristics. Therefore, market operators no longer need to guess the amount of
transmission that sets aside for reserve use.
REDUCED TRANSACTIONS COSTS: A large amount of transactional costs are incurred
for market participants and operators as the process of market operations
becomes more complex. With this formulation, transactions for market operators
and participants are straightforward and associated transaction costs can be
reduced. For instance, market participants need only submit one bid curve for
all three markets, thus simplifying the data processing and management.
Furthermore, iterations between the constrained and unconstrained market prices
can be eliminated or reduced.
MODELLING OF DISTRIBUTED RESERVE REQUIREMENT: The proposed formulation allows
modeling of the distributed reserve requirements at different locations.
Although the reserve requirements are often proportional to the energy demand,
there are cases that some areas may have more reserve requirements.
GENERATOR PAYMENT ADEQUACY: The total payment received by any generator is
equal to or more than its costs computed using its bid curve. This can be
observed from the bid curve in Figure 1. Assuming that a generator is dispatched
at z where z is between forward energy schedule x and total schedule x + y, the
total costs incurred are represented by the area OCFZ. The total payment,
represented by areas of OECX, ABCD and CGZX, is greater than OCFZ.
NETWORK SURPLUS ADEQUACY LEMMA: The network surplus is always positive and is
exactly the sum of branch congestion prices multiplied by their flows.
[GRAPHIC OMITTED]
[FORMULA OMITTED]
COROLLARY: When there is no congestion, the total consumer payment is the same
as generator credit payments. This can be derived by setting Bx = 0 and By = 0
in the above lemma.
IV. APPLICATIONS
This section presents numerical results of applying the new formulation for
computing clearing prices and schedules for integrated electricity markets. The
solutions to the example problems are obtained with a nonlinear optimization
solver.
The looped network used by the example is shown in Figure 2. The network has 3
buses and 5 generators. For simplicity, all three branches have equal
impedances. The bid price curve is represented as [Formula Omitted] where q is
the output quantity. The bid parameter, energy and capacity demands used in the
example are listed in Tables 1 and 2.
Table 3 shows 5 sets of results with different demand or transmission
parameters. The parameter changes from the base case are in bold fonts. These
results are discussed below. No market separation constraints are applied in
the example.
In the base case, transmission constraints are inactive, resulting in uniform
clearing prices for both energy and reserve markets. Although the transmission
constraints are inactive, the network price constraints are still playing an
important role. Before solving for the schedules, transmission flow is unknown
and congestion is undetermined. Simply allowing different locational prices, we
can come to the condition that different locational prices are computed without
any active transmission constraints. On the other hand, if we use pre-defined
congestion and restrict tradings between locations, we may not fully capture the
economic efficiency due to the inter-locational trading. To confirm this, we
remove the network price constraints in the base case and solve for the prices
and schedules. In the new solution (not shown in Table 3),
[GRAPHICS OMITTED]
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[GRAPHIC OMITTED]
the objective is reduced to be 2707.89, and the energy prices at the three buses
are (26.00, 28.37, 28.81). Locational prices are different although there are no
congestions. If 28.81 are chosen as the clearing price, the total energy
consumer payment alone will be 2881.
In case A, we derate the flow limit of Branch 3 from 30 to 25. Branch 3 becomes
congested since the unconstrained flow is 27.9437. This leads to reduction of
output in Bus 1 and increase of more expensive units in Buses 2 and 3.
Consequently, higher zonal clearing prices at Buses 2 and 3 are computed.
Let's examine the energy price at Bus 3. When energy is delivered from Bus 1 to
Bus 3, two thirds flow through Branch 1 between Buses 1 to 3 and one third goes
through the parallel path on Branches 1 and 2. At market equilibrium, we note
that Px3=px1+2/3 * Bx. That is, the clearing price at Bus 3 is exactly the
clearing price at Bus 1 plus the branch congestion costs from Bus 1 to Bus 3.
The network surplus 94.4444 can be computed either by using the difference
between demand payments and generator credits, or by multiplying the flows with
their corresponding congestion prices.
In Case B, we examine the solution behavior when energy demand at Bus 3
increases from 50 to 60. The objective function is increased by 407.0084. This
increase is due to three factors: network surplus, cost increase for additional
generation, and additional payment due to the price increases. Because of the
increased demand, energy clearing prices at all buses have increased and the
largest increase occurs at bus 3. However, reserve prices remain the same as in
Case A.
Case C is rather interesting. We increase the reserve demand at Bus 3 from 3 to
8. Branch 3 is unconstrained for energy schedules. Hence, uniform energy
clearing prices are computed for all three buses. However, there is not enough
transmission to support the reserve use from 1 to 3. Therefore, a congestion
price of 0.3222 is assessed to the capacity reservation for this branch.
Consequently, different clearing prices for reserve are computed and a network
surplus of 0.6626 (the product of reserve usage 2.0563 and congestion price
0.3222) is computed due to the insufficient transmission for reserve.
Finally, we simultaneously increase both the reserve demand and energy demand at
Bus 3 as shown in Case D. Branch 3 is constrained for energy delivery. This
results in different energy and reserve clearing prices for all three buses. The
largest price increases for reserve is also at Bus 3. Because the energy demand
is the same as in Case B, the energy clearing prices are the same as in Case B.
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V. CONCLUSIONS
In this paper we proposed a new formulation that solves for schedules and
clearing prices for integrated electric energy, transmission and generation
capacity reserve markets simultaneously. The direct use of price variables and
innovative network price constraints are the most significant aspects of the
formulation. Moreover, this new formulation incorporates a DC transmission
network models and yields locational clearing prices when transmission
congestion is present.
The new formulation has several advantages over traditional pricing and
scheduling methods. We overcomed the problem of clearing price computation in
the traditional unit commitment and optimal power flow techniques by
introducing price based dispatch methods. Because prices are used as control
variables, post processing for price calculation is no longer required and
there are no price ambiguity to market participants. All bidders can easily
verify that their profits are maximized. Furthermore, competition for
transmission allocation between energy and reserve market is addressed. The new
formulation offers the ability to integrate three different, but inter-related,
markets for better economic efficiency as well as for reducing market
transactions costs. The formulation can be adopted for real market applications.
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BIOGRAPHIES
SHANGYOU HAO received his B.S. degree from Wuhan Institute of Hydraulic and
Electrical Engineering, China, in 1982, and his M.S. and Ph.D. degrees in
Electrical Engineering from Ohio State University in 1984 and 1988,
respectively. He was with the Pacific Gas and Electric Company from 1988 to
1997, working on development of analytical methodologies for California
electricity industry restructuring. He has been with Perot Systems Corporation
since 1997, developing information system and business protocols for the
California ISO and Power Exchange.
DR. DARIUSH SHIRMOHAMMADI is the Director of Energy Infrastructure Services
with Perot Systems Energy Group. He has been with Perot Systems since 1996
where he has been developing and integrating information technology solutions
for market operations and settlements of emerging energy market players
worldwide including the California Independent System Operator and California
Power Exchange. Dariush has around 25 years of experience with the electric
utility industry mainly in the development and implementation of methodologies
and computer models for operations, planning, costing, pricing and automation
of electric power systems. His career path includes positions as researcher for
two years with Hydro Quebec, transmission planner for three years with Ontario
Hydro, systems engineer, transmission planner, automation engineer, and the
Director of the Energy Systems Automation Organization with Pacific Gas and
Electric Company for 11 years and the principal consultant with Shir Power
Engineering Consultants, Inc., for more than a year. Dariush has authored
numerous technical paper and reports and has lectured in academic, industrial
and regulatory forums all around the world. Dariush has a Ph.D. in Electric
Power Engineering from the University of Toronto.
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