Copyright@ Feng Zhao
March 10, 2006 ~
1
Appendix for the PES2007 Paper - Bid Cost
Minimization vs. Payment Cost Minimization:
A Game Theoretic Study of Electricity Markets
Feng Zhao, Peter B. Luh, Fellow, IEEE, Ying Zhao, Joseph H. Yan, Senior Member, IEEE, Gary A.
Stern, and Shi-Chung Chang, Member, IEEE

I. DERIVATIONS FOR TWO-SUPPLIER GAMES (DEC. 9)
Consider an energy market with T hours indexed by t (1 t
T) with demand given for each hour and with I suppliers
indexed by i (1 i  I). Supplier i submits one single-block bid
to the market containing the min/max bid levels pi min/pi max, the
bid price ci $/MWh, and the startup cost Si $/Start for all
hours. The supplier’s incremental production cost is ci,0
$/MWh and the actual startup cost is Si,0 $/Start. Auctions are
solved to select bids and their levels based on bid cost
minimization or payment cost minimization auction models.
The selected bids are then paid at uniform market clearing
prices, and the startup costs incurred during the auction hours
are fully compensated. Denote the market clearing price at
Hour t is MCP(t). The selection status of Supplier i's bid at is
denoted by a zero-one variable xi(t) with “1” representing
“On” or “Selected” and “0” representing “Off” or “Not
selected,” and the selected power level is denoted by pi(t).
Now consider the game with two suppliers under the
payment cost minimization. The initial status of each supplier
is assumed to be “Off.” The Nash equilibria are obtained as
below. Depending on each supplier’s ability to meet the total
demand, three cases are considered to represent different
market situations: both suppliers can supply the system
demand; only one can do so; and neither can. For each case,
corresponding aspects under payment cost minimization and
bid cost minimization are compared. For concise presentation,
the time index t is omitted for the rest of this section since only
one hour is considered.
Case 1: PD[p1 min, p1 max] and PD [p2 min, p2 max]
In this case, both suppliers can meet the system demand.
Therefore, no one is guaranteed to be selected. To have a
compact solution form, denote by Ai the actual cost and Bi the
bid cost of Supplier i for supplying PD, i.e.,
Ai  ci ,0 p D  S i ,0 ,  i  1, 2 ,
(1)
Bi  ci p D  S i ,  i  1, 2 .
(2)
Payment Cost Minimization
Supplier 1’s reaction is:
Feng Zhao is currently with the Department of Electrical and Computer
Engineering, University of Connecticut, Storrs, CT 06269-1157 USA
Feng Zhao, Proprietary Confidential,
feng.zhao@uconn.edu
B2  , if B2  A1 2,

r1 : B1   A1  , if B2  A1 ,
 A  , if B  A .
2
1
 1
(3)
By symmetry, Supplier 2’s reaction is:
 B1  , if B1  A 2 2,

(4)
r2 : B2   A2  , if B1  A 2 ,
 A  ,
if
B

A
.
1
2
 2
Without loss of generality, let A1<A2. The reaction curves r1
and r2 are depicted in Figure II-1.
B2
(A2, A2+)
r1
r2
A2+
A1+
A 1+ 
A2+
B1
Fig. 1. Reaction curves of the two suppliers with A1<A2 in Case 1.
The intersection point, which is the Nash equilibrium is
obtained as:
( A2 , A2  ε ), if A1  A 2 ,

*
*
(5)
( B1 , B2 )  ( A1  ε, A2  ε ), if A1  A 2 ,
 ( A  ε, A ), if A  A .
1
1
2
 1
The expected production costs for the two suppliers at
equilibrium are:
( A2  A1 ,0), if A1  A 2 ,

(6)
( E1 , E 2 )   (ε / 2, ε / 2), if A1  A 2 ,
 (0, A  A ), if A  A .
2
1
1
2

Bid Cost Minimization
It can be verified that the two auctions will yield the same
solution under the current Case 1. As a result, the equilibrium
solutions are the same as (II-5) and (II-6).
Case 2: PD > p1 max and PD[p2 min, p2 max]
In this case, Supplier 2 has to be selected since Supplier 1
alone cannot provide PD.
Copyright@ Feng Zhao
March 10, 2006 ~
Payment Cost Minimization
To obtain Supplier 1’s reaction with Supplier 2’s strategy
given, Supplier 1 is considered to be selected or not selected.
For Supplier 1 to be selected, it has to bid zero startup cost and
a lower price than Supplier 2. Then Supplier 1 will be selected
at p1 max, and MCP is Supplier 2’s bid price. Supplier 1’s
profit maximization problem is
Max (c1 s1) (c2 - c1,0)p1 max + S1-S1,0,
(7)
s.t. c1 < c2,
(8)
S1 = 0,
(9)
c1 > c1,0+S1,0/p1 max.
(10)
Equations (8)-(9) guarantee the selection of Supplier 1.
Equation (10) ensures the positive profit. Note that the
objective (7) is not affected by c1. Thus, any feasible solution
of the above maximization problem is an optimal solution. If
the problem has no solution, Supplier 1 will choose to stay off
by bidding high, e.g., c1 > c2. Supplier 1’s reaction can be
represented by
S
S

(c1,0  1,0   , S1,0 ), if c2  c1,0  1,0 ,

p1max
p1max

(c1 , S1 ) : 
(11)
S1,0
(c1 , 0), for c1  [c1,0 
, c2   ], otherwise.

p1max

Given Supplier 1’s strategy, Supplier 2 should bid its startup
cost as high as possible since the cost is always fully
compensated and does not affect the amount of power
selected. Since Supplier 1 will not be selected if S1 is positive.
To select c2, S1 is assumed to be zero. One option is to choose
a value as high as possible to set MCP. Another option is to
bid a lower price than Supplier 1 so that more power, i.e., PD
MW, is selected. Denote the price and startup cost caps by ccap
and Scap, respectively. Therefore, Supplier 2’s profit under the
first option is
2 = (ccap – c2,0)( PD - p1 max) + Scap -S2,0.
(12)
Supplier 2’s profit under the second option is
2 = (c1- – c2,0)PD + Scap -S2,0.
(13)
Then Supplier 2’s reaction can be represented by
c1  c2,0
 cap cap
P D  p1max

,
(c , S ), if cap
( c2 , S 2 ) : 
(14 )
c  c2 , 0
PD
 (c   , S cap ), otherwise.
1

(8)
Let c1,0+S1,0/p1 max < ccap(1–p1 max/PD)+. The two reaction
curves are then depicted in Figure II-2.
c2
2
The intersection point, which is the Nash equilibrium, is
obtained as:
(c , c cap ), for c1  [ K 1 , K 2 ] , if K 1  K 2 ,
(15)
(c1* , c 2* )   1
None, if K 1  K 2 ,

where
K1  c1,0 
S1,0
p1 max
, K 2  (c cap  c 2,0 )(1 
p1 max
PD
)  c 2, 0 .
The expected production costs for the two suppliers at
equilibrium are:
c p
 S , c ( P D  p1max )  S 2,0 , if K1  K 2 ,
( E1 , E2 )   1,0 1max 1,0 2,0
N / A, elsewise,

(16)


Bid Cost Minimization
To obtain Supplier 1’s reaction curve, the supplier’s profit
maximization problem with given Supplier 2’s strategy as is
presented as follows:
Max( c1 , S1 ) MCP  p1  S1  c1,0 p1  S1,0   x1 .
(17)
Given Supplier 2’s strategy, Supplier 1 is selected to obtain
positive profit by bidding a lower price than Supplier 2. Then
Supplier 1 will be selected at p1 max, and MCP equals Supplier
2’s bid price. Therefore, the Supplier 1’s profit maximization
problem is represented as
max c1 , S1  c2  c1, 0  p1,max  S1  S1,0 .
(18)
When both Supplier 1 and Supplier 2 are selected, the bid cost
is smaller than the bid cost when only Supplier 2 is selected.
The following (19) guarantees that Supplier 1 is selected
c2 d  p1,max   S 2  c1 p1,max  S1     c2 d  S 2 .
(19)
From (19), we get
S1  c2  c1  p1,max    .
(20)
In order to simplify the resolution, we assume=p1,max.
Therefore, (20) is represented as:
S1  c2  c1    p1,max .
(21)
Equation (22) ensures the positive profit.
p1,maxc1, 0  S1, 0  p1,max c2     S1 .
(22)
Note that the objective (18) is only affected by S1.
S1
A
(0,(c2-)p1max
)
L1
p1max(c1,0-c2-)+S1,0
Equilibria
L2
ccap
c1p1max
r2
Fig. 3.
K1
r1
K2
ccap
c1
Fig. 2. Reaction curves in Payment Cost Minimization Case 2.
Feng Zhao, Proprietary Confidential,
feng.zhao@uconn.edu
In Fig.3, the Lines L1 and L2 present the condition that
Supplier 1 selected (21) and Supplier 2’s positive profit (22),
respectively. At point A, the maximum value of S1 is obtained.
Therefore, the maximum profit is obtained when S1 = (c2-)
p1,max and c1 = 0.
Adding (21) and (22), when c1 = 0 we get
Copyright@ Feng Zhao

1  S1,0
 c1,0     c2 .


2  p1,max

March 10, 2006 ~
(23)
When (23) is satisfied, the problem has solution. Otherwise,
Supplier 1 will choose to stay off by bidding high, e.g., c1> c2.
Therefore, Suppler 1’s reaction can be represented by


1  S1,0
 c1, 0     c 2 ,
0, c 2    p1,max  , if

r1 : 
2  p1,max




c


,
0
,
otherwise
.
 2
(24)
At a first glance, the solution seems unreasonable. Why the
bid price of Supplier 1 is zero? In case 2, Supplier 2 must be
selected and supplier 2’s will bid higher than supplier 1.
Therefore, Supplier 1’s bid price will not influent MCP. In the
bid cost minimization, Supplier 1 is paid as MCP. So Supplier
1 will maximize his profit by bidding high start up cost.
Supplier 2’s reaction with regard to Supplier 1’s strategy is
derived as follows.
Given Supplier 1’s strategy, Supplier 2’s profit
maximization problem can be represented as:
max c2 ,S2 p2 MCP  c2,0   S 2  S 2,0 .
(25)
Supplier 2 has two options. Under the first option, Supplier
2 bids as high as possible to set MCP when Supplier 1 is
selected. Under the second option, Supplier 2 can bid a lower
price than Supplier 1 so that all power is selected. Under both
options, Supplier 2 should bid its startup cost as high as
possible since the cost is always fully compensated and does
not affect the amount of power selected. Denote the price and
startup cost caps by ccap and Scap, respectively. We assume that
Supplier 1 and Supplier 2 have the same price and startup cost
caps.
Under the first option, Supplier 2’s profit maximization
problem is represented as
Maxc2 ,S2  c2  c2,0 d  p1,max   S 2  S 2,0 .
(26)
Supplier 2 must be selected to supply part of energy.
Therefore, Supplier 2 could bid as high as possible. When c2 =
ccap, Supplier 2 maximizes his profit. Therefore, Supplier 2’s
profit under the first option is represented as
c
cap

 c2,0 d  p1,max   S cap  S 2,0 .
(27)
Under the second option, Supplier 2’s profit maximization
problem is
 2  Maxc2 , S2  c2  c2,0 d  S 2  S 2,0 ,
(28)
s.t. c2    c1 .
(29)
Supplier 2’s profit under the second option is
 2  c1    c2,0 d  S cap  S 2,0 .
(30)
The following (31) guarantees that both Supplier 1 and
Supplier 2 are selected.
c cap d  p1,max   S cap  c1 p1,max  S1  c1   d  S cap (31)
The following (32) ensures the profit under the first option is
larger than the second one.
c cap  c2,0 d  p1,max   S cap  S 2,0  c1    c2,0 d  S cap  S 2,0


(32)
From (31) and (32), we get
Feng Zhao, Proprietary Confidential,
feng.zhao@uconn.edu
S1
p1,m ax
3
 c 2, 0  c1 .
(33)
Otherwise, the second option is selected.
Then Supplier 2’s reaction can be represented by
S1
 cap cap
 c2.,0  c1 ,
 c , S , if p
r2 : 
(34)
1,max
 c   , S cap , otherwise.
 1
In (20), there are three variables c2, c1, S1 and in (34) two
variables c1, S1 exits. The Nash solution is the intersection of
Suppliers’ reactions. We draw the figure separately when c1 =
0 and S1 = 0.




c2
ccap
r2
Nash Solution
r1
ccap
S1/p1 max
c1
Scap/p1 max
Fig. 4. Reaction curves with c2,0<(S1,0/p1max+c1,0+)/2 for Bid Cost
Minimization Case 2
If c2,0>(S1,0/p1max+c1,0+)/2, then there is no intersection.
Otherwise, there are multiple intersections and the dominant
one is {((S1,0/p1max+c1,0+)/2,0), ((S1,0/p1max+c1,0)/2, Scap)}. At
the equilibrium, Supplier 2 bids at (S1,0/p1max+c1,0)/2 so that
Supplier 1 has no chance to be selected without losing money.
It can be easily verified that the solution is unique.
Case 3: PD > p1 max, p2 max and PD< p1 max+ p2 max
In this case, both Suppliers have to be selected since one
supplier alone cannot provide PD.
Payment Cost Minimization
Given P1’s strategy, Supplier 2 has two choices: (a) Bid low
(c2<c1) so that p2 max MWs will be selected with profit (c1c2,0)p2 max+S2 - S2,0; (b) Bid at price cap so that MCP= ccap and
(PD -p1 max) MWs will be selected with profit (ccap - c2,0)(PD -p1
max)+ S2 - S2,0. Supplier 2’s reaction is obtained based on profit
maximization by comparing the above two options. Therefore,
Supplier 2’s reaction is obtained as:

c1  c2,0 P D  p1 max
cap

,
 ( c   , S ), if cap
r2 :  1
(35)
p2 max
c  c2 , 0
( c cap , S cap ), otherwise.

By symmetry, Supplier 1’s reaction is obtained as:

c2  c1,0 P D  p2 max
cap

,
 ( c2   , S ), if cap
r1 : 
(36)
p1 max
c  c1,0
( c cap , S cap ), otherwise.

The two reaction curves are depicted in Figure 5. It can be
seen that no Nash equilibrium exists in this case.
Copyright@ Feng Zhao
March 10, 2006 ~
c2
c2
r1
ccap
ccap
4
r2
r2
c1,0-(Scap-S1,0)/p1max+
ccap
r1
c2,0-(Scap-S2,0)/p2max+ ccap
c1
c1
Fig. 6. Reaction curves for bid cost minimization Case 3.
Fig. 5. Reaction curves in for Payment Cost Minimization Case 3.
There is no Nash solution in case 3. The two suppliers
could not get the maximum profit at the same time.
Bid Cost Minimization
The bid cost minimization problem is presented as
min c1 ,S1 ,S2 ,c2  c1 p1  S1  c2 p2  S 2 .
(37)
The profits of Supplier 1 and Supplier 2 are
 1  MCP  c1,0  p1  S1  S1,0 .
(38)
 2  MCP  c2, 0  p2  S 2  S 2,0
(39)
Given Supplier 1’s strategy (c1, S1), Supplier 2 has two
options: (a) Bid low (c1 > c2) so that p2,max MWs will be
selected; (b) Bid at price cap so that MCP = ccap and (PD p1,max) MWs will be selected. Supplier 2’s profit maximization
problem under the first option is represented as:
max c2 ,S2  c1  c2,0  p2,max  S 2  S 2,0 .
(40)
To maximize the profit of DM2, the start up cost of DM2
equals Scap. The bid cost under the first option is smaller than
the cost under the second option. Supplier 2 will be selected
to supply p2,max MWs.


c1 P D  p 2,max  S1  c2 p 2,max  S cap  


 c1 p1,max  S1  c 2 P D  p1,max  S cap
(41)
The following Eq. (42) guarantees that the profit of Supplier 2
is positive under the first option.
c1 p2,max  S cap  c2,0 p2,max  S 2,0   .
(42)
From (41) and (42), we get the constraints under the first
option.
P D    p1,max  p2,max .
(43)
c1  c2 .
(44)
S
cap
 S 2,0  c2,0  c1 p2, max   .
(45)
Therefore, the reaction of Supplier 2 is represented as

S cap  S 2,0    ,
cap
 c1   , S , if c1  c2,0 
r2 : 
 cap cap
 c ,S


p 2,max
(46)
otherwise.
By symmetry, Supplier 1’s reaction is:

S cap  S1,0    ,
cap
c2   , S , if c2  c1,0 
p1,max
r1 : 
 cap cap
c , S , otherwise.
The reaction curves r1 and r2 are shown in Figure 6.
Feng Zhao, Proprietary Confidential,
feng.zhao@uconn.edu
(47)