energy and reserve market designs with explicit
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003 53
Energy and Reserve Market Designs With ExplicitConsideration to Lost Opportunity Costs
Deqiang Gan and Eugene Litvinov
Abstract—The primary goal of this work is to investigate thebasic energy and reserve dispatch optimization (cooptimization)in the setting of a pool-based market. Of particular interest is themodeling of lost opportunity cost introduced by reserve allocation.We derive the marginal costs of energy and reserves under a va-riety of market designs. We also analyze existence, algorithm, andmultiplicity of optimal solutions. The results of this study are usedto support the reserve market design and implementation in ISONew England control area.
Index Terms—Electricity market, marginal pricing, optimiza-tion, power systems, spinning reserve.
NOMENCLATURE
Node energy demand, a vector.
System reserve demand (or requirement), a scalar.
Unit vector, every elements of is unity.
Lost opportunity cost function.
Transmission thermal limit vector.
Index set of generators (consists of 1, 2, 3, …).
Lost opportunity cost price.
Energy bid price/generation.
Reserve bid price/allocation.
Generation sensitivity factor matrix.
Equals to 1 or 0 indicating if a generator incurs lost
opportunity cost.Energy nodal price vector.
Reserve clearing price.
I. INTRODUCTION
THERE EXIST two school of thoughts for market design asderegulation in power industry proceeds. They are com-
monly known as pool model and bilateral model [1], [2]. ISONew England (ISO-NE) control area electricity market followsthe concept of pool model [3] and [4].
In a pool-based market, energy and reserves are centrally andoptimally allocated based on volunteer bids (in this paper re-
serve refers to 10-min spinning reserve). Under two-settlementdesign [4], the allocation of energy and reserves is implemented
in two steps—day-ahead scheduling and real-time dispatch. The
optimization concepts of the two steps are, broadly speaking,similar. In this study, we focus on the market design issues inthe real-time dispatch setting.
Manuscript received January 23, 2002; revised June 21, 2002.D. Gan was with ISO New England, Inc., Holyoke, MA 01040 USA.
He is now at Zhejiang University, Zhejiang 310027, China (e-mail:[email protected]).
E. Litvinov is with ISO New England, Inc., Holyoke, MA 01040 USA(e-mail: [email protected]).
Digital Object Identifier 10.1109/TPWRS.2002.807052
While the idea of locational marginal pricing advocated in
[5]–[7] seemingly dominates the pool-based energy markets,
there is little consensus on how to structure reserve markets.
In fact, the design of reserve markets is often a debatable
topic. Discussions on this topic can be found in, say, [8]–[12],
[22].
One of the current major tasks in the ISO New England, Inc. is
to study and possibly improve the existing reserve market. Aside
from a long-term forward market, which will not be discussed
here, the following four alternative designs received attention in
this effort:
• generators receive availability payments—model (A);
• generators receive lost opportunity cost payments—model
(L);
• generators receive availability and lost opportunity cost
payments—model (A L);
• generators receive either availability or lost opportunity
cost payment—model (A L).
Availability payment refers to the payment to generators that
offer reserve capability to the market. The definition of lost op-
portunity cost incurred in reserve allocation can be found in the
next section. More details about it can be found in New York,
PJM, or the New England ISO website. The existing ISO-NE
market, which is undertaking a major revision, follows model(A L). The revised market design will likely follow a variant
of model (A L) as outlined in [23]. We will briefly discuss this
revised market design subsequently.
The three electricity markets in the Northeastern U.S. all have
or will have energy and reserve markets where lost opportu-
nity costs are explicitly compensated. Whether or not generator
lost opportunity cost should be compensated is a policy issue. If
under certain market design generators do receive lost opportu-
nity cost compensation, then energy, reserve, and lost opportu-
nity cost are cooptimized. This is at present a common practice.
A systematic treatment on the formulation, solution algorithm,
and pricing analysis of cooptimization becomes a timely issue.
The primary goal of this work is to study the basic princi-ples of cooptimization, laying down the engineering founda-
tions of energy/reserve market design. The results can also be
valuable for market implementation. We do not investigate how
to choose the optimal market design which would require sub-
stantial economic analysis [13]–[15], [24]. Of particular interest
in this work is to present a general approach for modeling lost
opportunity cost. In the context of cooptimization, we derive
the formulae for calculating marginal costs under a variety of
market designs. We also investigate such issues as solution ex-
istence, algorithm, and multiplicity of cooptimization.
0885-8950/03$17.00 © 2003 IEEE
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GAN AND LITVINOV: ENERGY MARKET DESIGNS WITH CONSIDERATION TO LOST OPPORTUNITY COSTS 55
A. Optimization Formulation and Solution Existence
Let us define a constant vector as follows:
(8)
we have
(9)
The graph of a lost opportunity cost function is illustrated in
Fig. 2. Obviously, it is nondifferentiable but it is continuous.
The problem of energy and reserve dispatch with considera-
tion of lost opportunity cost is as follows:
(10-1)
(10-2)
(10-3)
(10-4)
(10-5)
(10-6)
(10-7)
The above model corresponds to market design model
(A L). To obtain the optimization model for market design
model (A), one only needs to assume that in the above
model. To obtain the optimization model for market design,
model (L), one needs to assume that in the above model.
In the recent proposal [23], generators that are able to provide
reserves are classified as tier 1 and tier 2 resources. In a nutshell,
tier 1 resources do not incur lost opportunity cost while tier 2 re-
sources do. The optimization formulation for tier 1 resource is
simply an energy-only optimization, and the optimization for-mulation for tier 2 resources is similar to that of model (A L).
By the continuity of objective function of the above problem,
the answer to the question of solution existence is a qualified
“yes,” provided the feasible set of the problem is nonempty [16,
Weierstrass Theorem].
Now let us briefly discuss the optimization formulation for
model (A L). It can be stated as follows:
(11-1)
(11-2)
(11-3)
(11-4)
(11-5)
(11-6)
(11-7)
The 0–1 variables are introduced to enforce the condition
that a generator either receives reserve availability payment or
lost opportunity cost payment. Mathematically, this model is
similar to model (A L), so we will not discuss it further. But the
results to be presented can be extended to deal with this model.
Before proceeding to study the solution algorithm, we note
that locational reserve requirements are not discussed here
Fig. 2. Lost opportunity cost function when a constant price is used.
but theoretically they could be incorporated into the presented
framework without conceptual difficulty using methods pre-
sented in the literature [17], [18].
B. Solution Algorithm
This section suggests a solution algorithm for the optimiza-
tion problem (10). First, let us convert the nondifferentiable lostopportunity cost function into a discrete function as follows:
(12)
With the above manipulations, the energy and reserve dispatch
problem can be reformulated as a 0–1 mixed integer program-
ming problem as
(13-1)
(13-2)(13-3)
(13-4)
(13-5)
(13-6)
(13-7)
(13-8)
The above 0–1 problem can be solved using the standard
branch- and-bound method [19]. Note that the integer variables
are associated with generators with reserve capabilities only. In
ISO-NE, the number of generators is approximately 350, but the
number of generators that have reserve capability is only about
50.
Whether or not the standard branch-and-bound algorithm can
meet the requirement of real-time application is out of the scope
of this study. However, the formulation (13) allows us to inves-
tigate the pricing issues of energy and reserves.
C. Optimality Conditions and Marginal Costs
Suppose we find the optimal integer solution . Now it is
straightforward to derive the marginal costs of energy and re-
serve. As usual, let , , , and be Lagrangian multipliers
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56 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003
of constraints in equations (15)–(17), we form the Lagrangian
function as follows:
(14)
If the th generatorhas lost opportunity cost, then we have that
, . By the standard Kuhn–Tucker optimality
conditions, we have that , and
(15-1)
(15-2)
The nodal energy prices are given by the marginal cost vector
[5], [6]. The reserve price is also set to the
marginal cost .
If the th generator does not have lost opportunity cost, we
have , . By the standard Kuhn–Tucker
optimality theory, . It follows that:
(16-1)
(16-2)
The energy price is again the marginal cost .
The reserve price is still given by . In the next section, we will
present a pricing analysis based on equations (15) and (16).
V. PRICING ANALYSIS FOR ALTERNATIVE MARKET DESIGNS
In this section, we present a pricing analysis for each of the
models mentioned in the previous section. Whenever possible,
we will indicate if multiple solutions exist.
A. Model (A)
Under this simple model, . The reserve clearing priceequals to the reserve availability price of the most expensive
generator that is designated to supply reserve. The past expe-
rience gained in ISO-NE is that bid prices for reserves are often
zero because for many generators reserve costs are sunk costs
and they are ensured lost opportunity costs. Under model (A), it
is less likely that generators will ask for zero reserve prices be-
cause they do not receive lost opportunity cost. However, there
is no guarantee that most generators will ask for reserve prices
that are greater than zero.
When the bid prices for reserves of many generators are equal
to zero (under uniform price auction many generators do bid
zero prices), then it is likely that and, generally, there are
multiple solutions in reserve allocation. In fact, there is a con-
tinuum of reserve solutions. For instance, consider the following
problem:
(17-1)
(17-2)
(17-3)
(17-4)
The optimal energy dispatch is , , but any
reserve dispatch that meets is an optimal reserve
dispatch. To find an unique solution, one method is to designate
reserve contributions to the generators with lowest energy bid
prices.
A major concern about this model is that when cheaper and
fast-start generators are backed down to provide reserve, they
have a disincentive to follow ISO dispatch instructions.
B. Model (L)
Under thismodel, . Let usconsider two situations. Inthefirst situation, the optimal dispatch does not require paying gen-
erators lost opportunity cost. This is just the situation in model
(A) when the bid prices for reserves are all equal to zero. So it is
quite possible that there are multiple solutions for reserve allo-
cation. To resolve this problem, again, one could designate re-
serve contributions to generators with lowest energy bid prices.
Now let us consider the second situation where, in the optimal
dispatch, some generators are paid lost opportunity cost. In the
sequel, we show that the marginal cost of producing reserves,
, can still be greater than zero. To get a feel of what could
be, let us derive an alternative expression of from equations
(15-1)–(15-2)
(18)
When there is no congestion, it is obvious that . Since
,
(19)
Recall that the marginal cost of a product is equal to the
change of production cost as the demand increases by a small
amount [20]. Based on this principle, let us verify the result (19)
by looking at an example noncongested five-generator system as
illustrated in Figs. 3 and 4.
Suppose reserve requirement is increased by a infinitesimal .
Suppose generator is the only one that is capable of supplying
reserve, the change of total production cost would consist of
three components
• energy cost increase of generator , which is ;
• energy cost decrease of generator , which is ;
• lost opportunity cost increase of generator , which is
.
Note that happens to be equal to the clearing price. The
change of production cost would be
. This indicates that the reserve marginal cost is equal to
which is consistent with the result in (19) if one considers
that and in this example.
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GAN AND LITVINOV: ENERGY MARKET DESIGNS WITH CONSIDERATION TO LOST OPPORTUNITY COSTS 57
Fig. 3. Clearing energy-only market in a five-generator system.
Fig. 4. Clearing energy and reserve market in a five-generator system (theshaded area is designated for reserve).
C. Model (A L)
This model, which is being used in the existing ISO-NE
market, is quite similar to model (L) except that the problem
of solution multiplicity in allocating reserves does not happen
very often. The reason is that ISO has availability bids. Fol-
lowing the observation derived from equations (18) and (19),if generator is designated to supply reserve and it has lost
opportunity cost, we have
(20)
D. Model (A L)
The pricing analysis for this model is similar to that of model
(A L) so we willnot proceed further. Under such a model, there
can still exist multiple solutions in reserve allocation, but this
does not happen very often because ISO has availability bids.
E. A Variant of Model (L)
This model is the same as model (L) except that lost opportu-
nity cost is not explicitly included in the objective function (but
generators do receive lost opportunity cost payment). The opti-
mization formulation is as follows:
(20-1)
(20-2)
(20-3)
(20-4)
(20-5)
(20-6)
When there is no congestion, the solution of the above
problem coincides with the solution of the problem where lost
opportunity cost is explicitly included in objective function.
This can be illustrated using Fig. 4 in which generator or
can provide reserve (assume that they are the only generators
that can provide reserves) but they also demand lost opportunity
cost. Based on formulation (20), generator would be desig-
nated to provide reserve and it would be paid lost opportunitycost. Apparently, if lost opportunity cost is included explicitly
into (20-1), the optimal solution would still be the same.
As in the model (L), there are multiple solutions for reserve
allocation when there is ample reserve capacity margin.
VI. EXAMPLES
First let us consider how to solve the two-generator coopti-
mization problem described in Section II. This example is re-
stated below for convenience
First, notice that , , and .
The next step is to convert the above nondifferentiable optimiza-
tion problem into mixedinteger programming problem based on
equation (10) as follows:
To solve the above mixed integer programming problem, one
only needs to solve two linear programs derived by assumingand , respectively. The optimal solution is
, , , and .
Now we briefly describe an example of an IEEE 118-bus
system. There is a congestion across line 89–92. We assume
that only the two generators at bus 100 and bus 112 have re-serve capability. The parameters of these two generators and the
energy-only optimization results are illustrated in Table I. The
reserve requirement is set to be 200 MW.
Under market model (A L), the cooptimization result is that
generator at bus 100 is designated to supplyreserve, the dispatch
of generator at bus 112 is 350 MW (which is the same as its en-
ergy-only dispatch). Under market model (A), the result is the
opposite. Generator at bus 112 is designated to supply reserve,
its generation dispatch is backed off to 150 MW. The above so-
lutions are hardly surprising because under model (A L), gen-
erator at bus 112 is more “expensive” due to lost opportunity
cost payment.
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58 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003
TABLE IPARAMETERS OF TWO GENERATORS OF 118-BUS TEST SYSTEM
VII. FORMULATION WHEN VARIABLE PRICE IS USED IN
CALCULATING LOST OPPORTUNITY COST
In this section, we assume that lost opportunity cost function
takes the most general form (6). This optimization is self-ref-
erential because the lost opportunity cost depends on variable
price which is not known until the final solution is obtained.
In this section, we suggest a method to get around this impasse.
Let us suppose temporarily that the reserve dispatch is given,
let itbe . Then, itis trivialto compute optimal energy dispatchand resultant energy prices
(21-1)
(21-2)
(21-3)
(21-4)
The Lagrangian function of the above optimization problem
is easily obtained as
(22)The spot prices are given by . The key to
solving the problem is to find out an energy dispatch that is
optimized taking into account reserve requirements. Consider
the following bilevel optimization problem:
(23-1)
(23-2)
(23-3)
(23-4)
(23-5)
(23-6)
(23-7)
(23-8)
In the above formulation, is a variable in the upper-level
optimization and a parameter in the lower-level optimization.
stays in the lower-level optimization subproblem. The primal
solution of the above bilevel optimization problem is the desired
optimal dispatch of energy and reserves. The dual solution con-
tains locational marginal prices.
In what follows we describe briefly how to solve the bilevel
optimization problem (23). By the standard theory of linear
programming, the primal and dual solutions of the lower-level
subproblem depend on the optimal basis of the lower-level
subproblem, that is
(24-1)
(24-2)
where is the right-hand-side of the lower-level sub-
problem, it is a linear function of .
To attack the bilevel optimization problem, observe that and
do not depend on the specific value of , rather they depend
on which constraints are included in optimal basis only. As a
result, suppose temporarily that the optimal basis is known,
then energy prices are known. Let it be and be as defined
based on (8). Now the bilevel optimization problem can be cast
as
(25-1)
(25-2)
(25-3)
(25-4)
The above problem possesses a structure that is similar to that
of (10). It can be solved using the general algorithm described
in Section IV-B.
The question remains to be how to find out the optimal basis.
One obvious solution is to enumerate all of the combinations
of bases of the lower-level subproblem. For example, in the
five-generator system illustrated in Fig. 3, where each gener-
ator is required to submit a single block bid, there are only five
possible energy prices. A better solution is to apply a standardbranch-and-bound algorithm [19]. Since a branch-and-bound al-
gorithm is fairly familiar to the power engineering audience,
we will not discuss it here. The readers are referred to [ 19] for
details.
VIII. FINAL REMARKS
In this paper, we studied four alternative energy/reserve
market designs that received attention in ISO-NE. We presented
a fairly detailed analysis on the basic formulation, solution
algorithm, and pricing formulae of cooptimization under these
market designs. The results of the research have been used
to support, from engineering perspective, the reserve marketdesign and implementation in ISO-NE. Many of the questions
raised during discussions on reserve market design at ISO-NE
are answered. The main finding is that energy, reserve, and lost
opportunity cost cooptimization are, in general, a nondifferen-
tiable and possibly bilevel optimization problem. This problem
can be further converted into a mixed integer programming
problem. A standard algorithm for solving these problems is
that of branch and bound. This algorithm can be efficient or
exceedingly slow, depending upon the size of the problem.
Whether or not a standard branch-and-bound algorithm can
meet engineering requirements is thus a subject of additional
research.
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GAN AND LITVINOV: ENERGY MARKET DESIGNS WITH CONSIDERATION TO LOST OPPORTUNITY COSTS 59
ACKNOWLEDGMENT
The authors have benefitted from the discussions in ISO-NE
Reserve Market Design Working Group. The opinions described
in the paper do not necessarily reflect those of ISO New Eng-
land, Inc. The authors remain solely responsible for errors.
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Deqiang Gan received the Ph.D.degree in electrical engineering from XianJiaotong University, Xian, China, in 1994.
Currently, he is with Zhejiang University, Zhejiang, China. He was a SeniorAnalyst in ISO New England, Inc., Holyoke, MA, where he worked on is-sues related to the design, implementation, and economic analysis of electricitymarkets. Prior to joining ISO New England, Inc., he held research positions atseveral universities in the U.S. and Japan.
Eugene Litvinov obtained the B.S. and M.S. degrees from the Technical Uni-versity, Kiev, Ukraine, U.S.S.R., and the Ph.D. degree from Urals PolytechnicInstitute, Sverdlovsk, Russia.
Currently, he is a Director of Technology with the ISO New England,Holyoke, MA. His main interests are power system market clearing models,system security, computer applications to power systems, and informationtechnology.