- Email: [email protected]
Long-run marginal cost based pricing of interconnected system wheeling
Electric Power Systems Research 50 (1999) 205 – 212
Long-run marginal cost based pricing of interconnected system wheeling C.W. Yu Department of Electrical Engineering, The Hong Kong Polytechnic Uni6ersity, Hung Hom, Hong Kong Received 10 June 1998; received in revised form 1 September 1998; accepted 1 October 1998
Abstract Each utility in an interconnected system has an obligation to guarantee sufficient transmission capability to maintain an efficient, economical, reliable and secure system during peak scenarios. Security is an important consideration underlying network investment. The standards of service have a direct impact on investment burdens and therefore definition and consensus among participants in respect of security standards are necessary. Charging for transmission services, ensuring the investment levels and recovery of sunk capital are new problems now receiving attention in the context of electricity supply industry unbundling. In this paper a method for long run marginal cost (LRMC) based pricing in multi-area interconnected system, based on the incremental use of each area’s transmission network at times of peak flow, is proposed. The LRMC of transmission capacity is based on long term costs of transmission investment requirements. The marginal wheeling costs, with security taking into account, are computed using the sensitivities of the MW-mile of each area with respect to the bus power demand. These sensitivities are calculated using a linear expansion of the Kuhn–Tucker conditions of the investment cost optimization problem. Contingency ranking method is used to speed up the computation. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Transmission pricing; Deregulation; Long-run marginal cost; Interconnected system
1. Introduction Interconnection of control areas is almost universal throughout the world. In an interconnected multi-area system, joint operation of generation resources can result in significant operational cost savings and reliability enhancement. Wheeling is one of the approaches to implementing open access conditions in interconnected systems. For instance open access has been mandated by the Federal Energy Regulatory Commission (FERC) in the US. Conventional wheeling is when one utility simply wishes to transport power through the transmission network of a neighbor to a third more distant utility. The more complex case is when a customer decides not to purchase power from the utility in whose franchise region he is located and instead contracts to purchase from another neighboring, or more distant, producer. Each utility in an interconnected system has the obligation of guaranteeing sufficient transmission capability to maintain efficient, economical, reliable and secure operation during peak scenarios.
The costs of operating, maintaining and expanding the transmission system, that is pricing transmission services, needs to be addressed in the context of industry deregulation [1]. Long-run marginal cost based pricing seeks to determine the present value of future investments required to support a marginal increase in demand at different locations in the system, based on peak scenarios of future demand and supply growth. The users of transmission pay a charge that is geographically differentiated to the provider of the service. Regarding long-run marginal based transmission pricing, Tabors [2] provides an extensive summary of longrun marginal cost philosophies. Munasinghe et al. [3] give basic economic principles underlying transmission services based on LRMC pricing. Calviou et al. [4] propose an approach, based on the transportation principle, to calculate the charge for the use of transmission system tailored for Britain’s National Grid Company. In this paper the evaluation of long-run marginal wheeling costs in a multi-area interconnected power system using sensitivity analysis is described. In a multi-
0378-7796/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 9 8 ) 0 0 1 4 4 - 8
206
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
area interconnected power system the long-run marginal wheeling cost of an area is the change in that area’s MW-mile (investment cost) due a unit change in the ‘transacted power’ between two buses. Wheeling occurs when change of power transport between two buses anywhere in the system causes a noticeable rearrangement of power flow in the area under study. The evaluation of marginal wheeling costs for each wheeling area using sensitivity analysis is described in Section 2. The charge levied to cover the cost of providing a secure system is described in Section 3. In Section 4 the implementation of the security related cost analysis is illustrated through a case study.
2. Long run marginal wheeling cost evaluation The marginal wheeling costs of transporting power between buses will be computed in this paper. The bus to bus wheeling include: 1. Intra-area; supplier to customer wheeling. A generator sells to a user. Both the generator and the user are located in the wheeling utility’s service territory. 2. Inter-area; supplier to customer wheeling. A generator sells to a customer, usually wholesale, located in another utility’s services territory. The long-run marginal wheeling costs are evaluated for each wheeling area (utility) on the basis of minimization of the investment costs required to supply the generation/load pattern during a defined peak scenario. A DC load flow model, supplemented by sensitivity analysis, is a versatile tool for long-run transmission capacity cost analysis in the pooling arrangement. As the system is planned for system peak and transmission rates for capacity are calculated at peak system conditions, the peak power injection at each bus is first forecast. The lines are divided into two groups, heavily and lightly loaded lines. The lightly loaded lines have ‘excess’ installed capacity. They will always be loaded well below their rating and hence investments along these rights of way are not necessary. Investments are only required on the heavily loaded routes. The evaluation of the incremental investment costs here uses the MW-mile concept combined with a linear programming (LP) approach [4]. Assume that the existing capacity of the heavily loaded lines can be changed continuously and there are no new rights of way. If in addition, generation redispatch is not allowed, the optimal capacity of each circuit is the same as the circuit flow resulting from the power flow solution. Let: A N Pgi Pdi
set of area indices in the interconnected system set of bus indices in the interconnected system power generation at bus i power demand at bus i
Fig. 1. IEEE 30-bus tested system,
net power injection at bus i the phase angle at node i the circuit susceptance of the circuit c between buses i and j
Pi ui bc
Table 1 Generation and load data Bus
Demand
Gen.
Bus
Demand
Gen.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 217 24 78 942 0 228 300 0 58 0 112 0 62 82
1750 480 0 0 214 0 0 230 0 0 125 0 123 0 0
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
35 90 32 95 22 175 0 32 87 0 35 0 0 24 107
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
207
Table 2 Transmission lines data Line
From bus
To bus
React (pu)
Length (mile)
Capacity (MW)
Area
CONT LINE
−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15 −16 −17 −18 −19 −20 −21 −22 −23 −24 −25 −26 −27 −28 −29 −30 −31 −32 −33 −34 −35 −36 −37 −38 −39 −40 −41
1 1 2 3 2 2 4 5 6 6 6 6 9 9 4 12 12 12 12 14 16 15 18 19 10 10 10 10 21 15 22 23 24 25 25 27 27 27 29 8 6
2 3 4 4 5 6 6 7 7 8 9 10 11 10 12 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 28 29 30 30 28 28
0.0575 0.1852 0.1737 0.0379 0.1983 0.1763 0.0414 0.0116 0.0820 0.0420 0.2080 0.5560 0.2080 0.1100 0.2560 0.1400 0.2559 0.1304 0.1987 0.1997 0.1932 0.2185 0.1292 0.0680 0.2090 0.0845 0.0749 0.1499 0.0236 0.2020 0.1790 0.2700 0.3292 0.2800 0.2087 0.3960 0.4153 0.6027 0.4533 0.2000 0.0599
5.75 18.52 17.37 3.79 19.83 17.63 4.14 11.60 8.20 4.20 20.80 55.60 20.80 11.00 25.60 14.00 25.59 13.04 19.87 19.97 19.32 21.85 12.92 6.80 20.90 8.45 7.49 14.99 2.36 20.20 17.90 27.00 32.92 28.00 20.87 39.60 41.53 60.27 45.33 20.00 5.90
1300 1300 650 1300 1300 650 900 700 1300 320 650 320 650 650 650 650 320 320 320 160 160 160 160 320 320 320 320 320 320 160 160 160 160 160 160 650 160 160 160 320 320
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 3 4 4 4 4 4 4 4 4 3 3
2 1 2 1 9 5 1 5 5 41 15 14 7 15 7 4 18 17 14 18 15 25 25 15 15 15 36 27 27 36 36 36 36 1 36 15 38 37 38 41 10
fc tc L Lh Lg li NAREA K(a)
end terminal phase angle difference of circuit c, i.e. (ui–uj ) the length of the circuit between buses i and j set of lines in the whole interconnected system set of heavily loaded lines set of lightly loaded lines shadow cost for power balance at bus i (dual variable of Eq. (6), gi =0) number of areas in the interconnected system the set of transmission lines belonging to area a
Ma
da Ma, c M fc f *c
capacityc
the total MW-mile of area a (also doubles up as a pseudo dual variable of Eq. (7), da = 0) a dummy variable MW-mile of area a due to circuit c vector of dimension A with elements Ma power flow of circuit c threshold transmission capacity of circuit c at which the line is considered stressed required capacity of the circuit c
Mathematically the objective of optimization is to minimize the investment costs in the interconnected
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
208
system. As the cost of a line is proportional to its length, using the MW-mile concept, the formulation is: Minimize CT= % tcbcfc
(1)
c Lh
c
s.t.
n
gi = %bij (ui −uj ) − Pi =0 j
Öi N
(2)
where bij = bc,
bii = % −bij j"i
The Lagrangian to be minimized over voltage angles u and the power generation at the reference bus (say bus 1, Pg1), is given by, L =CT+ % li gi
(3)
iN
Since generation redispatch is not allowed, the formulation of the ‘optimization’ problem in this way has
zero degrees of freedom. This linear program is actually solving the DC load flow problem. However this restatement of the DC load flow as an optimization problem makes it possible to determine the shadow costs from the dual solution. The shadow cost for power balance at bus i, li, gives the increase in pool MW-mile transport cost on the heavily loaded lines for a unit increase in the power demand at bus i. It should be noted that 1. the shadow cost for power generation at bus i is equal and opposite to the shadow cost for power demand at bus i; 2. the shadow cost for bus 1 must be equal to zero throughout, because bus 1 has been chosen as the reference bus. However, the choice of the slack bus will not change the transport cost between two buses. The unit of these marginal costs is miles of incremental transmission capacity and can be transformed to $/MW-mile per year by multiplying by annualized line construction costs [5].
Table 3 MW-mile sensitivity Change demand at bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Area MW-mile sensitivity 1
2
3
4
Total
0.0 293.8 344.5 386.9 835.6 471.5 723.5 515.5 613.3 913.8 405.3 852.4 852.4 855.5 857.9 878.0 902.9 877.5 889.0 895.1 900.1 895.7 850.8 841.3 729.5 729.5 658.7 524.2 658.7 658.7
0.0 1.6 −5.0 −6.0 6.8 9.3 7.1 10.5 38.7 97.8 38.7 −32.1 −172.1 208.2 314.2 218.5 75.7 456.0 539.8 515.9 107.3 110.3 243.1 148.0 101.5 101.5 72.0 16.1 72.0 72.0
0.0 −0.6 1.8 2.2 −2.5 −3.4 −2.6 48.6 −14.3 −20.0 −14.2 152.5 152.5 172.3 187.7 115.8 80.1 211.9 226.2 165.8 178.6 148.4 235.5 299.4 325.9 325.9 342.8 225.1 342.8 342.8
0.0 0.6 −1.8 −2.2 2.5 3.4 2.6 1.1 62.4 93.5 62.3 31.2 31.2 14.6 1.6 57.2 82.4 33.7 52.7 62.7 126.2 136.4 114.7 265.9 688.3 968.3 956.1 −10.1 1964.2 2416.3
0.0 295.3 339.5 380.9 482.5 480.9 730.7 575.5 700.1 1085.2 492.1 1004.1 864.1 1250.7 1361.5 1269.6 1141.3 1579.2 1707.9 1639.7 1312.3 1291.1 1444.2 1554.8 1845.4 2125.4 2029.7 755.3 3037.8 3489.9
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212 Table 4 Long run marginal wheeling cost in MW-mile incurred in area 1 From bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
To bus 1
16
21
30
5
– −294 −344 −387 −836 −472 −724 −515 −613 −914 −405 −852 −852 −856 −858 −878 −903 −877 −889 −895 −900 −869 −851 −841 −730 −730 −659 −524 −659 −659
878 584 534 491 42 407 155 363 265 −36 473 26 26 23 20 – −25 1 −11 −17 −22 −18 27 37 149 149 219 354 219 219
900 606 556 513 64 429 177 385 287 −14 495 48 48 45 42 22 −3 23 11 5 – 4 49 59 171 171 241 376 241 241
659 265 314 272 −177 187 −65 143 45 −255 253 −194 −194 −197 −199 −219 −244 −219 −230 −236 −241 −237 −192 −183 −71 −71 0 134 0 –
836 542 491 449 – 364 112 320 222 −78 430 −17 −17 −20 −22 −42 −67 −42 −53 −60 −64 −60 −15 −6 106 106 177 311 177 177
(L =0 (Pg1 gi = 0
From bus
(4) (5)
Öi N
The area MW-mile equations are:
n
% tcbcfc − Ma = 0;
c K(a)
Öa A
(7)
Table 5 Sum of the marginal wheeling cost incurred in all four areas (in MW-mile)
The sensitivity analysis below enables us to find the sensitivity of small changes in the power demand vector Pd to small changes in the problem variables and shadow cost vectors (u, Pg1, l, M) necessary to maintain optimum operation. These sensitivities are calculated using a linear expansion of the Kuhn – Tucker conditions for the multi-area pool model in the vicinity of the solution to the problem Eqs. (1) – (3). At the optimum solution the Kuhn – Tucker conditions are i (2…, N)
Now when the system Eqs. (4)–(7), whose solution is no different from the power flow solution, are to be perturbated while retaining optimality in the post-perturbation state, the changes of the variables and shadow cost vectors (u, Pg1, l, M) can be expressed in terms of changes in the power demand vector Pd. Now what is being done may be explained as follows. A small perturbation of power variables (Eq. (6) type constraints) is introduced but optimality is to retained after the perturbation. This requires that Eqs. (4) and (5) remain true, that is equal to zero. Eq. (7) is also included so as to enable us to calculate the change in M after the perturbation. Mathematically these points may be explained as follows:
2.1. Sensiti6ity analysis
(L =0 (ui
da =
209
(6)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
To bus 1
16
21
30
5
– −295 −339 −381 −842 −481 −731 −576 −700 −1085 −492 −1004 −864 −1251 −1362 −1270 −1141 −1579 −1708 −1640 −1312 −1291 −1444 −1555 −1845 −2125 −2030 −755 −3038 −3490
1270 974 730 889 427 789 539 694 569 184 777 265 405 19 −92 – 128 −310 −438 −370 −43 −21 −175 −285 −576 −856 −760 514 −1768 −2220
1312 1017 973 731 470 831 582 737 612 227 820 308 448 62 −49 43 171 −267 −396 −327 – 21 −132 −242 −533 −813 −717 557 −1725 −2178
3490 3195 3150 3109 2647 3009 2759 2914 2790 2405 2998 2486 2626 2239 2128 2220 2349 1911 1782 1850 2178 2199 2046 1935 1644 1364 1460 2735 452 –
842 547 503 462 – 362 112 267 142 −243 350 −162 −22 −408 −519 −427 −299 −737 −865 −797 −470 −449 −602 −712 −1003 −1283 −1187 87 −2195 −2647
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
210
ÇÃ ÃÃ
Æ (L D Æ Du Ç Ã (u à à à DP g1 (L S=à à = ÃD Dl (P g1 Ã Ã Ã È DM É Ã Dg È Dd
Æ 0 Ç Ã Ã 0 =à à DP à à dÃ È 0 É Ã
(8)
3. Interconnected system secure transportation
É
S, sensitivity matrix= Æ ( 2L ( 2L ( 2L à ( 2L à (u(u (P g1(u (l(u (M(u 2 2 à ( 2L ( L ( L ( 2L à (u(P g1 (P g1(Pg1 (l(P g1 (M(P g1 à (g (g (g à (g (u (P (l (M g1 à (d (d (d à (d à (u (l (M (P g1 È
Security is a prominent factor driving network investment. The standards of service have a direct impact on investments burdens and therefore definition and consensus among participants in respect of security standards are necessary. It is possible to extend the above long run marginal cost evaluation to take security into account. In the following a contingency is defined as the loss of any one circuit. The charge is levied to cover the cost of providing a secure system. To extend the above method to take security into account, the mathematical formulation is:
Ç Ã Ã Ã Ã Ã Ã Ã Ã Ã É
Minimize
These coefficients are relatively simple and a sample collection is provided in Appendix A. Eq. (8) can be written concisely as SDU =DV
(9)
cLh
(12)
c
The constraints to be satisfied, during the normal and any contingency conditions, are: gi = %bij (ui − uj )− Pi = 0 (bij (ui − uj ) 5 capacityc
(N −1)+ 1+(N) + (NAREA) 2. S is a square matrix Now by expressing DU as functions of the differentials DV, it will be possible to assess the effect of changes in demand on the solution. Pre-multiplying both sides of Eq. (9) by the inverse of the sensitivity matrix, we have (10)
In particular, the change in area MW-mile is of prime interest for marginal wheeling cost calculation. The total differential of MW-mile in area a will be: DMa = %Skm DVm
% tc capacityc
ÖiN
(13)
j
notice that 1. The dimensions of both DV and DU are:
DU =S − 1DV
in power demand at the receiving bus. The corresponding DMa can be easily found by setting the two appropriate elements of DPd to 9 1 and all other elements to zero.
(11)
m
where Skm are the elements of row k of the inverse sensitivity matrix (S − 1) such that element k of DU is DMa, DVm is the m-th element of DV. Skm is the MW-mile change in area a per unit change in Vm. Elements DPd in the column vector DV can be manipulated to determine the marginal cost of wheeling for specific buses. For a bus to bus transaction the marginal wheeling cost of an area (a measure of what it costs the area to maintain the wheel) is the sum of the MW-mile changes in the area due to a unit decrease in power demand at the supplying bus and a unit increase
(14)
Since generation redispatch is not allowed, the ‘optimization’ problem has zero degrees of freedom and hence the problem can be solved on a ‘line by line’ basis as follows:
3.1. Step 1 Calculate the flows on each line for every contingency. Record the contingency which produces the highest flows on each line.
3.2. Step 2 Consider a circuit c and use the network with the contingency which produces the highest flow on the circuit c found in step 1. Now capacityc = bij (ui –uj ) can be found by solving Eq. (13). Calculate the N sensitivities of each area’s MW-mile with respect to the bus demands at all N buses. The calculations methodology is the same as that in Section 2 except now we have only circuit c in the objective function (Eq. (1)) and the area MW-mile equation (Eq. (7)). This means that only the change of capacity of the circuit c is considered when the sensitivity calculation is carried out. The sensitivity of the change of MW-mile in area a to the power demand change at bus i is
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
((Ma, c ) (Pdi
aA
(15)
This sensitivity calculation is repeated for every circuit in the interconnected system.
3.3. Step 3 The summation of these sensitivities for all the circuits in the whole interconnected system gives the overall sensitivities of the change of MW-mile in area a to the power demand change at bus i. It is given by: ((Ma, c ) % (Pdi c
aA
(16)
3.4. The contingency ranking method A line outage will increase the stress on the transmission system. Step 1 in the above algorithm requires a large amount of computing for large systems. In practice, the stressing effects of some lines outages are small as compared with outage of the other lines. Therefore, contingency ranking [6] can be carried out according to the probability of system stress caused by different line outages. If the flow during the contingency ranking process exceeds a prescribed threshold, f *c (taken as 80% of the rating of the line), a line is considered to be stressed. Step 1 is only performed on the lines with higher probability of stressing the system and hence the amount of computing can be significantly reduced. In order to reflect the system stress, a system performance index is defined as: L
II= %
c=1
fc f*c
2
(17)
When there is no stress fc /f *c is not greater than one; the index is small. When there is stress in the system, fc /f *c for the stressing line is greater than one and the positive exponential element makes the index large. Therefore, this index reflects the system security.A sensitivity analysis of the index with respect to the change of a line admittance will reveal the impact of a line outage on the system security. When a line k fails, the change in the index is given by DPk =
(P DB (Bk k
(18)
where DBk =Bk is the admittance of the line k. The bigger the DPk is, the larger the increase in the P will be, which indicates that the probability of a faulted line k causing system stress becomes higher. Reference [6] provides detailed mathematics for calculating DPk when circuit k, which spans buses i and j, fails. A brief summary follows: DPk =
tk f 2k 2uk fk f 2k + − (1− Bkxk )2 (1 − Bkxk ) (1 − Bkxx )2f *2 k
(19)
211
where xk X tk T W A Wd uk PN
Xii+Xjj+2Xij system reactance matrix Tii+Tjj+2Tij XWX AWd A T branch to bus incidence matrix 2 *2 2 *2 diag [B 21/f *2 1 B 2/f 2 ...... B L/f L ] T A XWXPN bus real power injection vector
The process of contingency ranking is to compute the value of DP for all lines using Eq. (19) and rearrange in descending order of magnitude. During the contingency analysis in step 1, load flow calculations are carried out on the lines with the large values of DP. The lines with small values of DP are not subjected to analysis because the probability of system stress caused by these outages is small.
4. Case study The IEEE 30-bus system as depicted in Fig. 1 is used to demonstrate the performance of the method. The system has four interconnected areas, 41 transmission lines and six generators. Each tie line is assumed to belong to one area and the area that owns the line is indicated by a dot. The MW generation and MW demand at each bus are given in Table 1. The transmission line data are given in Table 2. All the lines are assumed to be heavily loaded. The most right column of Table 2 gives the line outage (contingency) which gives the maximum flow on a particular line, e.g. line 2 outage will give the maximum flow on line 1. This column can be obtained by running load flow for all contingencies. The column can also be obtained by the contingency ranking analysis as discussed in Section 3.4. It was found that running load flow for the contingencies which has the higher ranking (upper 46% of the DP magnitude descending order list) can give the column. Hence significant computer time can be saved. The sensitivities of the area MW-mile (Ma ) with respect to the change in bus demands (Pd ) for the test system are given in Table 3. Note that bus 1 was chosen as the reference bus (i.e. if the demand of any bus is increased by 1 MW, the demand at bus 1 will decrease by 1 MW accordingly). For example, the last row of Table 3 shows that if the power demand of bus 30 is increased by 1 MW, the MW-miles of areas 1, 2, 3 and 4 will increase 658.7, 72.0, 342.8, 2416.3, respectively. The overall increase is 3489.9 MW-miles. For a bus to bus transaction the marginal wheeling cost of an area is the sum of the MW-miles changes in the area due to a
212
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
unit decrease in power demand at the supplying bus and a unit increase in power demand at the receiving bus. The marginal wheeling costs of area 1 and the sum of the marginal wheeling costs incurred in all four areas are given in Tables 4 and 5, respectively. Each entry in the tables gives the long-run marginal wheeling cost for the secure transportation of another MW of electricity from the source on the left-hand side to a receiving end listed at the top of the table. The transport costs may be either positive or negative. Negative means that wheeling causes a decrease in MW-miles in an area.The total transport cost is seen to be positive if the flow direction of the power transaction is from area 1 to 4 while the cost is negative if the direction is reversed. This is because generation is concentrated in area 1, the power flow in mainly from area 1 to 4. In other words, if the power flow of a transaction further stress the transmission system, the cost will be positive while if it relieves the transmission system, the cost will be negative.
5. Conclusions The marginal wheeling costs of transporting power between buses, with security taking into account, are computed using the sensitivities of the MW-mile of each area with respect to the bus power demand. These sensitivities are calculated using a linear expansion of the Kuhn–Tucker conditions of the investment cost optimization problem. The marginal wheeling cost is proportional to the impact on system transmission investment requirement and provides economically efficient price for location. The marginal wheeling cost can be positive or negative. If the power flow of a transaction further stress the transmission system, the cost will be positive while if it relieves the transmission system, the cost will be negative. The securer the system, the higher will be the wheeling cost. Prices set equal to long-run transmission costs are stable and foreseeable. They correctly signal to users the long-run incremental cost of transmission service over several transmission
expansion cycles and hence facilitate long-term contractual decisions. The capacity charge can be applied as an annual charge and does not interfere with the short-run marginal cost economic signals for recovering the operating cost.
Appendix A. Typical coefficients of the sensitivity matrix are as follows: ( 2L =0 (ui(uj ( 2L = bij (ui(lj
Öi, j Öi, j
( 2L ( 2L = =0 (Pg1(ui (ui(Pg1 (gi = bij (uj
Öi
Öi, j
(g (g (d (d = =0 = = (l (l (Pg1 (M
References [1] C.W. Yu, A.K. David, Pricing transmission services in the context of industry deregulation, IEEE Trans. Power Syst. 12 (1) (1997) 503 – 510. [2] R.D. Tabors, Transmission system management and pricing: new paradigms and international comparisons, IEEE Trans. Power Syst. 9 (1994) 206 – 215. [3] M. Munasinghe, J.J. Warford, Electricity Pricing: Theory and Case Studies, John Hopkins Press, Baltimore, MD, 1982. [4] M.C. Calviou, R.M. Dunnett, P.H. Plumptre, Charging for use of a transmission system by marginal cost methods, in: Proceeding of the Eleventh PSCC, Avignon, France, 1993, pp. 385 –391. [5] C.W. Yu, A.K. David, Long run marginal cost evaluation of transmission capacity, Int. Power Eng. Conf. Singapore May (1997) 425 – 430. [6] T.A. Mikolinnas, B.F. Wollenberg, An advanced contingency selection algorithm, IEEE Trans. Power Appar. Syst. 100 (2) (1981) 608 – 615.
.
Long-run marginal cost based pricing of interconnected system wheeling C.W. Yu Department of Electrical Engineering, The Hong Kong Polytechnic Uni6ersity, Hung Hom, Hong Kong Received 10 June 1998; received in revised form 1 September 1998; accepted 1 October 1998
Abstract Each utility in an interconnected system has an obligation to guarantee sufficient transmission capability to maintain an efficient, economical, reliable and secure system during peak scenarios. Security is an important consideration underlying network investment. The standards of service have a direct impact on investment burdens and therefore definition and consensus among participants in respect of security standards are necessary. Charging for transmission services, ensuring the investment levels and recovery of sunk capital are new problems now receiving attention in the context of electricity supply industry unbundling. In this paper a method for long run marginal cost (LRMC) based pricing in multi-area interconnected system, based on the incremental use of each area’s transmission network at times of peak flow, is proposed. The LRMC of transmission capacity is based on long term costs of transmission investment requirements. The marginal wheeling costs, with security taking into account, are computed using the sensitivities of the MW-mile of each area with respect to the bus power demand. These sensitivities are calculated using a linear expansion of the Kuhn–Tucker conditions of the investment cost optimization problem. Contingency ranking method is used to speed up the computation. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Transmission pricing; Deregulation; Long-run marginal cost; Interconnected system
1. Introduction Interconnection of control areas is almost universal throughout the world. In an interconnected multi-area system, joint operation of generation resources can result in significant operational cost savings and reliability enhancement. Wheeling is one of the approaches to implementing open access conditions in interconnected systems. For instance open access has been mandated by the Federal Energy Regulatory Commission (FERC) in the US. Conventional wheeling is when one utility simply wishes to transport power through the transmission network of a neighbor to a third more distant utility. The more complex case is when a customer decides not to purchase power from the utility in whose franchise region he is located and instead contracts to purchase from another neighboring, or more distant, producer. Each utility in an interconnected system has the obligation of guaranteeing sufficient transmission capability to maintain efficient, economical, reliable and secure operation during peak scenarios.
The costs of operating, maintaining and expanding the transmission system, that is pricing transmission services, needs to be addressed in the context of industry deregulation [1]. Long-run marginal cost based pricing seeks to determine the present value of future investments required to support a marginal increase in demand at different locations in the system, based on peak scenarios of future demand and supply growth. The users of transmission pay a charge that is geographically differentiated to the provider of the service. Regarding long-run marginal based transmission pricing, Tabors [2] provides an extensive summary of longrun marginal cost philosophies. Munasinghe et al. [3] give basic economic principles underlying transmission services based on LRMC pricing. Calviou et al. [4] propose an approach, based on the transportation principle, to calculate the charge for the use of transmission system tailored for Britain’s National Grid Company. In this paper the evaluation of long-run marginal wheeling costs in a multi-area interconnected power system using sensitivity analysis is described. In a multi-
0378-7796/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 9 8 ) 0 0 1 4 4 - 8
206
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
area interconnected power system the long-run marginal wheeling cost of an area is the change in that area’s MW-mile (investment cost) due a unit change in the ‘transacted power’ between two buses. Wheeling occurs when change of power transport between two buses anywhere in the system causes a noticeable rearrangement of power flow in the area under study. The evaluation of marginal wheeling costs for each wheeling area using sensitivity analysis is described in Section 2. The charge levied to cover the cost of providing a secure system is described in Section 3. In Section 4 the implementation of the security related cost analysis is illustrated through a case study.
2. Long run marginal wheeling cost evaluation The marginal wheeling costs of transporting power between buses will be computed in this paper. The bus to bus wheeling include: 1. Intra-area; supplier to customer wheeling. A generator sells to a user. Both the generator and the user are located in the wheeling utility’s service territory. 2. Inter-area; supplier to customer wheeling. A generator sells to a customer, usually wholesale, located in another utility’s services territory. The long-run marginal wheeling costs are evaluated for each wheeling area (utility) on the basis of minimization of the investment costs required to supply the generation/load pattern during a defined peak scenario. A DC load flow model, supplemented by sensitivity analysis, is a versatile tool for long-run transmission capacity cost analysis in the pooling arrangement. As the system is planned for system peak and transmission rates for capacity are calculated at peak system conditions, the peak power injection at each bus is first forecast. The lines are divided into two groups, heavily and lightly loaded lines. The lightly loaded lines have ‘excess’ installed capacity. They will always be loaded well below their rating and hence investments along these rights of way are not necessary. Investments are only required on the heavily loaded routes. The evaluation of the incremental investment costs here uses the MW-mile concept combined with a linear programming (LP) approach [4]. Assume that the existing capacity of the heavily loaded lines can be changed continuously and there are no new rights of way. If in addition, generation redispatch is not allowed, the optimal capacity of each circuit is the same as the circuit flow resulting from the power flow solution. Let: A N Pgi Pdi
set of area indices in the interconnected system set of bus indices in the interconnected system power generation at bus i power demand at bus i
Fig. 1. IEEE 30-bus tested system,
net power injection at bus i the phase angle at node i the circuit susceptance of the circuit c between buses i and j
Pi ui bc
Table 1 Generation and load data Bus
Demand
Gen.
Bus
Demand
Gen.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 217 24 78 942 0 228 300 0 58 0 112 0 62 82
1750 480 0 0 214 0 0 230 0 0 125 0 123 0 0
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
35 90 32 95 22 175 0 32 87 0 35 0 0 24 107
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
207
Table 2 Transmission lines data Line
From bus
To bus
React (pu)
Length (mile)
Capacity (MW)
Area
CONT LINE
−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15 −16 −17 −18 −19 −20 −21 −22 −23 −24 −25 −26 −27 −28 −29 −30 −31 −32 −33 −34 −35 −36 −37 −38 −39 −40 −41
1 1 2 3 2 2 4 5 6 6 6 6 9 9 4 12 12 12 12 14 16 15 18 19 10 10 10 10 21 15 22 23 24 25 25 27 27 27 29 8 6
2 3 4 4 5 6 6 7 7 8 9 10 11 10 12 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 28 29 30 30 28 28
0.0575 0.1852 0.1737 0.0379 0.1983 0.1763 0.0414 0.0116 0.0820 0.0420 0.2080 0.5560 0.2080 0.1100 0.2560 0.1400 0.2559 0.1304 0.1987 0.1997 0.1932 0.2185 0.1292 0.0680 0.2090 0.0845 0.0749 0.1499 0.0236 0.2020 0.1790 0.2700 0.3292 0.2800 0.2087 0.3960 0.4153 0.6027 0.4533 0.2000 0.0599
5.75 18.52 17.37 3.79 19.83 17.63 4.14 11.60 8.20 4.20 20.80 55.60 20.80 11.00 25.60 14.00 25.59 13.04 19.87 19.97 19.32 21.85 12.92 6.80 20.90 8.45 7.49 14.99 2.36 20.20 17.90 27.00 32.92 28.00 20.87 39.60 41.53 60.27 45.33 20.00 5.90
1300 1300 650 1300 1300 650 900 700 1300 320 650 320 650 650 650 650 320 320 320 160 160 160 160 320 320 320 320 320 320 160 160 160 160 160 160 650 160 160 160 320 320
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 3 4 4 4 4 4 4 4 4 3 3
2 1 2 1 9 5 1 5 5 41 15 14 7 15 7 4 18 17 14 18 15 25 25 15 15 15 36 27 27 36 36 36 36 1 36 15 38 37 38 41 10
fc tc L Lh Lg li NAREA K(a)
end terminal phase angle difference of circuit c, i.e. (ui–uj ) the length of the circuit between buses i and j set of lines in the whole interconnected system set of heavily loaded lines set of lightly loaded lines shadow cost for power balance at bus i (dual variable of Eq. (6), gi =0) number of areas in the interconnected system the set of transmission lines belonging to area a
Ma
da Ma, c M fc f *c
capacityc
the total MW-mile of area a (also doubles up as a pseudo dual variable of Eq. (7), da = 0) a dummy variable MW-mile of area a due to circuit c vector of dimension A with elements Ma power flow of circuit c threshold transmission capacity of circuit c at which the line is considered stressed required capacity of the circuit c
Mathematically the objective of optimization is to minimize the investment costs in the interconnected
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
208
system. As the cost of a line is proportional to its length, using the MW-mile concept, the formulation is: Minimize CT= % tcbcfc
(1)
c Lh
c
s.t.
n
gi = %bij (ui −uj ) − Pi =0 j
Öi N
(2)
where bij = bc,
bii = % −bij j"i
The Lagrangian to be minimized over voltage angles u and the power generation at the reference bus (say bus 1, Pg1), is given by, L =CT+ % li gi
(3)
iN
Since generation redispatch is not allowed, the formulation of the ‘optimization’ problem in this way has
zero degrees of freedom. This linear program is actually solving the DC load flow problem. However this restatement of the DC load flow as an optimization problem makes it possible to determine the shadow costs from the dual solution. The shadow cost for power balance at bus i, li, gives the increase in pool MW-mile transport cost on the heavily loaded lines for a unit increase in the power demand at bus i. It should be noted that 1. the shadow cost for power generation at bus i is equal and opposite to the shadow cost for power demand at bus i; 2. the shadow cost for bus 1 must be equal to zero throughout, because bus 1 has been chosen as the reference bus. However, the choice of the slack bus will not change the transport cost between two buses. The unit of these marginal costs is miles of incremental transmission capacity and can be transformed to $/MW-mile per year by multiplying by annualized line construction costs [5].
Table 3 MW-mile sensitivity Change demand at bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Area MW-mile sensitivity 1
2
3
4
Total
0.0 293.8 344.5 386.9 835.6 471.5 723.5 515.5 613.3 913.8 405.3 852.4 852.4 855.5 857.9 878.0 902.9 877.5 889.0 895.1 900.1 895.7 850.8 841.3 729.5 729.5 658.7 524.2 658.7 658.7
0.0 1.6 −5.0 −6.0 6.8 9.3 7.1 10.5 38.7 97.8 38.7 −32.1 −172.1 208.2 314.2 218.5 75.7 456.0 539.8 515.9 107.3 110.3 243.1 148.0 101.5 101.5 72.0 16.1 72.0 72.0
0.0 −0.6 1.8 2.2 −2.5 −3.4 −2.6 48.6 −14.3 −20.0 −14.2 152.5 152.5 172.3 187.7 115.8 80.1 211.9 226.2 165.8 178.6 148.4 235.5 299.4 325.9 325.9 342.8 225.1 342.8 342.8
0.0 0.6 −1.8 −2.2 2.5 3.4 2.6 1.1 62.4 93.5 62.3 31.2 31.2 14.6 1.6 57.2 82.4 33.7 52.7 62.7 126.2 136.4 114.7 265.9 688.3 968.3 956.1 −10.1 1964.2 2416.3
0.0 295.3 339.5 380.9 482.5 480.9 730.7 575.5 700.1 1085.2 492.1 1004.1 864.1 1250.7 1361.5 1269.6 1141.3 1579.2 1707.9 1639.7 1312.3 1291.1 1444.2 1554.8 1845.4 2125.4 2029.7 755.3 3037.8 3489.9
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212 Table 4 Long run marginal wheeling cost in MW-mile incurred in area 1 From bus
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
To bus 1
16
21
30
5
– −294 −344 −387 −836 −472 −724 −515 −613 −914 −405 −852 −852 −856 −858 −878 −903 −877 −889 −895 −900 −869 −851 −841 −730 −730 −659 −524 −659 −659
878 584 534 491 42 407 155 363 265 −36 473 26 26 23 20 – −25 1 −11 −17 −22 −18 27 37 149 149 219 354 219 219
900 606 556 513 64 429 177 385 287 −14 495 48 48 45 42 22 −3 23 11 5 – 4 49 59 171 171 241 376 241 241
659 265 314 272 −177 187 −65 143 45 −255 253 −194 −194 −197 −199 −219 −244 −219 −230 −236 −241 −237 −192 −183 −71 −71 0 134 0 –
836 542 491 449 – 364 112 320 222 −78 430 −17 −17 −20 −22 −42 −67 −42 −53 −60 −64 −60 −15 −6 106 106 177 311 177 177
(L =0 (Pg1 gi = 0
From bus
(4) (5)
Öi N
The area MW-mile equations are:
n
% tcbcfc − Ma = 0;
c K(a)
Öa A
(7)
Table 5 Sum of the marginal wheeling cost incurred in all four areas (in MW-mile)
The sensitivity analysis below enables us to find the sensitivity of small changes in the power demand vector Pd to small changes in the problem variables and shadow cost vectors (u, Pg1, l, M) necessary to maintain optimum operation. These sensitivities are calculated using a linear expansion of the Kuhn – Tucker conditions for the multi-area pool model in the vicinity of the solution to the problem Eqs. (1) – (3). At the optimum solution the Kuhn – Tucker conditions are i (2…, N)
Now when the system Eqs. (4)–(7), whose solution is no different from the power flow solution, are to be perturbated while retaining optimality in the post-perturbation state, the changes of the variables and shadow cost vectors (u, Pg1, l, M) can be expressed in terms of changes in the power demand vector Pd. Now what is being done may be explained as follows. A small perturbation of power variables (Eq. (6) type constraints) is introduced but optimality is to retained after the perturbation. This requires that Eqs. (4) and (5) remain true, that is equal to zero. Eq. (7) is also included so as to enable us to calculate the change in M after the perturbation. Mathematically these points may be explained as follows:
2.1. Sensiti6ity analysis
(L =0 (ui
da =
209
(6)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
To bus 1
16
21
30
5
– −295 −339 −381 −842 −481 −731 −576 −700 −1085 −492 −1004 −864 −1251 −1362 −1270 −1141 −1579 −1708 −1640 −1312 −1291 −1444 −1555 −1845 −2125 −2030 −755 −3038 −3490
1270 974 730 889 427 789 539 694 569 184 777 265 405 19 −92 – 128 −310 −438 −370 −43 −21 −175 −285 −576 −856 −760 514 −1768 −2220
1312 1017 973 731 470 831 582 737 612 227 820 308 448 62 −49 43 171 −267 −396 −327 – 21 −132 −242 −533 −813 −717 557 −1725 −2178
3490 3195 3150 3109 2647 3009 2759 2914 2790 2405 2998 2486 2626 2239 2128 2220 2349 1911 1782 1850 2178 2199 2046 1935 1644 1364 1460 2735 452 –
842 547 503 462 – 362 112 267 142 −243 350 −162 −22 −408 −519 −427 −299 −737 −865 −797 −470 −449 −602 −712 −1003 −1283 −1187 87 −2195 −2647
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
210
ÇÃ ÃÃ
Æ (L D Æ Du Ç Ã (u à à à DP g1 (L S=à à = ÃD Dl (P g1 Ã Ã Ã È DM É Ã Dg È Dd
Æ 0 Ç Ã Ã 0 =à à DP à à dÃ È 0 É Ã
(8)
3. Interconnected system secure transportation
É
S, sensitivity matrix= Æ ( 2L ( 2L ( 2L à ( 2L à (u(u (P g1(u (l(u (M(u 2 2 à ( 2L ( L ( L ( 2L à (u(P g1 (P g1(Pg1 (l(P g1 (M(P g1 à (g (g (g à (g (u (P (l (M g1 à (d (d (d à (d à (u (l (M (P g1 È
Security is a prominent factor driving network investment. The standards of service have a direct impact on investments burdens and therefore definition and consensus among participants in respect of security standards are necessary. It is possible to extend the above long run marginal cost evaluation to take security into account. In the following a contingency is defined as the loss of any one circuit. The charge is levied to cover the cost of providing a secure system. To extend the above method to take security into account, the mathematical formulation is:
Ç Ã Ã Ã Ã Ã Ã Ã Ã Ã É
Minimize
These coefficients are relatively simple and a sample collection is provided in Appendix A. Eq. (8) can be written concisely as SDU =DV
(9)
cLh
(12)
c
The constraints to be satisfied, during the normal and any contingency conditions, are: gi = %bij (ui − uj )− Pi = 0 (bij (ui − uj ) 5 capacityc
(N −1)+ 1+(N) + (NAREA) 2. S is a square matrix Now by expressing DU as functions of the differentials DV, it will be possible to assess the effect of changes in demand on the solution. Pre-multiplying both sides of Eq. (9) by the inverse of the sensitivity matrix, we have (10)
In particular, the change in area MW-mile is of prime interest for marginal wheeling cost calculation. The total differential of MW-mile in area a will be: DMa = %Skm DVm
% tc capacityc
ÖiN
(13)
j
notice that 1. The dimensions of both DV and DU are:
DU =S − 1DV
in power demand at the receiving bus. The corresponding DMa can be easily found by setting the two appropriate elements of DPd to 9 1 and all other elements to zero.
(11)
m
where Skm are the elements of row k of the inverse sensitivity matrix (S − 1) such that element k of DU is DMa, DVm is the m-th element of DV. Skm is the MW-mile change in area a per unit change in Vm. Elements DPd in the column vector DV can be manipulated to determine the marginal cost of wheeling for specific buses. For a bus to bus transaction the marginal wheeling cost of an area (a measure of what it costs the area to maintain the wheel) is the sum of the MW-mile changes in the area due to a unit decrease in power demand at the supplying bus and a unit increase
(14)
Since generation redispatch is not allowed, the ‘optimization’ problem has zero degrees of freedom and hence the problem can be solved on a ‘line by line’ basis as follows:
3.1. Step 1 Calculate the flows on each line for every contingency. Record the contingency which produces the highest flows on each line.
3.2. Step 2 Consider a circuit c and use the network with the contingency which produces the highest flow on the circuit c found in step 1. Now capacityc = bij (ui –uj ) can be found by solving Eq. (13). Calculate the N sensitivities of each area’s MW-mile with respect to the bus demands at all N buses. The calculations methodology is the same as that in Section 2 except now we have only circuit c in the objective function (Eq. (1)) and the area MW-mile equation (Eq. (7)). This means that only the change of capacity of the circuit c is considered when the sensitivity calculation is carried out. The sensitivity of the change of MW-mile in area a to the power demand change at bus i is
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
((Ma, c ) (Pdi
aA
(15)
This sensitivity calculation is repeated for every circuit in the interconnected system.
3.3. Step 3 The summation of these sensitivities for all the circuits in the whole interconnected system gives the overall sensitivities of the change of MW-mile in area a to the power demand change at bus i. It is given by: ((Ma, c ) % (Pdi c
aA
(16)
3.4. The contingency ranking method A line outage will increase the stress on the transmission system. Step 1 in the above algorithm requires a large amount of computing for large systems. In practice, the stressing effects of some lines outages are small as compared with outage of the other lines. Therefore, contingency ranking [6] can be carried out according to the probability of system stress caused by different line outages. If the flow during the contingency ranking process exceeds a prescribed threshold, f *c (taken as 80% of the rating of the line), a line is considered to be stressed. Step 1 is only performed on the lines with higher probability of stressing the system and hence the amount of computing can be significantly reduced. In order to reflect the system stress, a system performance index is defined as: L
II= %
c=1
fc f*c
2
(17)
When there is no stress fc /f *c is not greater than one; the index is small. When there is stress in the system, fc /f *c for the stressing line is greater than one and the positive exponential element makes the index large. Therefore, this index reflects the system security.A sensitivity analysis of the index with respect to the change of a line admittance will reveal the impact of a line outage on the system security. When a line k fails, the change in the index is given by DPk =
(P DB (Bk k
(18)
where DBk =Bk is the admittance of the line k. The bigger the DPk is, the larger the increase in the P will be, which indicates that the probability of a faulted line k causing system stress becomes higher. Reference [6] provides detailed mathematics for calculating DPk when circuit k, which spans buses i and j, fails. A brief summary follows: DPk =
tk f 2k 2uk fk f 2k + − (1− Bkxk )2 (1 − Bkxk ) (1 − Bkxx )2f *2 k
(19)
211
where xk X tk T W A Wd uk PN
Xii+Xjj+2Xij system reactance matrix Tii+Tjj+2Tij XWX AWd A T branch to bus incidence matrix 2 *2 2 *2 diag [B 21/f *2 1 B 2/f 2 ...... B L/f L ] T A XWXPN bus real power injection vector
The process of contingency ranking is to compute the value of DP for all lines using Eq. (19) and rearrange in descending order of magnitude. During the contingency analysis in step 1, load flow calculations are carried out on the lines with the large values of DP. The lines with small values of DP are not subjected to analysis because the probability of system stress caused by these outages is small.
4. Case study The IEEE 30-bus system as depicted in Fig. 1 is used to demonstrate the performance of the method. The system has four interconnected areas, 41 transmission lines and six generators. Each tie line is assumed to belong to one area and the area that owns the line is indicated by a dot. The MW generation and MW demand at each bus are given in Table 1. The transmission line data are given in Table 2. All the lines are assumed to be heavily loaded. The most right column of Table 2 gives the line outage (contingency) which gives the maximum flow on a particular line, e.g. line 2 outage will give the maximum flow on line 1. This column can be obtained by running load flow for all contingencies. The column can also be obtained by the contingency ranking analysis as discussed in Section 3.4. It was found that running load flow for the contingencies which has the higher ranking (upper 46% of the DP magnitude descending order list) can give the column. Hence significant computer time can be saved. The sensitivities of the area MW-mile (Ma ) with respect to the change in bus demands (Pd ) for the test system are given in Table 3. Note that bus 1 was chosen as the reference bus (i.e. if the demand of any bus is increased by 1 MW, the demand at bus 1 will decrease by 1 MW accordingly). For example, the last row of Table 3 shows that if the power demand of bus 30 is increased by 1 MW, the MW-miles of areas 1, 2, 3 and 4 will increase 658.7, 72.0, 342.8, 2416.3, respectively. The overall increase is 3489.9 MW-miles. For a bus to bus transaction the marginal wheeling cost of an area is the sum of the MW-miles changes in the area due to a
212
C.W. Yu / Electric Power Systems Research 50 (1999) 205–212
unit decrease in power demand at the supplying bus and a unit increase in power demand at the receiving bus. The marginal wheeling costs of area 1 and the sum of the marginal wheeling costs incurred in all four areas are given in Tables 4 and 5, respectively. Each entry in the tables gives the long-run marginal wheeling cost for the secure transportation of another MW of electricity from the source on the left-hand side to a receiving end listed at the top of the table. The transport costs may be either positive or negative. Negative means that wheeling causes a decrease in MW-miles in an area.The total transport cost is seen to be positive if the flow direction of the power transaction is from area 1 to 4 while the cost is negative if the direction is reversed. This is because generation is concentrated in area 1, the power flow in mainly from area 1 to 4. In other words, if the power flow of a transaction further stress the transmission system, the cost will be positive while if it relieves the transmission system, the cost will be negative.
5. Conclusions The marginal wheeling costs of transporting power between buses, with security taking into account, are computed using the sensitivities of the MW-mile of each area with respect to the bus power demand. These sensitivities are calculated using a linear expansion of the Kuhn–Tucker conditions of the investment cost optimization problem. The marginal wheeling cost is proportional to the impact on system transmission investment requirement and provides economically efficient price for location. The marginal wheeling cost can be positive or negative. If the power flow of a transaction further stress the transmission system, the cost will be positive while if it relieves the transmission system, the cost will be negative. The securer the system, the higher will be the wheeling cost. Prices set equal to long-run transmission costs are stable and foreseeable. They correctly signal to users the long-run incremental cost of transmission service over several transmission
expansion cycles and hence facilitate long-term contractual decisions. The capacity charge can be applied as an annual charge and does not interfere with the short-run marginal cost economic signals for recovering the operating cost.
Appendix A. Typical coefficients of the sensitivity matrix are as follows: ( 2L =0 (ui(uj ( 2L = bij (ui(lj
Öi, j Öi, j
( 2L ( 2L = =0 (Pg1(ui (ui(Pg1 (gi = bij (uj
Öi
Öi, j
(g (g (d (d = =0 = = (l (l (Pg1 (M
References [1] C.W. Yu, A.K. David, Pricing transmission services in the context of industry deregulation, IEEE Trans. Power Syst. 12 (1) (1997) 503 – 510. [2] R.D. Tabors, Transmission system management and pricing: new paradigms and international comparisons, IEEE Trans. Power Syst. 9 (1994) 206 – 215. [3] M. Munasinghe, J.J. Warford, Electricity Pricing: Theory and Case Studies, John Hopkins Press, Baltimore, MD, 1982. [4] M.C. Calviou, R.M. Dunnett, P.H. Plumptre, Charging for use of a transmission system by marginal cost methods, in: Proceeding of the Eleventh PSCC, Avignon, France, 1993, pp. 385 –391. [5] C.W. Yu, A.K. David, Long run marginal cost evaluation of transmission capacity, Int. Power Eng. Conf. Singapore May (1997) 425 – 430. [6] T.A. Mikolinnas, B.F. Wollenberg, An advanced contingency selection algorithm, IEEE Trans. Power Appar. Syst. 100 (2) (1981) 608 – 615.
.