A Lost Opportunity Cost Model for Energy and Reserve Co-optimization
Deqiang Gan, and Eugene Litvinov
Abstract – The primary goal of this work is to investigate the
basic energy and reserve dispatch optimization (cooptimization) in the setting of a pool-based market. Of
particular interest is the modeling of lost opportunity cost
introduced by reserve allocation. We derive the marginal costs
of energy and reserves under a variety of market designs. We
also analyze existence, algorithm, and multiplicity of optimal
solutions. The results of this study are utilized to support the
reserve market design and implementation in ISO New England
control area.1
Index Terms – Electricity Market, Optimization, Power
Systems, Spinning Reserve, Marginal Pricing
I. NOMENCLATURE
DP
- node energy demand, a vector;
DR
e
f
- system reserve demand (or requirement), a scalar;
- unit vector, every elements of e is unity;
- lost opportunity cost function;
F
I
l
p/P
r/R
T
γ
ϕ
- 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;
- energy nodal price vector;
- reserve clearing price.
II. INTRODUCTION
There exist two school of thoughts for market design as
deregulation in power industry proceeds. They are commonly
known as pool model and bilateral model [1,2]. ISO New
England (ISO-NE) control area electricity market follows the
concept of pool model [3,4].
In a pool-based market, energy and reserves are centrally
and optimally allocated based on volunteer bids (in this paper
reserve refers to 10-minute spinning reserve). Under twosettlement design [4], the allocation of energy and reserves
are implemented in two steps, day-ahead scheduling and realtime dispatch. The optimization concepts of the two steps are,
broadly speaking, similar. In this study, we focus on the
market design issues in the real-time dispatch setting.
While the idea of locational marginal pricing advocated in
[5,6,7] 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].
1
The authors are with ISO New England, Inc., One Sullivan Road, Holyoke,
MA, 01040, USA. Email: [email protected]; [email protected]
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 opportunity cost incurred in reserve allocation can be
found in next section. More details about it can be found in
New York, PJM, or New England ISO web site. 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 Northeastern USA all
have or will have energy and reserve markets in which lost
opportunity costs are explicitly compensated. From social
welfare point of view, energy, reserve, and lost opportunity
cost ought to be co-optimized. 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
principles of co-optimization, laying down the engineering
foundations 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 substantial economic analysis [13-15]. Of
particular interest in this work is to present a general
approach for modeling lost opportunity cost. In the context of
co-optimization, we derive the formulae for calculating
marginal costs under a variety of market designs. We also
investigate such issues as solution existence, algorithm, and
multiplicity of co-optimization.
III. LOST OPPORTUNITY COST FUNCTIONS
Suppose in a pool-based power market each generator
submits a single bid block. The energy-only dispatch for one
dispatch interval (5 minutes in the ISO-NE) can be expressed
as [4-6]:
Min p T P
(0-1)
P
S .T .
eT ( P − DP ) = 0 , T ( P − DP ) ≤ F , P ≤ P ≤ P . (0-2)
The dispatch solution of the above model, let it be P̂ ,
does not need to respect reserve requirement. Some
generators may have to be backed down to provide reserve.
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The lost opportunity cost of a generator can be defined as
follows:
max{0, (γ − p )( Pˆ − P)}, γ > p
f ( P, γ ) = 
.
(1)
γ≤p
0,
The choice of specific form of f (⋅) is controversial. We
consider two alternative cases: γ is a constant vector
obtained from energy-only dispatch optimization (0-1)~(0-2);
γ is the final energy price which itself is a variable of the
optimization. For ease of exposition, the multi-block bids are
not considered in this paper, the results could be easily
extended to a multi-block case.
IV. FORMULATION WHEN CONSTANT PRICE IS USED
IN CALCULATING LOST OPPORTUNITY COST
In this section, the lost opportunity cost function is
assumed to take the following form:
max{0, (γˆ − p )( Pˆ − P)}, γˆ > p,
f ( P, γ ) = 
(2)
γˆ ≤ p,
0,
where γˆ is the constant energy price vector obtained from
energy-only dispatch optimization (0).
A. Optimization Formulation and Solution Existence
Let us define a constant vector l as follows:
γˆ − p, γˆ > p,
l=
γˆ ≤ p,
0,
We have:
f = max{0, l( Pˆ − P)} .
(3)
(4)
The graph of a lost opportunity cost function is illustrated
in Figure 2. Obviously, it is non-differentiable but it is
continuous.
f
slope = - l
P̂
P
Figure 2. Lost Opportunity Cost Function when a Constant
Price Is Used
The problem of energy and reserve dispatch with
consideration of lost opportunity cost is as follows:
Min
P,R
S .T .
T
T
p P + r R + ∑ fi
T
e ( P − DP ) = 0 ,
(5-1)
(5-2)
eT R = DR ,
(5-3)
T ( P − DP ) ≤ F ,
(5-4)
P≤ P+R≤ P ,
(5-5)
f i = max{0, l i ( Pˆi − Pi )} , i ∈ I ,
P ≥ 0, R ≥ 0 .
(5-6)
(5-7)
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 l = 0 in the above
model. To obtain the optimization model for market design
Model (L), one needs to assume that r = 0 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 resources do. The optimization formulation
for Tier 1 resource is simply an energy-only optimization,
and the optimization formulation 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:
Min
p T P + ∑ [vi ri Ri + (1 − vi ) f i ]
(6-1)
S .T .
e T ( P − DP ) = 0 ,
(6-2)
P,R
T
e R = DR ,
(6-3)
T ( P − DP ) ≤ F ,
(6-4)
P≤ P+R≤ P ,
(6-5)
f i = max{0, l i ( Pˆi − Pi )} , i ∈ I ,
P ≥ 0, R ≥ 0 ,
v ∈ {0,1} .
(6-6)
(6-7)
The 0-1 variables v 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 but theoretically they could be incorporated into the
presented framework without conceptual difficulty using
methods presented in the literature [17,18].
B. Solution Algorithm
This section suggests a solution algorithm for the
optimization problem (5). First, let us convert the non-
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differentiable lost opportunity cost function into a discrete
function as follows:
f i = ui l i ( Pˆi − Pi ) , ui ( Pˆi − Pi ) ≥ 0 , (1 − ui )( Pi − Pˆi ) ≥ 0 ,
u ∈ {0,1}
With the above manipulations, the energy and reserve
dispatch problem can be re-formulated as a 0-1 mixed integer
programming problem as:
∂Γ
( )
= pi − l i − λ − T T µ − τ i + τ i = 0 ,
∂Pi
(9-1)
∂Γ
( )
= ri − ϕ − τ i + τ i = 0 .
∂Ri
(9-2)
The energy price is the marginal cost γ = λe + T T µ [5,6].
The reserve price is also set to the marginal cost ϕ .
If the i-th generator does not have lost opportunity cost,
we have u i = 0 , Pˆi − Pi < 0 . By the standard Kuhn-Tucker
Min
p T P + r T R + ∑ ui l i ( Pˆi − Pi )
(7-1)
optimality theory, υ i = 0 . It follows that:
S .T .
e T ( P − DP ) = 0 ,
(7-2)
eT R = DR ,
(7-3)
T ( P − DP ) ≤ F ,
(7-4)
P≤ P+R≤ P ,
u ( Pˆ − P ) ≥ 0 ,
(7-5)
∂Γ
( )
= pi − λ − T T µ − τ i + τ i = 0 ,
∂Pi
∂Γ
( )
= ri − ϕ − τ i + τ i = 0 .
∂Ri
P,R
i∈I ,
ˆ
(1 − ui )( Pi − Pi ) ≥ 0 , i ∈ I ,
P ≥ 0, R ≥ 0 ,
u ∈ {0,1} .
i
i
(7-6)
i
(7-7)
(7-8)
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
about 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 (7) allows us to
investigate the pricing issues of energy and reserves.
C. Optimality Conditions and Marginal Costs
Suppose we find the optimal integer solution u * . Now it
is straightforward to derive the marginal costs of energy and
reserve. As usual, we form the Lagrangian function as
follows:
i∈I
- λ [eT ( P − DP )] - ϕ (eT R − DR ) + µ T [T ( P − DP ) − F ]
(
)
+ τ T ( P − P − R) + τ T ( P + R − P )
- ∑ ω u * ( Pˆ − P ) - ∑υ (1 − u * )( P − Pˆ ) .
(8)
i i
i
i
i∈I
i
i
(10-2)
The energy price is again the marginal cost
γ = λe + T T µ . The reserve price is still given by ϕ . In the
next section, we will present a pricing analysis based on
equations (9)-(10).
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 there exist multiple solutions.
A. Model (A)
Under this simple model, l = 0 . The reserve clearing
price equals to the reserve availability price of the most
expensive generator that is designated to supply reserve.
When the bid prices for reserves of many generators are all
equal to zero (this happens quite often), then ϕ = 0 and
generally there are multiple solutions in reserve allocation. In
fact, there is a continuum of reserve solutions. For instance,
consider the following problem:
Min 10 PA + 20 PB + 0 R A + 0 RB
PA + PB = 70 , R A + RB = 20 ,
0 ≤ PA + R A ≤ 100 , PA ≥ 0 , R A ≥ 0 ,
0 ≤ PB + RB ≤ 100 , PB ≥ 0 , RB ≥ 0 ,
The optimal energy dispatch is PA = 70, PB = 0 , but any
reserve dispatch that meet R A + RB = 20 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.
S .T .
Γ = p T P + r T R + ∑ ui*l i ( Pˆi − Pi )
i∈I
(10-1)
i
i
If the i-th generator has lost opportunity cost, then we
have that ui = 1 , Pˆi − Pi > 0 . By the standard Kuhn-Tucker
optimality conditions, we have that ω i = 0 , and:
B. Model (L)
Under this model, r = 0 . Let us consider two situations.
In the first situation, the optimal dispatch does not require
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paying generators 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 allocation. To resolve this problem,
again, one could designate reserve 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, ϕ , can still be greater than zero. To get
a feel of what ϕ could be, let us derive an alternative
expression of ϕ from equations (9-1)-(9-2):
ϕ = ri + l i + γ i − pi
(11)
When there is no congestion, it is obvious that γ i ≥ γˆi .
Since r = 0 ,
ϕ = l i + γ i − p i ≥ 2l i > 0
(12)
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
(12).
Price
Suppose reserve requirement is increased by a
infinitesimal ε . The change of total production cost would
consist of three components:
•
•
•
Energy cost increase of generator B, which is p B ε ;
Energy cost decrease of generator A, which is
− p Aε ;
Lost opportunity cost increase of generator A, which
is ( p B − p A )ε .
Note that p B happens to be equal to the clearing price.
The
change
of
production
cost
would
be
p B ε − p Aε + ( p B − p A )ε = 2l Aε . This indicates that the
reserve marginal cost is equal to 2l A which is consistent
with the result in (12).
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.
Following the observation derived from equations (11)-(12),
if generator i is designated to supply reserve and it has lost
opportunity cost, we have:
Load
ϕ = ri + l i + γ i − pi ≥ ri + 2l i .
D. Model (A|L)
clearing price
A
B
Quantity
Figure 3. Clearing Energy-only Market in a 5-Generator
System
Price
Load
The pricing analysis for this model is similar to that of
model (A+L) so we will not 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.
F. A Variant of Model (L)
This model is the same as Model (L) except that lost
opportunity cost is not explicitly included in the objective
function (but generators do receive lost opportunity cost
payment). The optimization formulation is as follows:
Min
pT P
(14-1)
S .T .
e T ( P − DP ) = 0 ,
(14-2)
P
clearing price
T
X
(13)
A
B
Quantity
Figure 4. Clearing Energy and Reserve Market in a 5Generator System (the shaded area is designated for reserve)
e R = DR ,
(14-3)
T ( P − DP ) ≤ F ,
(14-4)
P≤ P+R≤ P ,
P ≥ 0, R ≥ 0 .
(14-5)
(14-6)
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
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function! This can be illustrated using Figure 4 in which
generator A or X can provide reserve but they also demand
lost opportunity cost. Based on formulation (14), generator A
would be designated to provide reserve and it would be paid
lost opportunity cost. Apparently, if lost opportunity cost is
included explicitly into (14-1), 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. 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 (1). This optimization is
self-referential because the lost opportunity cost depends on
variable price γ 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 it be R * . Then it is trivial to compute optimal
energy dispatch and resultant energy prices:
Min
pT ⋅ P
(15-1)
S .T .
e T ( P − DP ) = 0 ,
(15-2)
P
T ( P − DP ) ≤ F ,
*
(15-3)
*
P−R ≤ P≤ P −R .
(15-4)
The Lagrangian function of the above optimization
problem is easily obtained as:
Γ = p T P + λ [ eT ( DP − P) ] + µ T [ F − T ( P − DP )]
(
)
(16)
+ τ T ( P − R* − P) + τ T ( P + R* − P ) .
The spot prices are given by γ = λe + T T µ . The key to
solving the problem is to find out an energy dispatch that is
optimized taking into account reserve requirements. Consider
the following bi-level optimization problem:
Min
pT P + r T R + ∑ f i
(17-1)
S .T .
eT R = DR , R ≥ 0 ,
(17-2)
R
max{0, (γ i − pi )( Pˆi − Pi )}, γ i > pi
, i∈I ,
fi = 
γ i ≤ pi
0,
(17-3)
T
(17-4)
Min p ⋅ P
P
S .T .
e T ( P − DP ) = 0 ,
(17-5)
T ( P − DP ) ≤ F ,
(17-6)
P−R≤ P≤ P −R,
(17-7)
P≥0.
(17-8)
In the above formulation, R is a variable in the upperlevel optimization and a parameter in the lower-level
optimization. P stays in the lower-level optimization subproblem. The primal solution of the above bi-level
optimization problem is the desired optimal dispatch of
energy and reserves. The dual solution contains locational
marginal prices.
In what follows we describe briefly how to solve the bilevel optimization problem (17). By the standard theory of
linear programming, the primal and dual solutions of the
lower-level sub-problem depends on the optimal basis, B , of
the lower-level sub-problem. That is:
P = B −1b( R ) ,
(18-1)
λ 
T −1
  = (B ) p ,
µ
 
(18-2)
where b(R) is the right-hand-side of the lower-level subproblem, it is a linear function of R .
To attack the bi-level optimization problem, observe that
λ and µ do not depend on the specific value of R, rather
they depend on which constraints are included in optimal
basis only. As a result, suppose temporarily that the optimal
basis B is known, then energy prices are known. Let it be γ~ ,
~
and l be as defined based on equation (3). Now the bi-level
optimization problem can be cast as:
Min
pT P + r T R + ∑ f i
(19-1)
S .T .
eT R = DR , R ≥ 0 ,
~
f i = max{0, li ( Pˆi − Pi )} , i ∈ I ,
(19-2)
R
−1
P = B b( R ) .
(19-3)
(19-4)
The above problem possesses a structure that is similar to
that of (5). 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 the
combinations of bases of lower level sub-problem. For
example, in the 5-generator system illustrated in Figure 3
where each generator is required to submit single block bid,
there are only five possible energy prices. A better solution is
to apply a standard branch-and-bound algorithm [19]. Since
branch-and-bound algorithm is fairly familiar to the power
engineering audience, we will not discussed 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,
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solution algorithm, and pricing formulae of co-optimization
under these market designs. The results of the research have
been utilized to support, from engineering perspective, the
reserve market design and implementation in ISO-NE. The
main finding is that energy, reserve, and lost opportunity cost
co-optimization is, in general, a non-differentiable and
possibly bi-level 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 requirement is thus a subject of additional
research.
IX. ACKNOWLEDGEMENT
The authors have benefited from the discussions in ISONE Reserve Market Design Working Group. The opinions
described in the paper do not necessarily reflect those of ISO
New England, Inc. The authors remain solely responsible for
errors.
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Deqiang Gan is currently a Senior Analyst in ISO New England, Inc. where
he works on issues related to the design, implementation, and economic
analysis of electricity markets. Prior to joining ISO New England, Inc., he
held research positions at several universities. Deqiang received a Ph.D. in
Electrical Engineering from Xian Jiaotong University, China, in 1994.
Eugene Litvinov obtained the BS and MS degree from the Technical
University in Kiev, Ukraine, and Ph.D. from Urals Polytechnic Institute in
Sverdlovsk, Russia. He is currently a Director of Technology in the ISO New
England. His main interests are power system market clearing models,
system security, computer applications to power systems, information
technology.
0-7803-7519-X/02/$17.00 (C) 2002 IEEE