bus Transmission Network Cost Allocation - IEEE Xplore

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 1, FEBRUARY 2007

Zbus Transmission Network Cost Allocation

Antonio J. Conejo, Fellow, IEEE, Javier Contreras, Senior Member, IEEE, Delberis A. Lima, and Antonio Padilha-Feltrin, Senior Member, IEEE

Abstract—This paper addresses the problem of allocating the cost of the transmission network to generators and demands. A physically-based network usage procedure is proposed. This procedure exhibits desirable apportioning properties and is easy to implement and understand. A case study based on the IEEE 24-bus system is used to illustrate the working of the proposed technique. Some relevant conclusions are finally drawn.

B. Results Electrical distance between bus and line (adimensional). Total transmission cost allocated to the demand . located at bus Total transmission cost allocated to the generator . located at bus Transmission cost of line allocated to the demand . located at bus Transmission cost of line allocated to the . generator located at bus Active power flow through line associated with the nodal current (W). Cost rate for line & . Usage of line (W). Usage of line allocated to the demand located at bus (W). Usage of line allocated to the generator located at bus (W). Usage of line associated with nodal current (W).

Index Terms—Network usage, transmission cost allocation, bus .

NOTATION The notation used throughout this paper is stated below for quick reference.

A. Data Cost of line . Nodal current (A). Current through line (A). Number of buses. Active power consumed by the demand located at bus (W). Active power produced by the generator located at bus (W). Active power flow through line (W). Complex power flow through line calculated at bus (VA). Nodal voltage at bus (V). Series admittance of the equivalent circuit of line (S). Shunt admittance of the equivalent circuit of line (S). Impedance matrix . Element of the impedance matrix . Set of all transmission lines. Manuscript received April 3, 2006; revised September 27, 2006. The work of A. J. Conejo and J. Contreras was supported in part by the Ministry of Science and Technology of Spain under CICYT Project DPI2003-01362 and in part by Junta de Comunidades de Castilla-La Mancha under Project PBI-05-053. The work of D. A. Lima was supported in partl by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant 201153/2004-1. The work of A. Padilha-Feltrin was supported in part by CNPq under Grant 473108/2004-6. Paper no. TPWRS-00173-2006. A. J. Conejo and J. Contreras are with Universidad de Castilla-La Mancha, Ciudad Real, Spain (e-mail: [email protected]; Javier.Contreras@ uclm.es). D. A. Lima and A. Padilha-Feltrin are with Universidade Estadual Paulista (UNESP) “Júlio de Mesquita Filho,” Ilha Solteira, São Paulo, Brazil (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2006.889138

I. INTRODUCTION A. Motivation and Approach

T

HIS PAPER provides a methodology to apportion the cost of the transmission network to generators and demands that use it. How to allocate the cost of the transmission network is an open research issue as available techniques embody important simplifying assumptions (see the literature review below), which may render controversial results. This paper contributes to seek an appropriate solution to this allocation problem using an usage-based procedure that relies on circuit theory. The proposed technique consists of the following steps. 1) The active power flow of any transmission line is apportioned among all nodal currents. 2) Based on the above apportioning, the cost of any line is allocated to all generators and demands. 3) The procedure is repeated for all lines. B. Literature Review A brief description of the most significant proposals reported in the technical literature on the allocation of the cost of the transmission network among generators and demands follows. In the traditional pro rata method, reviewed in [1] and [2], both generators and loads are charged a flat rate per megawatt-hour, disregarding their respective use of individual transmission lines. Other more elaborated methods are flow-based [3]. These methods estimate the usage of the lines by generators and de-

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TRANSMISSION NETWORK COST ALLOCATION

mands and charge them accordingly. Some flow-based methods use the proportional sharing principle [4], [5], which implies that any active power flow leaving a bus is proportionally made up of the flows entering that bus, such that Kirchhoff’s current law is satisfied. Other methods that use generation shift distribution factors [6], are dependent on the selection of the slack bus and lead to controversial results. The usage-based method reported in [7] and [8] uses the so-called equivalent bilateral exchanges (EBEs). To build the EBEs, each demand is proportionally assigned a fraction of each generation, and conversely, each generation is proportionally assigned a fraction of each demand, in such a way as both Kirchhoff’s laws are satisfied. The technique presented in this paper is related to the alloca, prevition of the cost of transmission losses based on the ously reported and explained in [9]. It should be emphasized that all transmission lines must be modeled including actual shunt presents an admittances. Doing so, the impedance matrix appropriate numerical behavior. A salient feature of the proposed technique is its embedded proximity effect, which implies that a generator/demand uses mostly the lines electrically close to it. This is not artificially imposed but a result of relying on circuit theory. This proximity effect does not take place if the EBE principle is used [7], as this principle allocates the production of any generator/demand proportionally to all loads/generators, which implies treating similarly “close by” and “far away” lines. Other techniques require stronger assumptions, which diminish their practical interest. Applying the proportional sharing principle implies imposing that principle, and using the pro rata criterion implies disregarding altogether network locations. Particularly, it should be noted that the proposed methodology simply relies on circuit laws while the proportional sharing technique, on top of circuit laws, relies on the proportional methodology. sharing principle, not needed by the

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Fig. 1.

equivalent circuit of line jk .

illustrate how the proposed technique works and to show its salient features. Section III provides and analyzes results from a case study based on the IEEE 24-bus Reliability Test System (RTS). Finally, Section IV gives some relevant conclusions. II.

NETWORK COST ALLOCATION METHODOLOGY

A. Problem Statement The purpose of the methodology presented in this paper is to allocate the cost pertaining to the transmission lines of the network to all the generators and demands. Once a load flow solution is available, the proposed method determines how line flows depend on nodal currents. This result is then used to allocate network costs to generators and demands. B. Background Consider the complex power flow computed at bus and flowing through the line connecting bus to bus , as shown in Fig. 1 [10]. As the power flow solution is known, we select . The the direction of the complex power flow so that is complex power flow (1)

C. Contributions The contributions of this paper are stated below. The proposed technique: 1) uses the contributions of the nodal currents to line power flows to apportion the use of the lines; 2) shows a desirable proximity effect; that is, the buses electrically close to a line retain a significant share of the cost of using that line; 3) is slack independent. 4) does not require an a priori definition of the proportion in which to split transmission costs between generators and demands. Specifically, the main contribution of this paper is a physical-based technique to identify how much an individual power injection “uses” the network. D. Paper Organization The remaining of this paper is organized as follows. Section II formulates the problem and describes the proposed apportioning technique. A simple four-bus example is used to

Using the

matrix, the voltage at node is given by (2)

The current through the line

is obtained as (3)

Substituting (2) in (3) (4) Rearranging (4)

(5)

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Note that the first term of the product in (5) is constant, as it depends only on network parameters. Thus, (5) can be written as

If bus contains only generation, the usage allocated to genis eration pertaining to line (14)

(6)

On the other hand, if bus contains only demand, the usage is allocated to demand pertaining to line

where (15) (7) provides a measure Observe that the magnitude of parameter of the electrical distance between bus and line . This notion is further elaborated in Section II-D below. Substituting (6) in (1)

Else, if bus contains both generation and demand, the usage allocated to the generation at bus pertaining to line is (16) and the usage allocated to the demand at bus pertaining to line is

(8) (17) Then, the active power through line

is (9)

For the sake of simplicity and for each line, we consider a total annualized line cost in , which includes operation, maintenance, and building costs. Note that how this cost is computed is outside the scope of this paper. The corresponding cost is then rate for line

or, equivalently (18) (10)

In this way, the cost of line at bus is

allocated to the generator located

Thus, the active power flow through any line can be split and associated to the nodal currents in a direct way. Then, the active power flow through line associated with nodal current is

(19) Similarly, the cost of line bus is

allocated to the demand located at

(11) (20) C. Transmission Cost Allocation due to nodal Following [7], we define the usage of line current as the absolute value of the active power flow component , i.e., (12) That is, we consider that both flows and counter-flows do use the line. The total usage of line is then

Finally, the total transmission cost of the network allocated to the generator located at bus is (21) In addition, similarly, the total transmission cost allocated to the demand located at bus is

(22)

(13) D. Effect of Flow Directions Then, we proceed to allocate the use of transmission line to any generator and demand. Without loss of generality, we consider at most a single generator and a single demand at each node of the network. apportioned to the generator or Then, the usage of line demand located at bus is stated below.

Note that (8) is written in such a manner that , that is, in the direction of the active power flows. Note also that this way . Howto write (8) leads to electrical distance parameters ever, (8) can also be written in the direction of the active power . Observe counter-flows, which leads to distance parameters that both ways to write (8) are correct. However, (7) shows that

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distance parameters are not generally symmetrical with respect , which results in different usage to line indexes, i.e., allocations depending on whether (8) is written in the direction of the active power flows or counter-flows [see (11)–(12)]. It naturally raises the following question: which active power directions should be selected to write (8)? To properly address -based techniques. this question, we propose two The first one is denoted and is based on (8) written in the direction of the active power flows. This is a natural choice as the actual active power flows directions are used. Nevertheless, this selection generally results in higher usage allocation to generators versus demands. , provides The second technique proposed, denoted by the average value of allocated cost (usage) using 1) the protechnique with (8) written in the direction of the posed active power flows and 2) with (8) written in the direction of the active power counter-flows. This second technique smooths the trend of allocating higher network usage to generators versus demands. E. Building Tariffs

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Fig. 2. Four-bus system.

TABLE I TRANSMISSION COST ALLOCATION OF LINE 1 (1; 2) FOR EACH BUS

Given a converged power flow, the technique allows computing the actual usage that any generating unit or demand makes of any network line. This is raw information that needs to be processed to generate appropriate network usage tariffs. A similar situation occurs with electric energy LMPs, which have to be properly aggregated to produce stable consumer prices. In order to avoid hourly volatility, network usage values can be aggregated by peak and off-peak periods and perhaps also by season within a yearly framework. Given the annualized costs of all network lines, usage charges are then computed ex post for all considered periods within the year. Alternatively, usages can be predicted and aggregated, and tariff constructed ex ante. This second procedure has to be complemented with yearly adjustments due to prediction errors. Using the above aggregating techniques, generating units or demands face stable network usage charges within a given year, which allows them to properly develop strategies (pool bidding, forward contract involvements, and the like) taking into account network usage costs.

TABLE II TRANSMISSION COST ALLOCATION OF LINE 2 (1; 3) FOR EACH BUS

F. Example

TABLE III TRANSMISSION COST ALLOCATION OF LINE 3 (1; 4) FOR EACH BUS

To illustrate the working of the and the methods for transmission cost allocation, we consider the four-bus system depicted in Fig. 2. Note that all buses are similar in terms of generation/demand. The five lines in the system have the same values of series resistances and reactances: 0.01275 and 0.097 p.u., respectively, and the shunt admittance is identical for the five lines: 0.4611 p.u. Fig. 2 provides the active power generated and consumed at each bus and the active power flow through the five lines. Finally, note that the cost of each line is considered to be proportional to its series reactance; thus, . This four-bus system allows visualizing the proximity effect, as it is expected that buses directly connected to a line would be apportioned most of the usage of that line. Tables I–VI provide the results of the transmission cost allocation to each bus. The results obtained are compared with

those obtained using other methods, namely, EBE [7], proportional sharing (PS) [4], and pro rata (PR) [1]. Observing Tables I–V, it can be noted that, for all the lines, and methods have the property that they allocate the a significant amount of the cost of each line to the buses directly connected to it. For lines 1, 2, 3, and 5, the two buses with the

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TABLE IV TRANSMISSION COST ALLOCATION OF LINE 4 (2; 4) FOR EACH BUS

TABLE V TRANSMISSION COST ALLOCATION OF LINE 5 (3; 4) FOR EACH BUS

TABLE VI TOTAL TRANSMISSION COST ALLOCATION FOR EACH BUS

Fig. 3. IEEE 24-bus RTS system.

highest line usage are these at the ends of the corresponding line. Taking into account that the power injected and extracted at each bus is very similar, the results reflect the location of each bus in the network. Note that the behavior of other procedures and the allocate most is different. For instance, the of the usage of line 5 (between buses 3 and 4) to buses 3 and 4, while the EBE to buses 1, 2, and 3 and the PS to buses 2 and 4. Note also that, for line 4 (between buses 2 and 4), the results method are somewhat different, since the provided by the allocation to bus 1, not directly connected to line 4, is also relevant. This happens, mostly, because the power injected at bus 1 is greater than the power extracted at bus 4: 261.3 and 250.0 MW, respectively. In addition, the absolute values of the elecand are identical, as well as the trical distance terms and , which makes buses 1 and 4 being at the values of same electrical distance to line 2–4. Nevertheless, the cost allocated to bus 4 is significant and similar to the cost allocated to bus 1. It should also be noted that for line 4, the results provided variant allocate the highest portion of line usage to by the buses 2 and 4, the terminal buses of line 4. and , it can be concluded that Comparing methods method smooths the trend of the one (as well as the of other methods) to allocate a higher portion of usage to generating buses versus demand buses. Finally, Table VI provides

the total cost allocated to each bus for the use of the entire network using the different methods considered. Note that results are significantly different. Note also the similar pattern of alloand EBE. cation provided by methods We conclude this example stating that the above results illusmethodology in relation trate adequately the features of the to other methods and show its appropriate behavior. III. CASE STUDY The IEEE 24-bus RTS [11] depicted in Fig. 3 is considered for this case study. The same five methods considered in the previous example are used in this section. The converged power flow corresponds with the IEEE RTS peak load, taking place on the Tuesday of week 51 from 5 P.M. to 6 P.M. All required data pertaining to the IEEE RTS can be found in [11]. Note also that the costs of the lines are considered to be proportional to their respective series reactances. A. Results Tables VII–X provide the transmission cost allocation to generators and demands for lines 23 (bus 14 to bus 16) and 11 (bus 7 to bus 8), respectively. These lines, highlighted in Fig. 3, are selected for the two reasons below. In terms of transmission cost allocation, line 23 behaves as most lines throughout the system do, thus being a representative line of the network. Conversely, line 11, which is peripheral, exhibits clearly the proximity effect discussed in the example above.

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TABLE VII LINE 23 TRANSMISSION COST ALLOCATION TO GENERATORS

TABLE IX LINE 11 TRANSMISSION COST ALLOCATION TO GENERATORS

TABLE VIII LINE 23 TRANSMISSION COST ALLOCATION TO DEMANDS

TABLE X LINE 11 TRANSMISSION COST ALLOCATION TO DEMANDS

Additionally, Tables XI and XII show the total transmission cost allocation for all the generators and demands, respectively.

Tables IX and X show that, for the and methods, almost 100% of the cost of line 11 is allocated to bus 7, split between its generation and demand. This happens because the only way in which bus 7 can inject to or extract power from the network is through line 11, as it can be seen in Fig. 3. Regarding the other methods, the EBE method splits the power generated at bus 7 proportionally to all the demands of the system; thus, no significant proximity effect takes place. Because of the existence of counter-flows, the PS method allocates no cost to demand 7. This last result is not desirable, as demand 7 uses line 11. and methods, it can be Additionally, for the noted that a relatively small portion of the total network cost is allocated to bus 7, because this bus is placed at the network boundary (see Tables XI and XII). Note also that for the and methods, the amount of the cost of line 11 allocated

B. Result Analysis Table VII shows that all methods allocate most of the costs of using line 23 to generators 21, 22, and 23. This is expected because all these generators are electrically close to that line, and their productions are comparatively high. As a result of eliminating counter-flows, the PS procedure does not allocate any cost of line 23 to generator 23, which is a questionable result. , EBE, and PS methods Table VIII shows that the allocate most of the cost of line 23 to demand 14. This is also reasonable because that demand is comparatively high and is directly connected to line 23. However, observe the significant allocation differences among methods.

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TABLE XI TOTAL TRANSMISSION COST ALLOCATION TO GENERATORS

Table XII. This happens because buses 3 and 8 have the highest demands, and they are located far away from the main generators: 21, 22, and 23. Therefore, buses 3 and 8 use many of the lines in the network. Finally, observe that all methodologies tend to allocate signiftechnique icantly higher usage to generators. The proposed follows this trend being the average allocation of net generating , while the method smooths this trend allobuses 177 to generating buses. The EBE, PS, cating on average 172 and PR procedures result in average allocations of 151, 172, and 144, respectively. Average allocation for net demand buses of , EBE, PS, and PR procedures are, respectively, 95, . 99, 81, 114, and 83 IV. CONCLUSION

TABLE XII TOTAL TRANSMISSION COST ALLOCATION TO DEMANDS

and the procedures to allocate the cost of Both the the transmission network to generators and demands are based on circuit theory. They generally behave in a similar manner as other techniques previously reported in the literature. However, they exhibit a desirable proximity effect according to the underlying electrical laws used to derive them. This proximity effect is more apparent on peripheral rather isolated buses. For these buses, other techniques may fail to recognize their particular lovariant smooths the trend of the method cations. The (as well as of other techniques) to allocate a higher line usage to generators versus demands. We have performed extensive numerical simulations and induced ill-conditioning encountered neither numerical nor unreasonable results. Thus, we conclude that the proposed methods are appropriate for the allocation of the cost of the transmission network to generators and demands, complement existing methods, and enrich the available literature. REFERENCES

to bus 8 (0.000117 and 0.0784 , respectively, demand only) , is much smaller than that allocated to bus 7 (61.4 and 59.1 respectively, demand plus generation). However, total network , respectively, usage allocated to bus 8 (174.5 and 179.96 demand only) is almost as high as the allocation to bus 7 (179.9 , respectively, demand plus generation). This and 180.74 can be considered a reasonable result and a consequence of bus 8 being is a less isolated spot of the network, which allows bus 8 a more intensive use of the network. and methods allocate most Table XI shows that the of the total cost of the network to generators 21, 22, and 23, just like the other methods. Considering that these generators are the highest producers in the network and that they feed a significant amount of the demand of the system, this is an appropriate reand methods, the netsult. For the demands, using the work costs are mostly allocated to demands 3 and 8, as shown in

[1] M. Ilic, F. Galiana, and L. Fink, Power Systems Restructuring: Engineering and Economics. Norwell, MA: Kluwer, 1998. [2] D. S. Kirschen and G. Strbac, Fundamentals of Power System Economics. Chichester, U.K.: Wiley, 2004. [3] J. W. M. Lima, “Allocation of transmission fixed rates: An overview,” IEEE Trans. Power Syst., vol. 11, no. 3, pp. 1409–1418, Aug. 1996. [4] J. Bialek, “Topological generation and load distribution factors for supplement charge allocation in transmission open access,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1185–1193, Aug. 1997. [5] D. S. Kirschen, R. N. Allan, and G. Strbac, “Contributions of individual generators to loads and flows,” IEEE Trans. Power Syst., vol. 12, no. 1, pp. 52–60, Feb. 1997. [6] W. Y. Ng, “Generalized generation distribution factors for power system security evaluations,” IEEE Trans. Power App. Syst., vol. PAS-100, pp. 1001–1005, Mar. 1981. [7] F. D. Galiana, A. J. Conejo, and H. A. Gil, “Transmission network cost allocation based on equivalent bilateral exchanges,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1425–1431, Nov. 2003. [8] H. A. Gil, F. D. Galiana, and A. J. Conejo, “Multiarea transmission network cost allocation,” IEEE Trans. Power Syst., vol. 20, no. 3, pp. 1293–1301, Aug. 2005. [9] A. J. Conejo, F. D. Galiana, and I. Kockar, “Z-bus loss allocation,” IEEE Trans. Power Syst., vol. 16, no. 1, pp. 105–110, Feb. 2001. [10] A. R. Berger and V. Vittal, Power Systems Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000. [11] Reliability Test System Task Force, “The IEEE reliability test system 1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999.

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Antonio J. Conejo (F’04) received the M.S. degree from Massachusetts Institute of Technology, Cambridge, in 1987 and the Ph.D. degree from the Royal Institute of Technology, Stockholm, Sweden, in 1990. He is currently a full Professor at the Universidad de Castilla—La Mancha, Ciudad Real, Spain. His research interests include control, operations, planning, and economics of electric energy systems, as well as statistics and optimization theory and its applications.

Delberis A. Lima received the B.S. degree in electrical engineering and the M.Sc. degree from Universidade Estadual Paulista (UNESP) “Júlio de Mesquita Filho,” Ilha Solteira, São Paulo, Brazil, in 2000 and 2003, respectively. He is currently pursuing the Ph.D. degree at UNESP. He was a Research Visitor at the Universidad de Castilla—La Mancha, Ciudad Real, Spain, in 2005. His research interests include transmission and distribution operations and planning.

Javier Contreras (SM’05) received the B.S. degree in electrical engineering from the University of Zaragoza, Zaragoza, Spain, in 1989, the M.Sc. degree from the University of Southern California, Los Angeles, in 1992, and the Ph.D. degree from the University of California, Berkeley, in 1997. He is currently an Associate Professor at the Universidad de Castilla—La Mancha, Ciudad Real, Spain. His research interests include power systems planning, operations and economics, and electricity markets.

Antonio Padilha-Feltrin (SM’06) received the B.Sc. degree from Escola Federal de Engenharia de Itajubá (EFEI), Itajubá, Minas Gerais, Brazil, and the M.Sc. and Ph.D. degrees from Universidade Estadual de Campinas (UNICAMP), Campinas, São Paulo, Brazil. He is currently a full Professor at Universidade Estadual Paulista (UNESP) “Júlio de Mesquita Filho,” Ilha Solteira, São Paulo, Brazil. From 1995 to 1997, he was a Visiting Faculty Member at the Electrical and Computer Engineering Department of the University of Wisconsin–Madison. His main interests are in analysis and control of power systems.