1
Real and Reactive Power Prices and Market Power HyungSeon Oh and Robert J. Thomas
Abstract – The exercise of market power is a major concern in deregulating the electric energy business in the U.S. Exploitation of market power can destroy any market efficiencies and award unwarranted profits based solely on the lack of competition rather than on economic merit. There have been several attempts to quantify market power with much of the work focused separately on either real power or reactive power. This paper addresses a new possibility for a market participant to exploit an opportunity to increase revenue by gaming both real and reactive markets.
the earning of the generators. With a given relationship, they exploit markets for both real and reactive power by gaming. This paper addresses the issue of the relationship between two prices for a given dispatch. In this paper, generators submit offer for real power only, but they get paid for reactive power at the nodal pricing which is a shadow price.
Index Terms – Market power, nodal price, null space, Kuhn Tucker optimality conditions, reactive power
Based on offers submitted by participating generators, a system operator finds an optimal solution to minimize whole system cost while all constrains are met such as power . balance equations and voltage constraints etc.
I. Introduction The deregulation of the world-wide electric power industry has been on-going for several decades. A focus of deregulation has been real power adequacy. System operators decide the quantity of reactive power production and consumption of participating generators to preserve system operational reliability. Generators, however, did not charge financial reward since reactive power was traditionally thought as an integral part of the real power dispatch. In order to maintain a good voltage profile, reactive power should be considered. There have been several studies on a proper method for reactive power pricing [1 – 9]. Reactive power supplies may make subset of generators must-run. Such a case is often observed in contingencies, which results in market power. For real power, there have been developed to quantify market power; for example, index-based methods [10 – 13] and sensitivity matrix approach [14 – 16]. To estimate market power for reactive power, same index-based methods have been applied [17 – 21]. When the reactive power is only a part of constraints, the Kuhn Tucker multipliers associated with the constraints determine the price of the reactive power when binding. In such a case, the price of reactive power is determined based on location as well as the price of real power in general. If generators get paid for reactive power at the shadow prices, the relationship between two prices is important to increase
This project was support in part by the US Department of Energy through the Consortium for Electric Reliability Technology Solutions (CERTS) and in part by the National Science Foundation Power Systems Engineering Research Center (PSERC). H. Oh and R. J. Thomas are with the Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 (e-mail:
[email protected];
[email protected]).
II. Change in the Price of Reactive Power
min cT g g subject to
(1)
& f ( g , y )# f ( z, y) = $ 1 ! % f 2 ( y) " where f1(g,y) contains the network active power constraints at the generator buses, f2(y) contains the rest of binding constraints (other power flow constraints, transmission capacity constraints, voltage constraints), and y represents all remaining relevant variables in an OPF. After obtaining a solution for the optimization problem (1), one can reconstruct a problem whose solution is same as before:
min #T g g subject to PT2 + Q T2 = C 2 ! H gk "g + H qk "q = 0 1T g = PD + Ploss
(2)
1T q = Q D + Qloss V m = V mcr q w = q wcr ! q w = IIq = q wcr where ! is the price paid for real power and the H matrices relate changes in real and reactive power generation with flow over the k-congested lines having capacities C. II represents a matrix comprised of w-unit row vectors when w is the number of binding reactive constraints. Constraints (2) are only the binding constraints and are therefore are a subset of the inequality constraints in (1). The first constraint in
2 equation (1) is derived in Appendix I. The solution to (2) is the same as the solution to (1). Note that the problem is a Linear Program and therefore the subset of binding constraints does not change under small perturbations in the solution. This is easily proved by using the strong duality property of linear programs. Thus, only binding constraints are considered in the second optimization problem. The real and reactive dispatch sensitivity matrix relating real power to energy prices is given by:
& I 0# '1 !! M = B '1 V T (VB '1V T ) VB '1 ' I $$ %0 0"
[
]
(3)
where B and V are as derived in Appendix II. The sensitivity matrix shows how changes in real and reactive power dispatch are related to small price changes, i.e.,
& '. # - M gg & 'g # !=+ $$ !! = M $$ ! ' q % " % 'LMPq " , M qg
M gq *& '. # $ ! M qq ()$% 'LMPq !"
(4)
where Dx is a diagonal matrix whose elements are composed of the vector x. Since Mg and Mq are known, one can find the revenue sensitivity matrix N for a given R.
IV. Simulation results and discussion A modified IEEE 30 bus system with 6 generators and 20 loads was used for a simulation as shown in Fig. 1. In the system, there are three different areas divided due to line constraints. Due to strict line constraints between Area 2 and the rest of the system, Firm 5 and 6 have a locational benefit in case of heavy load in the Area 2, which implies a potentially duopoly situation. It is frequently observed that two lines connecting Area 2 and other areas are congested and several voltage constraints are binding. As a result, the reactive power constraints of some generators are binding. In that case, they have nonzero nodal price for reactive powers. Consequently they have financial reward if reactive powers are paid, while others do not.
where "! and "LMPq represent price changes for real and reactive power, respectively. The sensitivity matrix is useful for studying market power if market participants have an ability to affect both prices. However, under many current market designs, only changes in the price of real power are possible. Consequently, any change in the price of reactive power depends on the price change of real power. Therefore, it is more convenient to represent changes in dispatch with respect to the change in real power price in the following way:
( "g % . M gg M gq +( I % && ## = , )&& ##"! ' "q $ - M qg M qq *' R $ . M gg + M gq R + (M g % #"! =, "! = && ) # - M qg + M qq R * 'Mq $
(5)
"LMPq = R"!
(6)
Fig. 1. Modified IEEE 30 bus system with six suppliers. Capacities of lines connecting Area 1–Area 2 and Area 3–Area 2 are lower than those of other lines.
IV-I. Case A Detailed derivation for R matrix is listed in Appendix II. Then, given the M matrix, one can determine how revenue changes with respect to the change in real power price given the R matrix as follows:
"ri = "(!i g i + ! q ,i q i ) ( % = !i & ) M gij "! j # + g i "!i j ' $ ( % ( % + ! q ,i & ) M qij "! j # + q i & ) R ij "! j # ' j $ ' j $ = D! M g + D g + D! M q + Dq R i "!
[ "r = (D M !
= N"!
q
g
+ D g + D! M q q
] + D R )"! q
(7)
(8)
In Case A, the reactive constraints of Firm 2 and Firm 6 are binding, and their nodal prices for the reactive power are about $ 22.8/MVar-hr and $ 111/MVar-hr, respectively. For comparison, their real power prices are $ 73/MW-hr and $ 683/MW-hr, respectively. Table I shows the revenue sensitivity matrix obtained for a situation that generators receive financial reward for reactive power at the nodal price. Selection of columns and rows in N matrix corresponding to Firm 5 and Firm 6 gives a sub matrix that gives information of their potential cooperation.
& 'rG 5 # -. 1.444 1.892 *& '0G 5 # $$ !! = 10 4 / + !! (9) ($$ ' r , 1.893 . 2.477 )% '0G 6 " % G6 " Using Equation (9) one can construct a map showing how their earnings change with respect to the change in their offers as shown in Figure 2. The area where changes in
3 offers result in increase in their earnings is termed a “winwin” region. In this case, “win-win” region is placed in the first quadrant. In other words, Firm 5 and Firm 6 both increase their earnings by raising their offers, which would make the market less efficient.
price changes market power structure. Firm 6 has two ways to maximize its earning from both real and reactive power payment while Firm 5 has one. Table II. Revenue sensitivity matrix shown without financial reward for reactive power in the same unit used in Table I.
Table I. Revenue sensitivity matrix shown in equation (8) in the unit of 10,000 [MWh]
N
G1
G2
G3
G4
G5
G6
N
G1
G2
G3
G4
G5
G6
G1
-7.04
6.32
0.61
0.14
-0.23
0.21
G1
-7.04
6.32
0.61
0.14
-0.23
0.21
G2
6.93
-9.32
1.12
1.06
-0.44
0.68
G2
6.93
-9.32
1.12
1.06
-0.44
0.68
G3
0.19
0.32
-0.62
0.14
0.13
-0.17
G3
0.19
0.32
-0.62
0.14
0.13
-0.17
G4
0.07
0.51
0.24
-0.79
0.10
-0.13
G4
0.07
0.51
0.24
-0.79
0.10
-0.13
G5
-2.34
-4.13
4.20
1.91
-1.444
1.892
G5
-2.34
-4.13
4.20
1.91
-1.444
1.892
G6
2.18
6.31
-5.55
-2.46
1.895
-2.486
G6
2.18
6.31
-5.55
-2.46
1.893
-2.477
"!G5 "!G5
"rG6 > 0
“win-win” region
"rG5 > 0 "rG5 > 0 "!G6
"rG6 > 0
"!G6
Fig. 2. Schematic diagram to illustrate the relationship between the change in offers and that in earnings. “win-win” region is the area where changes in offers result in the increase in earnings of all the participants.
For comparison, the revenue sensitivity matrix is calculated in the case that the reactive power is not paid as listed in Table II. Similar sub matrix is formulated in a following way:
& 'rG 5 # -. 1.444 1.892 *& '0G 5 # $$ !! = 10 4 / + !! ($$ , 1.895 . 2.486)% '0G 6 " % 'rG 6 "
(10)
Since the reactive constraint of Firm 5 is not binding, the row in N matrix is not affect by the change in payment. A similar map can be constructed by rotating the line corresponding to Firm 6 whose reactive constraint is binding. Due to the rotation, “win-win” region is located in the third quadrant as shown in Figure 3. In this case both firms can increase their earnings by reducing their offers, which would make the market more efficient. Financial reward for reactive power
Fig. 3. Schematic diagram to illustrate the relationship between the change in offers and that in earnings.
Table III shows a reactive power price sensitivity matrix R. The reactive power constraints of Firm 2 and Firm 6 are binding, which implies that there are only two rows in the R matrix corresponding to firms 2 and 6 that are non-zero. Table III has positive diagonal elements for both firms, i.e., their reactive power prices increase if the real power prices increase. Diagonal element in R for Firm 2 is much smaller than that for Firm 6. Thus, the row in N corresponding to Firm 2 shows that Firm 2 is not affected by the financial reward for reactive power as much Firm 6 is. Table III. Reactive power price sensitivity matrix R in units of [W/Var] R
G1
G2
G3
G4
G5
G6
G1
0
0
0
0
0
0
G2
-0.11
0.09
0.01
-0.06
-0.06
0.12
4 generator. The triangular cone-shape of “win-win-win” region is a conjunctional area constructed by three planes of G2, G5 and G6, i.e., any change in price locates in the “winwin” region results in the increase in their revenue.
G3
0
0
0
0
0
0
G4
0
0
0
0
0
0
G5
0
0
0
0
0
0
G6
-0.51
-0.50
-0.62
-0.49
-0.62
2.62
Table V. Revenue sensitivity matrix without financial reward for reactive power in the same unit used in Table I.
IV-I. Case B In another case, the reactive constraints of Firm 2 and Firm 6 are again binding and their nodal prices for reactive power are about $ 72.3/MVar-hr and $ 865/Mvar-hr, respectively. For comparison, their real power prices are $ 112/MW-hr and $ 275/MW-hr, respectively. Table IV shows the revenue sensitivity matrix obtained for the situation where generators receive financial payments for reactive power equal to the nodal prices. Table IV. Revenue sensitivity matrix shown in equation (8) in the unit of 10,000 [MWh] N
G1
G2
G3
G4
G5
G6
G1
-5.43
6.02
-0.52
0.55
0.15
-0.77
G2
7.50
-9.19
-0.29
0.93
-0.13
1.22
G3
-0.06
-0.03
-0.01
0.02
0.07
0.01
G4
0.21
0.29
0.07
-0.52
0.10
-0.15
G5
0.13
-0.10
0.53
0.23
-1.10
0.27
G6
-2.15
3.15
0.04
-1.15
0.94
-0.80
N
G1
G2
G3
G4
G5
G6
G1
-5.43
6.02
-0.52
0.55
0.15
-0.77
G2
7.50
-9.19
-0.29
0.93
-0.13
1.22
G3
-0.06
-0.03
-0.01
0.02
0.07
0.01
G4
0.21
0.29
0.07
-0.52
0.10
-0.15
G5
0.13
-0.10
0.53
0.23
-1.10
0.27
G6
-2.34
3.01
0.20
-1.20
0.92
-0.59
"!6
"rG2 > 0 "rG5 > 0 "rG6 > 0
“win-win-win” region
"!2
By selecting columns and rows in the N matrix corresponding to Firms 2, 5 and 6, one can construct a sub matrix for obtaining information about their incentive for possible cooperation.
& 'rG 2 # - . 9.19 . 0.13 1.22 *& '0G 2 # $ ! ! + ($ 4 $ 'rG 5 ! = 10 / +. 0.10 . 1.10 0.27 ($ '0G 5 ! $ 'r ! +, 3.15 0.94 . 0.80()$% '0G 6 !" % G6 "
(11)
There is no positive "!’s to result in increase in the revenue, "r, of G2, G5 and G6’s. Consequently, “win-win” region does not exist where they can increase their revenues as well as their prices simultaneously. In other words, implicit cooperation with each other will not raise their revenue. The revenue sensitivity matrix shown in Table V indicates no financial reward for the reactive power. Based on the sub matrix obtained from proper selection, a similar map can be drawn as shown in Figure 4. A plane bisects 3-dimensional space into positive and negative revenue sub-spaces of each
"!5
Fig. 4. Schematic diagram to illustrate the relationship between the change in offers and that in earnings.
Since the reactive constraint of Firm 2 and Firm 5 are not binding, the two corresponding planes remain unchanged. Only one plane corresponding to Firm 6 will rotate around zero, and the resulting map shows “win-win” region by raising their real power prices simultaneously. Consequently, market power exists for Firm 2, Firm 5 and Firm 6, which potentially decreases market efficiency. Different from Case A, the financial reward for reactive power price helps to remove market power. The reason is that the decrease in earning for reactive power is larger than that for real power for Firm 6. Therefore, Firm 6 would rather decrease its price. Table VI shows a reactive power price sensitivity matrix R. Since the reactive power constraints of Firm 2 and Firm 6 are binding, only two rows in R matrix corresponding to the firms are non-zero. Different from Case A, the diagonal element for Firm 2 is positive while that for Firm 6 is negative. In other word, reactive power price of Firm 6 drops significantly as its real power increases. Consequently, the
5 earning from reactive power decreases more than the increase in that from real power with respect to increase in real power price. Table VI. Reactive power price sensitivity matrix R in units of [W/Var] R
G1
G2
G3
G4
G5
G6
G1
0
0
0
0
0
0
G2
3.60
2.61
-3.14
0.94
0.24
-4.14
G3
0
0
0
0
0
0
G4
0
0
0
0
0
0
G5
0
0
0
0
0
0
G6
45.90
33.11
-39.7
11.99
3.11
-53.0
B , ) * Ys + j c ' Y s 2 ' &Vi # & Ii # * (14) $ I ! = * .. * . * ' $V ! % k" * Ys Bc ' % k " Ys + j * ' . 2 ( + where #, Ys and Bc represent the turn ratio, the admittance and the line charging susceptance of the line, respectively. The flow over the line is expressed in terms of voltages between two buses:
S i = PT + jQT *
B & # $ Ys + j c ! (15) Y 2 V ' s V ! = Vi I i* = Vi $ i k * $ (2 ! ( $ ! % " where PT and Q T are real and reactive power flow over the line. Consequently, one finds: PT = G# Vi
2
$ G " Vi Vk cos(! i $ ! k )
(16)
+ B " Vi Vk sin (! i $ ! k )
V. Conclusion A new method to measure the dependence of changes in the price of reactive power on the price of real power was presented. Its use was illustrated on the IEEE 30 bus system. The relationship between the changes in real and reactive prices provides an opportunity for generators to increase their earnings by changing their offer strategy to accommodate market designs that incorporate reactive pricing.
QT = $ B# Vi
+ G " Vi Vk sin (! i $ ! k )
(17)
+ B " Vi Vk cos(! i $ ! k ) where
G! + jB! =
G " + jB " =
Appendix I
2
Ys + j
!
Bc 2
(18)
2
Ys !*
(19)
Taking derivative of Equations (16) and (17) gives:
For a congested line, the apparent power flow equals the line capacity: 2
S = P2 + Q2 = C 2
(12)
[
%PT = 2G$ Vi & G # Vk cos(! i & ! k )
]
+ B # Vk sin (! i & ! k )% Vi
[
]
+ & G # cos(! i & ! k )+ B # sin (! i & ! k ) ' Vi % V k
where C, P and Q stand for line capacity, real and reactive power flow over the line, respectively. Suppose that the line is congested after perturbation of price. Then, one can write an equation to describe the change of the flow as follows:
(
)
( )
! P 2 + Q 2 = 2 P!P + 2Q!Q = ! C 2 = 0
(13)
A lumped-parameter Pi-model is often used for a transmission line. Currents between bus i and bus k are calculated in a following way:
(20)
[
]
+ G # sin (! i & ! k )+ B # cos(! i & ! k ) ' Vi Vk (%! i & %! k ) = " 1 % Vi + " 2 % Vk + " 3 (%! i & %! k )
[
%QT = & 2 B$ Vi + G # Vk sin (! i & ! k )
]
+ B # Vk cos(! i & ! k )% Vi
[
]
+ G # sin (! i & ! k )+ B # cos(! i & ! k ) ' Vi % V k
[
(21)
]
+ G # cos(! i & ! k )& B # sin (! i & ! k ) ' Vi Vk (%! i & %! k ) = " 1 % Vi + " 2 % Vk + " 3 (%! i & %! k )
6 where |V| and $ represent magnitude and angle of voltage. Changes in the voltage magnitude and angle can be expressed in terms of the changes in real and reactive power injection in a following way: .1
& '/ # - B11 $' V ! = ++ % " , B21
B12 * &'G # ( B22 () $%'Q !"
(22)
# - X .g ! = ++ V " ,Xg
X q. X qV
(30)
d q = " # L = q cr ! IIq = 0
(31)
dV = " µ L = V
(32)
q
* &'g # ($ ! ( % 'q " )
(23)
where !PT = K g !g + Z g !q
PT !PT + QT !QT =
Perturbation of real price makes the conditions changed:
"f g = "% # "µ g (1 # ! g Ploss )+ µ g B g "g # 2(K gT D p + K qT D q )"$ # WVT "µ V = 0
3
T
3
1 T
3
1
T
V q i
T
3
2
T
, g i
T
2
T
V g
"f q = # II T "% q # "µ q (1 # ! q Qloss )+ µ q B q "q # 2(Z gT D p + Z qT D q )"$ # FVT "µ V = 0
k
2
T
2
, q k
T
V q
k
(24)
( ) !h = (# Q " 1) !q = 0 !fl = "2[(D K + D K )!g + (D Z + D Z )!q ]= 0 T
(36)
T
(37)
!d q = !(q cr " IIq ) = " II!q = 0
(39)
!d V = "! V = "(WV !g + FV !q ) = 0
(40)
q
loss
g
q
g
(
(
Appendix II
L = #T g + #Tq (q cr ! IIq ) + µ g (PD + Ploss ! 1T g )+ µ q (Q D + Qloss ! 1T q )
(
+ " (C ! P ! Q )+ µ V
)
T
2 T
2 T
T V
cr
!V
(
(
)
)
T
) !g
(41)
" !µ VT WV !g
(1 # $
fq = "q L
= (27)
! 2(Z D p + Z D q )# ! F µ V = 0 T V
h g = " µ L = PD + Ploss ! 1T g = 0
(28)
hq = " µ L = Q D + Qloss ! 1T q = 0
(29)
g
=
T
q
T V
(42)
FV !q = 0
g
Ploss
)
T
1 " # 2 K gT D p + K qT Dq ! # WVT µV µg
[
(1 # $
)
! 2 K gT D p + K qT Dq # ! WVT µ V = 0
T q
T
) !q B !q ) " !µ
From Equations (26) and (27), one obtains:
(26)
= ! II T $ q ! µ q (1 ! " q Qloss )
(
(25)
fg = "gL = $ ! µ g 1 ! " g Ploss
(
2!$ T Z gT D p + Z qT Dq = " !# q + µ q
Kuhn-Tucker optimality conditions are following::
q
(38)
q
Similarly Equations (35) and (37) allow:
One can formulate a Lagrange function from Equation (2):
T g
q
2!$ T K gT D p + K qT Dq = !# + µ g B g !g
2
q
Combining Equations (34) and (36) gives:
"
= H g !g + H q !q
T
(35)
where B q = ! qq Qloss
p
, g k
, q i
T
(34)
where B g = ! gg Ploss
p
( ) + (. P + - Q )(X ) + (. P + - Q )[(X ) + (X ) ]} !g + {(. P + - Q )(X ) + (. P + - Q )(X ) $ + (. P + - Q )*((X ) + (X ) '% #!q ) & + - 1QT ) X
1 T
(33)
! V = WV !g + FV !q
q
{(. P
! V =0
!hg = # g Ploss " 1 !g = 0
Combining Equations (13), (20), (21) and (23) gives:
V g i
cr
V
!QT = K q !g + Z q !q
Note that there are non-zero changes in real and reactive power injection for generation bus only due to the change in price. The column and the row of B matrix corresponding to the reference bus are removed and then the reduced B matrix is inversed. If the reference bus is chosen one of demand buses, Equation (22) can be simplified by choosing corresponding columns of inverse B matrix and corresponding elements of generation vector to generation bus only:
& '. $' V %
fl = " # L = C 2 ! PT2 ! QT2 = 0
q
Qloss
(
)
T
)
T
1 # " q # 2 Z gT D p + Z qT Dq ! # FVT µ V µq
[
(43)
]
(
One finds a following Equations (34) – (44):
)
equation
] T
by
(44)
combining
7 B.x + V T .y = '.1 and V.x = 0 where & .µ g # $ ! .µ q ! $ & .g # & .0 # .x = $$ !!, .y = $ ./ !, .1 = $$ !! . q $ ! % " % 0 " .µ $ V! $ .0 ! % q" µ B 0 - g g * B=+ ( and 0 µ B q q) , & ' 1 -0 ' 2(K gT Dp # * $ + ! 0 ( T T $ µ g + K q Dq )/ ' WV µV ( ! ) $ , ! T *! $ 1 -0q + 2(Z g Dp 0 + (! V =$ µ g +,+ Z qT Dq )/ + FVT µV )( ! $ $ ' 2(Dp K g + Dq K q ) ' 2(Dp Z g + Dq Z q ) ! $ ! ' WV ' FV $ ! $ ! 0 ' II % "
(45)
Note that B is an invertible matrix. Thus one can solve for "x:
'x = ( B (1V T 'y ( B (1 '+ 'y = ((VB V and (1
T
) VB (1
(1
and
V'x = 0
'- * )'(46)
& 'g # & ', # (1 'x = $$ !! = B (1 V T (VB (1V T ) VB (1 ( I $$ !! ' q % " % 0 "
[
]
By definition of dispatch sensitivity matrix M, one finds:
& I 0# '1 !! M = B '1 V T (VB '1V T ) VB '1 ' I $$ %0 0"
(47)
Let % be the last w-row vectors in & where w is the number of binding reactive power constraints, and then one finds a relation between the changes in real and reactive power price in a following way:
) #! & " 2w* n )'' $$ = " 1 #! (48) ( 0 % Note that all other reactive power prices do not change except for the w-rows. Therefore, one finds: #! q = (" 1w* n
$LMPq = II T $" q = II T ! 1 $" = R$" # R = II T ! 1
(49)
For convenience, fast decoupled approximation can be used to evaluate K, Z, W and F matrices.
References
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