Download price-based market clearing under marginal pricing: a

PRICE-BASED MARKET CLEARING UNDER MARGINAL PRICING: A
BILEVEL PROGRAMMING APPROACH
Ricardo Fernández-Blanco
José M. Arroyo
Natalia Alguacil
Universidad de Castilla – La Mancha
Ciudad Real, Spain
[email protected]
Universidad de Castilla – La Mancha
Ciudad Real, Spain
[email protected]
Universidad de Castilla – La Mancha
Ciudad Real, Spain
[email protected]
Pedro J. Muñoz
Universidad de Castilla – La Mancha
Ciudad Real, Spain
[email protected]
Abstract – Market clearing in restructured power systems is mostly implemented through an offer-based setting
with the goal of maximizing social welfare. This approach
leads to sound results from an economic viewpoint when
generation offers reflect true production costs. However,
offers may significantly differ from actual costs in practice, thus yielding undesired distortion. Under a marginal
pricing scheme, this paper presents a general bilevel programming formulation for alternative market-clearing
procedures dependent on market-clearing prices rather
than on offers. The resulting nonlinear mixed-integer
bilevel programming formulation is transformed into an
equivalent single-level mixed-integer linear program suitable for efficient off-the-shell software. The bilevel formulation is investigated through a particular instance of
price-based market clearing driven by consumer payment
minimization. This problem has recently received considerable attention due to the open challenges posed from
both modeling and computational perspectives. Numerical
results are provided to illustrate the performance of the
proposed approach.
Keywords: Offer cost minimization, bilevel programming, marginal pricing, consumer payment minimization, price-based market-clearing procedure
1 INTRODUCTION
In the framework of the nowadays competitive power
industry this paper considers a pool-based electricity
market for energy. The Independent System Operator
(ISO) receives energy offers from producers and energy
bids from consumers, and determines, for every hour,
the market-clearing price, the power productions, and
the consumption levels.
The market-clearing procedure used by the ISO is
typically formulated as a unit commitment problem [1]
in which generation offers replace production costs and
consumption bids are taken into account. In this problem, the objective function to be maximized is referred
to as the declared social welfare. When the demand is
inelastic, the objective function only includes generation offer information and the resulting problem becomes an offer cost minimization. Market-clearing
prices resulting from the unit commitment problem are
used for market settlement.
17th Power Systems Computation Conference
Under the assumption of perfect competition, the optimal market-clearing solution is equal to the equilibrium solution where the marginal value to consumers is
equal to the marginal cost to producers. This equilibrium maximizes the social welfare, which is the sum of
the consumer surplus and the producer surplus. As a
consequence, suppliers have no incentive to offer different from their production costs. Thus, the declared
social welfare based on generation offers reflects the
true social welfare and maximizing social welfare is
commonly accepted as the right goal in a market setting.
However, market-clearing procedures based on declared social welfare maximization are characterized by
several practical shortcomings [2]:
- The assumption of perfect competition does not
hold in real-life power markets. Consequently,
generation offers may not reflect actual costs and
the optimization might not maximize the true social
welfare or accurately reflect the natural behavior of
market participants.
- The scoring rule, which is used to select the accepted offers and bids, and the payment rule are
different. While the former is based on offer cost
minimization, the latter is determined in terms of
market-clearing prices. This discrepancy leads to
distortion since the solution does not necessarily
constitute an equilibrium where the objectives of
market participants are maximized.
This paper analyzes an alternative type of marketclearing procedure for energy trading in an electricity
pool. The alternative market clearing allows considering
the payment rule through the incorporation of marketclearing prices in the problem formulation. This procedure is hereinafter referred to as price-based market
clearing. Examples of price-based market clearing are
(i) procedures based on consumer payment minimization [3]-[6], where market-clearing prices appear in the
objective function, and (ii) revenue-constrained marketclearing procedures [3], [4], where market-clearing
prices appear in the constraints. Moreover, new procedures explicitly including generation and consumer
surplus maximization [7] would fall within this class of
problems.
Stockholm Sweden - August 22-26, 2011
The presence of market-clearing prices in the problem formulation complicates the solution of the optimization problem since market-clearing prices may themselves result from an optimization process.
Several definitions for the market-clearing price are
available [1], being the two most commonly adopted (i)
the price of the highest accepted generation offer, which
guarantees revenue adequacy for suppliers, and (ii) the
marginal price, which provides appropriate economic
signals. While the former has received considerable
attention in price-based market clearing [4], [5], and
references therein, few works have considered marginal
pricing [3], [6], [7].
In [3], payment minimization under marginal pricing
was addressed by a heuristic approach. In [6], Zhao et
al. first formulated consumer payment minimization
under marginal pricing as a bilevel programming problem, which was solved by augmented Lagrangian relaxation and surrogate optimization. In [7], agent-based
simulation was used to analyze an auction driven by
consumer and generation surplus maximization based
on marginal market-clearing prices.
Within the framework of marginal pricing, the main
purpose of this paper is to formulate a general pricebased market-clearing procedure as a bilevel programming problem [8], of which the consumer payment
minimization reported in [6] is a special case. Bilevel
programming is appropriate to model problems where
one agent, the leader, optimizes its objective function
(upper-level problem) considering that a second agent,
the follower, will react by optimizing its own objective
function (lower-level problem). These models are relevant in those situations where the actions of the follower affect the decision making of the leader. This is
the case in price-based market clearing: the selection of
accepted bids and offers (upper-level problem) depends
on market-clearing prices (lower-level problem), which
are in turn determined based on the set of accepted bids
and offers. Under marginal pricing, market-clearing
prices are the Lagrange multipliers or dual variables
associated with the power balance equations in an economic dispatch problem [1].
Another salient feature of this paper with respect to
[6] is the proposal of an equivalent single-level mixedinteger linear formulation based on the Karush-KuhnTucker (KKT) optimality conditions, duality theory, and
integer algebra results. The main advantages of expressing the original bilevel optimization as an equivalent
single-level mixed-integer linear program is the guaranteed convergence to the optimal solution in a finite
number of steps and the ready availability of efficient
commercial branch-and-cut software.
The main contributions of this paper are threefold:
- The ISO is provided with an optimization-based
framework for market clearing explicitly accounting for marginal market-clearing prices in the problem formulation.
- Bilevel programming is proposed as a suitable solution methodology.
17th Power Systems Computation Conference
- The applicability of bilevel programming is illustrated with the solution of an instance of pricebased market clearing, namely the consumer payment minimization problem.
The remainder of the paper is organized as follows.
Section 2 provides a general formulation for price-based
market clearing under marginal pricing. Section 3 presents the solution methodology. Section 4 describes the
application of the proposed framework for consumer
payment minimization. In Section 5, numerical results
illustrate the performance of the proposed approach.
Finally, relevant conclusions are drawn in Section 6.
2 PROBLEM FORMULATION
The price-based market-clearing procedure can be
formulated in a compact way as:
Min F(x, y, λ )
(1)
x , y ,λ
subject to:
x∈X
G e (x, y ) = 0
(2)
(3)
G i (x , y ) ≥ 0
(4)
(5)
H(x, y, λ ) ≥ 0 ,
where x represents the vector of scheduling variables
(status of generating units, start-up costs, etc.), y is the
vector of dispatching variables (generation levels, network-related variables, etc.), λ is the vector of marketclearing prices, F(x, y, λ ) is the objective function (declared social welfare, consumer payment, consumer and
generation surplus, etc.), X is the feasibility set for
scheduling variables x (integrality constraints, start-up
cost constraints, minimum up and down times, etc.),
G e (x , y ) denotes the equalities associated with power
balance equations, G i (x, y ) models all inequality constraints involving variables x and y (power limits, ramping limits, etc.), and H(x , y, λ ) represents all remaining
inequality constraints involving market-clearing prices
such as those imposing revenue adequacy.
Under marginal pricing, the vector of marketclearing prices λ is defined as the vector of marginal
costs. Mathematically, λ can be derived from the following economic dispatch problem for the optimum
vector of scheduling variables x * :
(
Min f x * , y
y
)
(6)
subject to:
( )
G (x , y ) ≥ 0 : (μ ) ,
G e x * , y = 0 : (λ )
i
*
(7)
(8)
Stockholm Sweden - August 22-26, 2011
where f (⋅) is the objective function, which may be
different from that of the price-based market-clearing
procedure F(⋅) ; the vector of market-clearing prices λ is
equal to the vector of Lagrange multipliers or dual variables associated with the power balance equations (7);
and µ is equal to the vector of Lagrange multipliers or
dual variables associated with constraints (8).
Note that under marginal pricing, the definition of λ
is an optimization problem itself. Thus, the price-based
market-clearing procedure can be recast as the following bilevel programming problem:
(
*
*
Min F x , y , λ
x
)
(9)
subject to:
x∈X
(
)
H x , y * , λ* ≥ 0 ,
*
(10)
(11)
*
where y and λ are obtained from:
(
Min f x * , y
y
)
( )
G (x , y ) ≥ 0 : (μ ) .
i
*
(
) (
)
(
G (x , y ).
L x * , y, λ, μ = f x * , y − λT G e x * , y
−μ
T
i
)
*
(15)
The optimal solution to (12)-(14) must satisfy the
KKT necessary optimality conditions:
(
)
(
)
(
∂L x * , y, λ, μ ∂f x * , y ⎛⎜ ∂G e x * , y
=
−
⎜
∂y
∂y
∂y
⎝
⎛ ∂G i x * , y
−⎜
⎜
∂y
⎝
(12)
)⎞⎟
⎟
⎠
)⎞⎟
⎟
⎠
T
λ
T
μ=0
(16)
μ≥0
(17)
e
*
(18)
i
*
T
i
( )
G (x , y ) ≥ 0
μ G (x , y ) = 0 ,
G x ,y = 0
(13)
(14)
The above bilevel problem consists of an upper-level
optimization (9)-(11) and a lower-level optimization
(12)-(14). The upper level controls the vector of scheduling variables x. Lower-level decision variables comprise the vector of dispatching variables y whereas the
vector of market-clearing prices λ is also associated
with the lower-level problem.
The goal of the upper level is to minimize the pricebased market-clearing objective function (9) evaluated
at the optimal values of the lower-level variables, y *
and λ* , subject to expressions exclusively constraining
upper-level decision variables x (10), to a set of constrained functions with y and λ as parameters (11), and
to the lower-level optimization (12)-(14).
The lower-level optimization is identical to the economic dispatch problem (6)-(8) associated with the
optimal upper-level decision variables x * .
In general, problem (9)-(14) is a mixed-integer nonlinear bilevel programming problem. It is worth noting
that the lower-level objective function (12) is typically
piecewise linear whereas constraints (13)-(14), including power balance equations as well as generation and
network limits, are usually linear. As a consequence, the
lower-level problem (12)-(14) is parameterized in terms
of the upper-level decision vector x in such a way that
the lower-level problem is linear and thus convex. As
17th Power Systems Computation Conference
3 SOLUTION APPROACH
To convert the original bilevel formulation (9)-(14)
into an equivalent single-level problem, the lower-level
optimization is first replaced by its KKT conditions.
Consider the Lagrangian function associated with the
lower-level problem for a given upper-level vector x * :
(
subject to:
G e x * , y = 0 : (λ )
described next, this feature allows the transformation of
problem (9)-(14) into an equivalent single-level mixedinteger problem.
(19)
*
(20)
where (16)-(17) are the dual feasibility constraints,
(18)-(19) are the primal feasibility constraints, and (20)
express the complementary slackness conditions.
Hence, problem (9)-(14) can be recast as the following single-level equivalent:
Min F(x , y, λ )
(21)
x , y , λ ,μ
subject to:
x∈X
H(x, y, λ ) ≥ 0
∂f (x , y )
∂y
μ≥0
(22)
(23)
T
T
i
⎛ ∂G e (x, y ) ⎞
⎛
⎞
⎟ λ − ⎜ ∂G (x, y ) ⎟ μ = 0
−⎜
⎜
⎟
⎜
⎟
∂y
∂y
⎝
⎠
⎝
⎠
(24)
(25)
G (x , y ) = 0
(26)
G (x, y ) ≥ 0
(27)
μ G (x , y ) = 0 .
(28)
e
i
T
i
Problem (21)-(28) falls into the category of what is
known in the literature as a mathematical program with
equilibrium constraints (MPEC) [8]. As can be seen, the
vectors of Lagrange multipliers λ and µ become decision variables of the resulting single-level equivalent. In
Stockholm Sweden - August 22-26, 2011
addition, the following types of nonlinearities are present in problem (21)-(28):
- Nonlinear expressions (23) typically involving
products of market-clearing prices and power outputs. These sets of nonlinear products can be transformed into equivalent mixed-integer linear programming expressions based on the KKT conditions as described in [9].
- Products of scheduling variables x, typically binary, and continuous variables λ and µ in (24).
These nonlinearities can be equivalently formulated as linear expressions using well-known integer algebra results [10].
- Products of Lagrange multipliers and lower-level
decision variables y in the complementary slackness conditions (28). As shown by Fortuny-Amat
and McCarl [11], complementary slackness conditions can also be formulated as mixed-integer linear programming expressions. It is worth mentioning that under the assumption of linearity of the
lower-level problem, complementary slackness
conditions (28) can be replaced by the equality associated with the strong duality theorem [10],
thereby leading to a more effective linearization in
terms of computational performance.
Thus, after some algebra, problem (21)-(28) is recast
as a mixed-integer linear programming problem suitable
for commercially available branch-and-cut software.
4 APPLICATION
To illustrate the above bilevel programming framework, we now consider an instance of price-based market clearing, namely the payment minimization problem
presented in [5], hereinafter referred to as PM. As done
in [5], we assume that (i) the demand is inelastic, (ii)
generation offers comprise a single block, and (iii)
minimum up and down times, ramping limits, transmission network, and ancillary services are ignored. Notwithstanding, these simplifications do not alter our main
conclusion, and results could be extended to include all
of them at the expense of an increased notational complexity. For quick reference, PM is formulated as:
Minimize
p jk , s jk , v jk ,λ k
∑λ
k Dk
+
k∈K
∑∑ s
jk
(29)
j∈J k∈K
subject to:
⎛
∂⎜
⎜
⎝
λk =
∑∑
j∈J
)
is the start-up cost of unit j in period k; v jk is a binary
variable that is equal to 1 if unit j is scheduled on in
period k, being 0 otherwise; λ k is the market-clearing
price in period k; K is the index set of time periods; D k
is the demand in period k; J is the index set of generatis the coefficient of the start-up cost of
ing units; C SU
j
unit j; P jk and Pjk are the lower and upper bounds for
the power output of unit j in period k, respectively; and
C Pjk represents the offer cost coefficient of unit j in
period k.
The objective function (29) represents the total payment by consumers and comprises two terms: the payment for energy and the payment for generation startups. Constraints (30) and (31) model the start-up costs.
Constraints (32) impose the integrality of variables v jk .
Constraints (33) are the power balance equations. Constraints (34) set the lower and upper bounds for the
power outputs. Finally, constraints (35) model the marginal pricing setting. For the sake of unit consistency,
hourly time periods are considered.
According to Section 2, the bilevel programming
formulation for problem (29)-(35) is:
Minimize
s jk , v jk
∑∑ s
+
jk
(36)
j∈J k∈K
(
)
s jk ≥ C SU
v jk − v j,k −1 ; ∀j ∈ J, ∀k ∈ K
j
(37)
s jk ≥ 0; ∀j ∈ J, ∀k ∈ K
(38)
v jk ∈ {0,1}; ∀j ∈ J, ∀k ∈ K ,
(39)
where market-clearing prices λ k are obtained from:
p jk
(31)
subject to:
v jk ∈ {0,1}; ∀j ∈ J, ∀k ∈ K
(32)
∑
(33)
∑p
j∈J
jk
∑∑ p
P
jk C jk
(40)
j∈J k∈K
s jk ≥ 0; ∀j ∈ J, ∀k ∈ K
= D k : (λ k ) ; ∀k ∈ K
(
(41)
)
v jk P jk ≤ p jk ≤ v jk Pjk : γ jk , θ jk ; ∀j ∈ J, ∀k ∈ K, (42)
j∈J
17th Power Systems Computation Conference
k Dk
subject to:
(30)
v jk P jk ≤ p jk ≤ v jk Pjk ; ∀j ∈ J, ∀k ∈ K
∑λ
k∈K
s jk ≥ C SU
v jk − v j,k −1 ; ∀j ∈ J, ∀k ∈ K
j
p jk = D k ; ∀k ∈ K
(35)
where p jk is the power output of unit j in period k; s jk
Minimize
(
⎞
p jk C Pjk ⎟
⎟
k∈K
⎠
; ∀k ∈ K ,
∂D k
(34)
Stockholm Sweden - August 22-26, 2011
and where γ jk and θ jk are the Lagrange multipliers
per bound for the dual variable γ jk , and θ jk is the
associated with the lower and upper bounds for the
power output of unit j in period k, respectively.
The upper-level optimization (36)-(39) determines
the scheduling variables v jk and the corresponding
lower bound for the dual variable θ jk .
start-up costs s jk that minimize the consumer payment.
In contrast, the lower-level problem (40)-(42) determines generation power outputs p jk and the marketclearing prices λ k associated with the upper-level
scheduling variables v jk by solving a multiperiod economic dispatch based on the minimization of the generation offer cost.
Applying the methodology described in Section 3,
the bilevel programming problem (36)-(42) is transformed into the following equivalent single-level
mixed-integer linear program:
∑ λ D + ∑∑ s
Minimize
k
a jk ,a Ajk , b jk , b Ajk , p jk ,
k∈K
s jk , v jk , γ jk ,θ jk ,λ k
k
jk
(43)
j∈J k∈K
subject to:
s jk ≥ C SU
v jk − v j,k −1 ; ∀j ∈ J, ∀k ∈ K
j
(
)
(44)
s jk ≥ 0; ∀j ∈ J, ∀k ∈ K
(45)
v jk ∈ {0,1}; ∀j ∈ J, ∀k ∈ K
γ jk + θ jk + λ k =
C Pjk ;
(46)
∀j ∈ J, ∀k ∈ K
(47)
γ jk ≥ 0; ∀j ∈ J, ∀k ∈ K
(48)
θ jk ≤ 0; ∀j ∈ J, ∀k ∈ K
(49)
∑p
(50)
jk
= D k ; ∀k ∈ K
j∈J
v jk P jk ≤ p jk ≤ v jk Pjk ; ∀j ∈ J, ∀k ∈ K
∑λ
k Dk
+
k∈K
∑∑ a
j∈J k∈K
∑∑b
(51)
∑∑p
(52)
a Ajk = γ jk − a jk ; ∀j ∈ J, ∀k ∈ K
(53)
0 ≤ a jk ≤ v jk γ jk ; ∀j ∈ J, ∀k ∈ K
(54)
0 ≤ a Ajk ≤ 1 − v jk γ jk ; ∀j ∈ J, ∀k ∈ K
(55)
b Ajk = θ jk − b jk ; ∀j ∈ J, ∀k ∈ K
(56)
v jk θ jk ≤ b jk ≤ 0; ∀j ∈ J, ∀k ∈ K
(57)
jk Pjk
j∈J k∈K
(
=
tively.
5 NUMERICAL RESULTS
The bilevel programming framework for PM has
been tested on two case studies described in [5]. For
both cases, the results of PM are compared with those
achieved by a conventional market-clearing procedure
based on the minimization of the sum of the generation
offer costs and start-up costs. This cost minimization
problem is denoted as CM. The model has been implemented on an Intel Core i7, 1.73-GHz processor with 8
GB of RAM using Cplex 12.0 under GAMS.
5.1 Illustrative Example
The first example considers two hours and four generating units which are initially scheduled off. Table 1
shows the data for both periods. This small example is
useful to highlight the differences in the results yielded
by PM and CM.
D1 = 100 MW
Hour
1
jk P jk
P
jk C jk
+
Expressions (43)-(46) correspond to the upper-level
problem while constraints (47)-(58) equivalently replace the lower-level problem. Constraints (47)-(49) are
the dual feasibility constraints, (50)-(51) represent the
primal feasibility constraints, and (52)-(58) are related
to the linearization of the strong duality equality. Constraint (52) is the linear expression where the objective
functions of the primal and dual problems are equated.
Constraints (53)-(55) and (56)-(58) model the linearization of the product terms v jk γ jk and v jk θ jk , respec-
Unit
1
2
3
4
P j1
[MW]
5
5
0
5
j∈J k∈K
)
(1 − v jk ) θ jk ≤ b Ajk ≤ 0; ∀j ∈ J, ∀k ∈ K ,
(58)
where a jk represents the nonlinear product v jk γ jk ,
b jk represents the nonlinear product v jk θ jk , a Ajk and
b Ajk are auxiliary continuous variables used in the lin-
earization of the above nonlinear terms, γ jk is the up-
17th Power Systems Computation Conference
Hour
2
Unit
1
2
3
4
C Pj1
Pj1
[MW] [$/MWh]
50
10
40
20
10
65
60
30
D 2 = 150 MW
C SU
j
[$]
0000
0000
0050
1800
P j2
P j2
C Pj2
C SU
j
[MW]
5
5
0
5
[MW]
060
060
030
100
[$/MWh]
15
20
65
30
[$]
0000
0000
0050
1800
Table 1: Data for the illustrative example.
For this illustrative example, the computing time required to achieve the optimal solution to PM using the
single-level equivalent was less than 1 s.
The optimal solutions to CM and PM are summarized in Table 2. Under conventional market clearing,
generators 1, 2, and 3 are dispatched at maximum capacity in both periods whereas generator 4 is not sched-
Stockholm Sweden - August 22-26, 2011
uled. Generator 3 is the marginal unit in both periods
thereby setting the corresponding market-clearing prices
equal to its offer cost, i.e., $65/MWh. This solution
costs $6050 and yields a payment equal to $16300.
Unit
p jk [MW]
1
2
3
4
λ k [$/MWh]
Offer Cost + Start-up Cost [$]
Payment [$]
CM
Hour
1
2
50 60
40 60
10 30
00 00
65 65
6050
16300
PM
Hour
1
2
50 60
40 60
00 00
10 30
30 30
6400
9300
tion in payment is achieved by slightly increasing the
cost by 2.6%.
Hour
01
02
03
04
05
06
07
08
09
10
11
12
D k [MW]
2500
2550
2570
2530
2650
2700
2680
2740
2850
2900
3500
3700
Hour
13
14
15
16
17
18
19
20
21
22
23
24
D k [MW]
3800
3900
4200
4250
4400
4500
4100
3850
3400
2700
2300
2350
Table 2: Results for the illustrative example.
In contrast, PM results in a different schedule in both
periods. While generators 1 and 2 do not experience any
change in their generation levels with respect to the
solution to CM, generators 3 and 4 exchange their
schedules and power dispatches. As a consequence,
generator 4 becomes the marginal unit in both periods,
setting both market-clearing prices at $30/MWh. The
optimal payment is equal to $9300 and the associated
cost is equal to $6400. In other words, a 42.9% reduction in payment is attained at the expense of a 5.8%
increase in cost.
In this case, the market-clearing price in each period
under marginal pricing is identical to the corresponding
highest accepted offer. Note, however, that both pricing
schemes lead in general to different market-clearing
prices irrespective of the objective function being minimized. As an example, let the demand at hour 1 be
reduced so that it belongs to the interval (50, 55] MW.
At the optimal solution to either CM or PM, generator 2
would be dispatched at its minimum power output while
generator 1 would be the marginal unit by supplying the
remaining demand. Under marginal pricing, the marketclearing price would be the offer cost of generator 1,
i.e., $10/MWh, whereas the highest accepted offer is
that of generator 2, i.e., $20/MWh.
5.2 Medium-Sized Test Case
The second case study considers 25 generating units
and 24 hours. The hourly system demand is shown in
Table 3. Data for generators are presented in Table 4. It
is assumed that generators do not modify their respective supply offers over the time span. In addition, all
units are initially scheduled off except units 1-8.
The execution of Cplex was stopped when the value
of the payment was below a specified threshold or when
this number appeared to have reached a lower bound.
With these stopping criteria, Table 5 shows the results attained by the proposed approach for PM. The
computing time required by this solution was equal to
78 s. Table 5 also lists the results corresponding to the
optimal solution to CM. As can be seen, a 7.0% reduc-
17th Power Systems Computation Conference
Table 3: System demand for the medium-sized test case.
Unit
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
P jk
Pjk
C Pjk
C SU
j
[MW]
100
060
050
030
040
040
030
030
020
020
020
020
020
020
020
020
020
020
020
020
020
020
020
015
010
[MW]
455
350
300
200
350
320
240
200
190
180
170
160
150
140
130
125
120
115
110
120
115
110
100
090
080
[$/MWh]
30
34
35
37
40
42
45
47
55
57
58
59
60
62
63
65
66
68
70
75
78
80
90
93
95
[$]
1200
1150
1100
1000
0350
0350
0300
0300
0180
0180
0175
0160
0250
0200
0150
0180
0140
0140
0160
0250
0250
0300
0050
0050
0040
Table 4: Generation data for the medium-sized test case.
0Payment [$]
Offer Cost + Start-up Cost [$]
PM
4764845
3482645
CM
5122905
3394415
Table 5: Results for the medium-sized test case.
Stockholm Sweden - August 22-26, 2011
100
8
90
7
80
6
70
5
60
4
50
System demand (GW)
Market-clearing price ($/MWh)
Hourly market-clearing prices associated with the
solutions presented in Table 5 are depicted in Fig. 1.
This figure also shows the hourly system demand. As
expected, market-clearing prices resulting from either
PM or CM follow the shape of the demand curve. Note,
however, that market-clearing prices corresponding to
PM are less than or equal to those resulting from CM
for all periods.
3
40
30
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Hour
Market-clearing
price for CM
Market-clearing
price for PM
System
demand
Figure 1: Market-clearing prices and system demand for the
medium-sized test case.
6 CONCLUSION
This paper has formulated the price-based marketclearing problem under marginal pricing as a general
bilevel programming problem, offering flexibility in the
problem definition. The resulting bilevel programming
formulation is transformed into an equivalent singlelevel mixed-integer linear program. This transformation
comprises two steps. First, the lower-level optimization
is replaced by a set of constraints based on the KKT
optimality conditions and duality theory. Subsequently,
a number of nonlinearities are converted to linear
equivalents using some well-known integer algebra
results.
The ultimate goal of this paper is to provide the ISO
with a tool that can be used for comparative analysis of
different market-clearing procedures, so that informed
decisions can be made.
This general bilevel formulation and its single-level
equivalent have been applied to an instance of pricebased market clearing in which consumer payment is
minimized. Numerical results reveal that bilevel programming is an effective approach to address pricebased market-clearing procedures.
Research is currently underway to solve other instances of price-based market clearing such as those
explicitly maximizing the surplus of consumers and
producers, as well as market-clearing procedures with
revenue adequacy constraints. Finally, further research
will also be devoted to the analysis of joint energy and
reserve electricity markets.
17th Power Systems Computation Conference
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ACKNOWLEDGMENT
The authors acknowledge the support from the Ministry of Science of Spain under project ENE200907836, and from the Junta de Comunidades de Castilla
– La Mancha under project PAI08-0077-6243.
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Stockholm Sweden - August 22-26, 2011