Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
Determination of Market Clearing Price in Pool
Markets with Elastic Demand
Bijuna Kunju K∗ and P S Nagendra Rao
Department of Electrical Engineering
Indian Institute of Science, Bangalore 560012
kbijuna@gmail.com, nagendra@ee.iisc.ernet.in
∗ Department of Electrical and Electronics Engg
T.K.M.College of Engineering, Kollam 691005
Abstract— Determination of Market Clearing Price is one of
the prime functions of a pool operator. In many of the existing
markets, market clearing is based on stepped bids received from
generators and consumers. However, quadratic bid functions
have more information and are more realistic. A simple closed
form solution scheme to compute the clearing price from such
supply and offer bids submitted to the operator is proposed in
this paper.
I. I NTRODUCTION
The most popular form of electricity market so far has
been the centralized auction, which has often mimicked the
procedures used for central dispatch in the past, reflecting
concerns over system security [1], [2], [3]. After the auction,
some pool-based electricity markets operate under a systemwide market price [3] while others have elected the approach
of nodal (or locational) marginal prices [4]. The bids supplied
by the generators/generating companies indicate the quantity
of power they are ready to supply at a certain price in
terms of a price function [1], [2]. In many pool markets, the
consumers also submit offers in terms of quantity of power
required and a price function (at which they are prepared to
purchase) [5]. In the single-price auction, the market-clearing
price is determined by a merit order algorithm [6]: a wellestablished technique where all participating generators bids
are ordered according to increasing levels while participating
consumers bids are ranked according to decreasing levels. The
market clearing price is then defined by the intersection of the
aggregate demand and supply curves. This method has the
advantage that it is very simple to implement.
With this process the total demand that can be met and the
individual generations accepted/ loads supplied can be decided
fairly easily. The stepped bids in use are approximations of the
true cost function of the generating units. A basic requirement
for the success of market dictated utility operation is that
the bids must reflect the true costs of operation. Hence, it
is easy to see that use of quadratic functions to represent the
cost of generation would provide a better representation of
the true cost of generation. It may also be noted that the use
of quadratic functions to represent the cost of generation has
been in use in power systems for a long time now. Hence,
continuing to use this form of cost functions (bid functions)
should be more acceptable to the industry rather than adopting
the step form of cost functions. A similar argument holds good
for the bids of loads also.
It must be pointed out that there has been a considerable
body of research which attempts to forecast the MCP [7], [8],
[9], [10]. Such forecasts are meant to be used by the bidders
in order to tune their bids to maximize their individual profits.
Techniques based on artificial intelligence and probability
theory have been proposed for this purpose. It is easy to see
that such techniques are obviously not intended to maximize
the societal benefit and hence, cannot be expected to achieve
the desired goals of restructured markets.
The aim of this paper is to develop a method for determining
the market clearing price in the case of pool markets under
the assumption of no congestion. In this development, it is
considered that generator bid functions and the load valuation
functions are quadratic functions of real power. The maximum
and minimum limits on both individual generations and loads
are also accounted in this process. The proposed method has
been developed by extending an earlier method [11], meant for
determining the generation schedules in vertically integrated
power systems. However the method in [11] does not consider
the bids of consumers and it also requires the total demand to
be specified.
In this approach, a formula is developed in Section II to
find the clearing price where the generator and load bids are
quadratic functions of real power. It is then shown how this
formula can be used in a systematic way to handle the limits
on loads and generators and arrive at the final schedule. A
detailed illustration of the algorithm is given in Section III
and the paper is concluded in the next Section. In Section III,
methods of handling generators bids and consumers offerswith and without limits, fixed demands (block bids) and their
combinations are illustrated.
II. A F ORMULA
FOR
M ARKET C LEARING P RICE
Consider a system with N generators and M consumers. Let
the generator bid function for the ith generator be
Ci (P gi ) = ai P gi2 + bi P gi + ci
and the consumer benefit function for the j th load be
214
Bfj (P dj ) = αj P d2j + βj P dj + γj
Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
The objective of the pool market operator is to maximise the
social welfare function
M
Bfj (P dj ) −
j=1
N
Ci (P gi )
where
(2ak P gk + bk ) = λ
Hence the value of λ is obtained as
i=1
λ=
subject to the power balance constraint,
N
(P gi ) =
i=1
M
(P dj )
(1)
L=
N
Ci (P gi ) −
i=1
M
N
M
Bfj (P dj ) − λ( (P gi ) −
(P dj ))
j=1
i=1
λ=
and
dL
dBfj
=0
=
dλ
dP dj
(14)
M
βi
Bd =
( )
α
i
i=1
(2)
(15)
Solving (12) and (13), we have
∀ j
(3)
PR
Equations (2)and (3) imply that for optimality, the incremental
cost functions of all the generation as well as the incremental
utility function of all loads must be equal to λ. The incremental
cost for generators can also be written as
dCi
= λ = 2ai P gi + bi ,
dP gi
i∈N
(4)
At the optimum, the incremental costs of all generators are
same and we have,
=
λ =
ABd − Ad B
2(Ad − A)
(16)
(B − Bd )
(A − Ad )
(17)
The schedules for each of the generators and the demand
of each consumer that can be met is obtained as,
P gi
=
λ − bi
2ai
2ai P gi +bi = 2ak P gk +bk = λ ∀ i ∈ N, for a particular k ∈ N
(5)
2ak P gk + bk − bi
λ − βi
,
∀i∈N
(6)
P gi =
P di =
2ai
2αi
Let the total demand (total generation) given by (1) be equal
A. Incorporating the Limits
to PR . Therefore,
N
(P gi ) =
i=1
2ak P gk
N
2ak P gk + bk − bi
2ai
i=1
= PR
N
N
N
1
1
bi
( ) + bk
( )−
( ) = 2PR .
a
a
a
i
i
i
i=1
i=1
i=1
(7)
(8)
Define two parameters A and B,
A=
N
1
( )
ai
i=1
(9)
B=
N
bi
( ).
ai
i=1
(10)
and
(13)
M
1
( )
αi
i=1
Ad =
j=1
∀ i
2PR + Bd
Ad
where,
where λ is the Lagrangian multiplier. The conditions for
optimality of L are given by
dCi
dL
=0
=
dλ
dP gi
(12)
Similarly, with the demand offers, it can be shown that
j=1
Hence, the augmented objective function for unconstrained
optimisation is
2PR + B
A
and (8) can be written as
(2ak P gk + bk )A = 2PR + B
(11)
∀ i
(18)
∀ i
(19)
The formula in (17) for the market clearing price is valid
only if the corresponding generators and loads given by (18)
and (19) are within their limits. However, this may not always
happen. If at any stage of calculation of the system λ using
(17), the corresponding schedule results in limit violations,
then the violating loads / generations must be constrained
at the limits that they are violating and the net difference
between such allocated loads and generations has to be first
optimally allocated among the non violating generators / loads.
For this step also, the formula derived above is made use
of. After this step, the cost functions of the non violating
generators / loads are modified to reflect the partial allocation
of load / generation that has been done to the particular entity.
The process of finding the system market clearing price is
continued considering only the non violating generators / loads
and the modified incremental cost functions of these generators
and loads.
215
Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
B. Case 2: Maximum Limit on Generator Output
III. I LLUSTRATION OF THE METHOD
Consider a system consisting of three generators and two
consumers with variable demands. The generator cost functions of the three units as given in [11] are,
C1 = 0.003P g12 + 2P g1 + 80
Rs/h,
C2 = 0.015P g22 + 1.45P g2 + 100
Rs/h,
C3 = 0.01P g32 + 0.95P g3 + 120
Rs/h.
Bf1 = −0.002P d21 + 5P d1 + 150 Rs/h,
Bf2 = −0.001P d22 + 6P d2 + 200 Rs/h.
A. Case 1: No Limit Constraints
For this system the values of A, B, Ad and Bd are calculated
using (9), (10), (14) and (15)as
B=
1
1
1
+
+
= 500
0.003 0.015 0.01
2
1.45
0.95
+
+
= 858.33
0.003 0.015 0.01
Ad =
1
1
+
= −1500
−0.002 −0.001
β2 = 2 × −0.001 × 400 + 6 = 5.2
5
6
Bd =
+
= −8500
−0.002 −0.001
Now, the total demand that can be supplied is obtained using
(16) as
PR =
(500 × −8500) − (−1500 × 858.33)
= 740.63M W.
2(−1500 − 500)
The Market Clearing Price is obtained using (12) or (13) as
λ=
Or
λ=
2 × 740.63 + 858.33
= 4.6792 Rs/M W
500
2 × 740.63 − 8500
= 4.6792 Rs/M W
−1500
The schedules are then found using (18) and (19)as
P g1 =
4.679 − 2
= 446.5M W
2 × 0.003
P g2 =
4.679 − 1.45
= 107.64M W
2 × 0.015
P g3 =
4.679 − 0.95
= 186.46M W
2 × 0.01
P d1 =
4.679 − 5
= 80.2M W
2 × −0.002
P d2 =
4.679 − 6
= 660.4M W
2 × −0.001
2 × 400 − 8500
= 5.1333 Rs/M W
−1500
The allocation of 400 MW among the loads would be
corresponding to this λint of 5.1333. This value turn out to
be
5.133 − 5
P d1 =
= −33.3 M W
2 × −0.002
5.133 − 6
= 433.3 M W
P d2 =
2 × −0.001
The negative value implies the violation of minimum limit
for P d1 . So, the full 400 MW is allocated to P d2 . Now, the
remaining generators (No. 2 and No. 3) and demand bids are
to be used in the next stage. However, the incremental cost
function of P d2 is to be changed to 2α2 P d2 + β2 to account
for the allocation of 400 MW and the modified value of β is
obtained as
β2 = 2 × α2 × P d2 + β2
λint =
The offer functions of the two consumers are,
A=
In addition to the conditions in case 1, suppose that generator 1 has a maximum limit of 400 MW. Hence, the generation
of this unit must be constrained to 400 MW, instead of 446.5
MW calculated above. In addition, this generation has to be
allocated optimally among the customers. P gx1 = 400M W .
The corresponding λint is computed (13) using the values of
Ad and Bd obtained earlier
With this procedure for accounting of the allocated component
of this demand, the slope of the incremental cost function of
P d2 does not change, only the constant value changes. So,
for further scheduling, only the value of Bd will change to
Bd (Ad will remain the same).
5
5.2
+
= −7700
−0.002 −0.001
Since, the set of available generators has changed, the values
of A and B (9,10) are to be recalculated and the new values
are
1
1
+
= 166.67
A =
0.015 0.01
1.45
0.95
+
= 191.67
B =
0.015 0.01
From the new set, the demand that can be supplied is found
out as
(166.67 × −7700) − (−1500 × 191.67)
= 298.7501M W
PRn =
2(−1500 − 166.67)
Bd =
The new system λ being
2 × 298.7501 + 191.67
= 4.7350 Rs/M W
166.67
The individual shares of the loads and generations are calculated using (18,19) and the schedules are
λ=
P g2 =
216
4.7350 − 1.45
= 109.5 M W
2 × 0.015
Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
P g3 =
4.7350 − 0.95
= 189.25 M W
2 × 0.01
The additional demand that can be met is
PR =
4.7350 − 5
= 66.25 M W
P d1 =
2 × −0.002
4.7350 − 5.2
= 232.5 M W
P d2 =
2 × −0.001
The final schedules for the three generators are 400.0, 109.5
and 189.25 MW respectively. The values for the two final
demands are 66.25 and 632.5 MW (i.e.400M W +232.5M W )
respectively. The Market Clearing Price is 4.735 Rs/MW.
Since the generator with a lower cost has been constrained
at its maximum limit, the total demand supplied reduces from
a possible 740.63 MW level to 698.75 MW while the clearing
price shows a rise from 4.6792 to 4.735 Rs/MW.
It is not difficult to see that the solution obtained by this
method satisfies the Kuhn Tucker conditions also. The optimal
incremental cost for the total system is 4.735 Rs/MW where
as the incremental cost of the generator constrained at its
maximum limit is
λ1 = 2 × 0.003 × 400 + 2 = 4.4 Rs/M W
C. Case 3: Maximum Limit on Load
Consider that the schedule is as in case 1, and suppose that,
in addition, there is a maximum limit on the load at node
2 (specified) of 600MW. This implies a constraint violation.
Hence, first this load (600MW) is to be optimally allocated
between generators. An intermediate value of λ for allocating
this 600MW among the generators is found using (12) as
λint =
2 × 600 + 858.3333
2 × P rn + A
=
= 4.1167 Rs/M W
B
500
and the corresponding allocation are
P g1 =
4.1167 − 2
= 352.76M W
2 × 0.003
P g2 =
4.1167 − 1.45
= 88.6667M W
2 × 0.015
P g3 =
4.1167 − 0.95
= 158.3333M W
2 × 0.01
Now, the incremental cost functions of the generators have to
be modified to reflect this allocation. The slope ai of all the
generators remain unchanged, and it is easy to see that bi of
all the generators must be now taken as 4.1167. Now for the
next stage of optimisation, the value of B need to be calculated
again and is denoted as B .
4.1167 4.1167 4.1167
+
+
= 2058.35
0.003
0.015
0.01
Now, the demand has to be met is of consumer 1. The new
values of Ad and Bd corresponding to this situation are
This turns out to be the demand of consumer 1 P d1 =
110.4125 MW. The new clearing price is obtained using (16)
as
2 × 110.4125 + 2058.35
λ=
= 4.5584 Rs/M W
500
Or
2 × 110.4125 − 2500
λ=
= 4.5584 Rs/M W
−500
The contribution of the three generators towards this additional
load are
P g1 = 73.61M W
P g2 = 14.72M W
P g3 = 22.08M W
Hence, the final schedules for the three generators are 426.377
(352.76 + 73.61), 103.3867 (88.6667+14.72) and 180.4133
(158.333 + 22.08) MW respectively. The two demands are
110.4125 and 600.0 MW. The Market Clearing Price is 4.5584
Rs/MW. Here since the demand with a higher valuation is
constrained at the maximum limit, the total demand met gets
reduced from 740.63 MW to 710.4125 MW while the clearing
price decreases from 4.735 to 4.5584 Rs/MW.
D. Case 4: Maximum Limits on Both Generators and Consumer Loads
Here, we consider both the constraints to be operative
simultaneously. The generation 1 has a maximum limit of 400
MW and the maximum demand of consumer 2 is 600 MW.
From the optimal schedule of case 1 it is seen that both limits
are violated. So P g1 is fixed at 400 MW and P d2 is fixed at
600 MW. The difference of these two values is 200 MW of
demand. This demand is optimally distributed among the two
generators not set at their limits as follows. The intermediate
value of λ is obtained using values of A and B already
calculated in (Case 2) as 166.67 and 191.67 respectively.
2 × P r + B 2 × 200 + 191.67
=
= 3.55Rs/M W
A
166.67
and the corresponding allocation to the two generators are
λint =
B =
Ad =
1
= −500
−0.002
Bd =
5
= −2500
−0.002
(500 × −2500) − (−500 × 2058.35)
= 110.4125 M W
2(−500 − 500)
P g2 =
3.55 − 1.45
= 70M W
2 × 0.015
P g3 =
3.55 − 0.95
= 130M W
2 × 0.01
The incremental cost function of the two generators now have
to be modified such that they have the same value of bi equal to
3.55 with their ai values remaining unchanged. So, the value
of B is calculated again as
B =
217
3.55
3.55
+
= 591.67
0.015 0.01
Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
Now, the demand that is to be met is only that of consumer
1. Hence, the new value of Ad and Bd are
Ad = −500
The values of A and B are then recalculated with three
generators only. The new values are A = 123.1429 and B =
216.
λint =
Bd = −2500
2 × P r + A
2 × 75.4 + 216
=
= 2.9787Rs/M W
B
123.1429
Hence, the additional load that can be met is
This value of λint is still lesser than the bi of the generators
at 13, 23 and 27 and their output has been correctly allocated
(166.67 × −2500) − (−500 × 591.67)
PR =
= 90.625 M W to be zero. Using this λint we get,
2(−500 − 166.67)
P g1 = 24.4675M W
Therefore, the demand of consumer 1 is also 90.625 MW.
P d1 = 90.625 MW. The new clearing price is
2 × 90.625 + 591.67
λ=
= 4.6375
166.67
P g2 = 35.1057M W
Rs/M W
P g22
= 15.8296M W
Or
The incremental cost function of these three generators now
have to be modified such that they have the same value of bi
as 2.9787 with their ai values remaining unchanged. So, the
value of B is calculated again as
2 × 90.625 − 2500
λ=
= 4.6375 Rs/M W
−500
The contribution of the two generators towards this load is
4.6375 − 3.55
= 36.25 M W
2 × 0.015
4.6375 − 3.55
P g3 =
= 54.375 M W
2 × 0.01
The final schedules are generations 400.0, 106.25 and 184.375
MW respectively. The final demands are 90.625 and 600.0
MW. The Market Clearing Price is 4.6375 Rs/MW.
P g2 =
B =
3.25
3
3
2.9787 2.9787 2.9787
+
+
+
+
+
= 998.3719
0.02
0.0175 0.0625 0.0083 0.025 0.025
The total demand of the nodes with price dependent loads is
then determined by (16)
PR =
(323.624 × −1410) − (−300 × 998.3719)
= 125.72M W.
2(−300 − 323.624)
The new clearing price is found by (17)
λ = 3.862Rs/M W
E. Case 5: Mixed Load - (No Limit violation)
This example is provided to illustrate the method when the
system has both fixed loads and price sensitive loads. The 30
bus system is used for this illustration. The bids and offers
details of the 30 bus system used here is given Tab. I. There
are loads which do not submit offer function implying that
they are ready to accept power at any cost. In this system the
total fixed demand by those consumers is 75.4 MW.
The individual share of generations and loads are then found
from the clearing price by substituting in (18,19)
TABLE II
T HE SCHEDULES OBTAINED FOR 30 BUS SYSTEM
Gen No.
1
2
22
27
23
13
Gen.
TABLE I
T HE BIDS OF 30 BUS SYSTEM
Gen No.
1
2
22
27
23
13
ai
0.02
0.0175
0.0625
0.0083
0.025
0.025
bi
2.00
1.75
1.00
3.25
3.00
3.00
Dem No.
2
7
8
12
21
30
αi
-0.02
-0.02
-0.02
-0.02
-0.02
-0.02
βi
6.0
4.4
4.8
4.2
4.8
4.0
P gi
24.47
35.10
15.83
0.0000
0.0000
0.0000
75.4
P gi
22.11
25.27
7.10
36.74
17.25
17.25
125.72
P gif inal
46.58
60.37
22.93
36.74
17.25
17.25
201.12
Dem No.
2
7
8
12
21
30
V ar.Dem.
P di
53.45
13.45
23.45
8.45
23.46
3.46
125.72
F. Case 6: Mixed Load with Limit Violation
Consider the system as in case 5, and suppose that in
addition there is a maximum limit of 50 MW on the demand
= 50M W . The steps of Case 5 are
of consumer 2, P dmax
2
repeated until its last stage, where it is seen that the load
assigned to consumer 2 is greater than its maximum value.
Hence the demand of consumer 1 is fixed to 50 MW, P d2 =
50M W . After the fixed demand allocation, this demand is
2 × P r + A
2 × 75.4 + 323.6248
λint =
=
= 2.928Rs/M W allocated. So the values of A and B are taken considering
B
847.5663
all six generators (323.6248 and 998.3719 respectively). Now,
It is clear that for three generators, 13, 23 and 27 their bi value this demand is distributed optimally among the generators as
is greater than this λint and hence their output is set to zero. demonstrated earlier. The intermediate λ considering all six
The value of A for the set of supply bids is 323.6248 and
B is 847.5662. The corresponding values for demand bids are
Ad is −300 and Bd is −1410.
In the first step, the fixed demand is allocated to the
generators. Considering all the six available generators share
the demand, the value of λint is first calculated
218
Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
generators, for this load of 50 MW turned out to be 3.3940
Rs/MW.
2 × P r + A
2 × 50 + 323.6248
λint =
=
= 3.394Rs/M W
B
998.3718
The schedules of generators are
P g1 = 10.3825M W,
P g2 = 11.8657M W
P g22
= 3.3224M W,
P g27
= 8.6403M W
P g23
= 7.88M W,
P g13
= 7.88M W
The next step of scheduling requires the modified cost
function for all the generators. The value of bi of all generators
is now 3.394. The values of A, B, Ad and Bd for the new set of
available demands and generators and with the modified cost
functions are 323.6248, 1098.4, -250 and -1110. The value of
λ obtained is 3.8499 Rs/MW and the additional demand is
73.762 MW. There is no additional constraint violation and so
the schedule is final. The generations and demands met are
λ = 3.8499Rs/M W,
P g1 = 46.288M W,
PR = 199.1M W
P g2 = 59.9985M W
P g22 = 22.7996M W,
P g27 = 36.0082M W
P g23 = 17.000M W,
P g13 = 17.000M W
P d2 = 50.00M W,
P d7 = 13.725M W
P d8 = 23.7525M W,
P d12 = 8.7525M W
P d21 = 23.7525M W,
P d30 = 3.7525M W.
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The demand with high valuation is limited in quantity and
hence the system incremental price reduces as compared with
the previous case.
In this case, the fixed demand can be considered as a block
bid with infinite offer. The offer can have a format of specified
quantity of power at a specified price also. However, in such a
case, the fixed demand will be supplied only when the system
incremental cost is same as the offer price of the demand (not
the first demand to be supplied as done in this case).
IV. C ONCLUDING R EMARKS
A method of determining the Market Clearing Price and
the schedules based on the bids submitted by the consumers
and generators has been proposed in this paper. The bids
are assumed to be quadratic functions of real power instead
of the stepped bids currently in use. A close form solution
(formula) for determining the system λ based on bids has
been developed. It is shown that this formula can be used
in a systematic way when limits on the generation/loads have
to be accounted. The algorithm is simple and give the global
optimal solution for this problem. The method can be used
to find the market schedules when network constraints need
not be considered; such results are also useful in assessing the
impact of network constraints on the market schedules and
prices.
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