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AbstractPower transfer distribution factors depend on the operating point and topology of an electric power system. However, in a recent paper, the power transfer distribution factors were shown theoretically to be relatively insensitive to the operating point. In this paper, we provide empirical corroboration of this theoretical result. The observations are consistent with those made in several recent papers; however, the systems considered in this paper are considerably larger than have been reported in other studies.

Index TermsPower transfer distribution factors, shift factors, DC power flow.

I. INTRODUCTION n (incremental) power transfer distribution factor (PTDF) is the sensitivity of the change in power flow on

a particular line to the change in the injection and corresponding withdrawal at a pair of busses. According to Kirchhoffs laws, PTDFs depend on the topology of the electric power system, the behavior of controllable transmission system elements as their limits are approached, and on the operating point [1]. That is, PTDFs change when an outage of a line occurs, if a controllable element reaches its control limits, and also as the pattern of injections and withdrawals change the loadings on the lines in the system.

In a recent paper [2], it was shown theoretically that the PTDFs are approximately constant in a system with losses and arbitrary fixed topology but having reactive compensation sufficient to keep voltage magnitudes constant at all busses. This result has also been demonstrated empirically in a number of small and medium sized test systems. See, for example, [3, 3.9][4] for empirical studies of the variation of PTDFs for certain test systems.

In this paper, we empirically evaluate the variation of PTDFs with loading by comparing the DC PTDFs, calculated from the DC power flow approximation, to the incremental PTDFs evaluated at a heavily loaded operating point. We consider the three principal interconnections in North America:

The Eastern Interconnection,

Ross Baldick and Krishnan Dixit are with the Department of Electrical and Computer Engineering, The University of Texas, Austin, TX78712 USA (e-mail: Ross.Baldick@engr.utexas.edu and krisdixit@ece.utexas.edu).

Thomas J. Overbye is with the Department of Electrical and Computer Engineering, The University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: overbye@ece.uiuc.edu).

The Western Interconnection, and The Electric Reliability Council of Texas (ERCOT)

Interconnection. In each case, we consider incremental PTDFs evaluated at a Summer peak base-case derived from FERC Form 715 data [5] or an outage case derived from the Summer peak base-case. We refer the PTDFs calculated for an outage of a transmission line as the outage transfer distribution factors (OTDFs).

Similar results to some of those we present here for an ERCOT peak base-case were also presented in [2]. However, the PTDF comparisons for the considerably larger Western and Eastern Interconnections represent, by far, the largest such PTDF studies reported to date. Moreover, the outage cases for ERCOT are the first examples of large scale testing of the variation of OTDFs under stressed system conditions. We also consider the PTDFs in ERCOT when significant reactive capacity from transmission system capacitors is unavailable. Both the line outage and the capacitor outage cases involve stressed system conditions that are outside of the strict region of validity of the theoretical results presented in [2].

DC PTDFs can be interpreted as being equivalent to incremental PTDFs at the (unrealistic) operating point where the net real power injection at each bus is zero. That is, where there are no net real power loads and no net real power injections. We find that these DC PTDFs are almost identical to the incremental PTDFs calculated at the Summer peak base-case operating points. Similarly, we find that the DC OTDFs are almost identical to the incremental OTDFs. Since PTDFs and OTDFs vary smoothly with the operating point except at the ultimate stability loading point [4], this means that the PTDFs and OTDFs evaluated at all intermediate loadings can also be expected to be very close to the corresponding DC PTDFs and OTDFs, respectively.

In summary, we show that the DC PTDFs are extremely close to the incremental PTDFs for the three large interconnections in North America at any operating point up to the Summer peak loading conditions, given the fixed topology of the Summer peak base cases. Similarly, we show that the DC OTDFs are close to the incremental OTDFs for the ERCOT system.

Empirical analysis of the variation of distribution factors with loading

Ross Baldick, Member, Krishnan Dixit, Student Member, and Thomas J. Overbye, Senior Member

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Many market clearing mechanisms and transmission rights schemes in use or proposed for North America make use of the DC PTDF approximation [6,7,8]. As discussed in [9], deviations between the DC and incremental PTDFs result in different market clearing prices and consequently in different market incentives. Empirical verification of the correspondence between the DC and incremental PTDFs validates the use of DC models for the market clearing mechanisms, at least under non-emergency conditions, and is consistent with the observation in [9] that the locational marginal prices calculated with DC and AC power flow models are typically close together.

The paper is organized as follows. Section II presents a brief literature survey. Section III discusses the study cases and the computation of the PTDFs. Section IV presents the results and we conclude in section V.

II. LITERATURE SURVEY PTDFs are often used in the context of approximating power flows. For example, Baughman and Schweppe use distribution factors to approximate flows as a function of injections and after a change in the topology of the network [10]. Sauer formulates PTDFs for linear load flows in [11]. Ng describes PTDFs for calculating the change in flows on lines given changes in generation and conforming changes in load at the busses [12]. Wood and Wollenberg describe the calculation of DC PTDFs using the DC power flow approximation in [13, Appendix 11A] and discuss the calculation of OTDFs [13, 11.3.2]. Sauer et al. also consider OTDFs in [11] and [14, 5].

The evaluation of incremental PTDFs (or OTDFs) at an operating point using the Jacobian of the power flow equations is described in [13, 13.3]. Grijalva analyzes in detail the variation of PTDFs in a three bus, three line example system with voltages maintained constant and also discusses how the PTDFs vary with loading. Grijalva shows that if voltages are maintained constant at all busses then, as loading increases from zero injection conditions, the incremental PTDFs that were largest at zero injection tend to decrease, while the PTDFs that were smallest at zero injection tend to increase [3, 3.9].

Liu and Gross conduct an empirical study of the variation of PTDFs with injections and with other changes [4]. They show that, for the IEEE 118 bus RTS system and for a system representing portions of the Eastern Interconnection, the PTDFs typically change by a relatively small amount as the levels of injections and withdrawals change.

Finally, Overbye, Cheng, and Sun study the effect on locational marginal prices (LMPs) of using the AC or DC power flow models [9]. They show that the differences in LMPs are usually small between the two models.

III. STUDY DATA AND COMPUTATION OF PTDFS We used data derived from FERC form 715 submissions to model Summer peak base-cases for the three principal interconnections in North America [5]. The DC and incremental PTDFs were calculated using PowerWorld Simulator [15]. The DC PTDFs were calculated on the basis of the complex part of the line admittances (instead of the inverse of the inductive reactances.) The incremental PTDFs were calculated at the Summer peak operating points.

Following the direct and adjoint approaches to sensitivity calculations [16, Chapter 9], the software provided a number of different approaches for calculating PTDFs. In this paper two approaches were used:

1. The PTDFs for each line in the system for injection at a selection of buses and withdrawal at the reference bus, and

2. The PTDFs for one selected line and for injection at each bus in the system and withdrawal at the reference bus.

In the first approach, the software allows for a weighted average shift factor over the group of selected busses, called an injection group. For example, a large generator might provide a larger percentage of the total injection than a smaller generator. We calculated PTDFs for injection groups that consisted of generators in particular regions to be able to cover a large number of combinations of busses and lines. However, we also calculated some PTDFs for combinations of individual generator busses to all lines to ensure that our results were not an artifact of, for example, averaging PTDFs over the injection group. In some cases, we used the second approach to calculate PTDFs from all generator busses in ERCOT to a particular line. In all cases, we will show PTDFs as percentages.

IV. RESULTS In this section, we describe the results for the ERCOT, Western, and Eastern Interconnections.

A. ERCOT Interconnection In this section, we compare the DC and incremental PTDFs for the Electric Reliability Council of Texas (ERCOT) system. We first consider PTDFs for a 2003 Summer peak case having 5165 busses and 5989 transmission lines. We refer to this as the base case.

As well as the PTDFs for the base case, we also calculate PTDFs for three line outage cases. That is, we consider OTDFs for these three outage cases [13, 11.3.2][14, 5]. Finally, we consider PTDFs for a case where considerable reactive support has been removed from the system.

1) Base case Figure 1 shows DC PTDFs versus the incremental PTDFs calculated for the base case for eleven different injection groups distributed across ERCOT and for the 5989 transmission lines in ERCOT. Each injection group consists of one or more generator busses associated with a particular

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entity in ERCOT. The injection groups are: AENXA, BPUB, CPS, DC East, HLP, LCRA, San Migul, STECA, TMPPA, TNMP, and Oncor. 1 Each injection group is illustrated with a different symbol in figure 1. For each PTDF, the point of withdrawal is the reference bus.

This case is similar to the one analyzed in [2] and the results here are similar. There are approximately 66,000 DC PTDFs and incremental PTDFs represented in figure 1.2 The PTDFs are shown as percentages and essentially all of them fall on a line with slope equal to one and intercept equal to zero.

Figure 1. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for injection groups in ERCOT system. PTDFs to each line in ERCOT and for each of eleven injection groups are shown. The PTDFs for each group are drawn with a different symbol. (Source: This figure was produced using the same methodology, but with a slightly different base case, as Figure 1 of [2].)

As discussed in [2, VII], figure 1 shows that for each injection group and almost all lines, the DC PTDF and corresponding incremental PTDF are essentially the same. The only exceptions are the eleven PTDFs calculated for the line that joins the reference bus to the rest of the system, which have DC PTDFs equal to 100% but incremental PTDFs greater than 100%. All power flowing to the reference bus flows through this line and consequently the incremental effect on losses throughout the system is reflected in this line.

To provide some perspective on the errors, we calculated the difference between the DC and incremental PTDFs and graphed them versus the incremental PTDFs in figure 2. Figure 2 shows that, except for the PTDFs involving the line connecting the reference bus to the rest of the system, the

1 DE East is the location of a DC intertie to the Eastern Interconnection.

There is no physical generation at this bus, but injection into ERCOT from the Eastern Interconnection is modeled as generation at this bus.

2 In fact, since most of the injection groups consist of several generator

busses, there are hundreds of thousands of DC and incremental PTDFs calculated and averaged across the injection groups to obtain the DC and incremental PTDFs shown in figure 1 and subsequent figures.

absolute value of the difference between the DC and incremental PTDFs is less than 4 percentage points. For most of the PTDFs in figure 2, the absolute value of the difference is less than 2 percentage points.

Figure 2. Error between DC and incremental PTDFs versus incremental PTDFs for 2003 Summer peak conditions for ERCOT system.

To further assess the error in using DC PTDFs instead of incremental PTDFs, we consider the relative error, defined by (DC PTDF incremental PTDF)/(incremental PTDF), and plot the relative error versus the incremental PTDF, as shown in figure 3. The points along the vertical axis in figure 3 show that the relative error can be fairly large for some PTDFs; however, these all correspond to cases where the incremental PTDF is itself small. In cases where the incremental PTDF is itself small, a large relative error will typically not cause significant error in the calculation of the effect of an injection and withdrawal on the flows on a line.

Figure 3. Relative error in PTDFs versus incremental PTDFs for 2003 Summer peak conditions for ERCOT system.

With the exception of the eleven PTDFs calculated for the line that joins the reference bus to the rest of the system, almost all of the points in Figure 3 that are not along the

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vertical axis lie along the horizontal axis. These points show that whenever the incremental PTDF is significantly different from zero then the relative error is small. That is, the relative error is small in the cases where the effect of an injection and withdrawal is significant. In summary, the relative error is small whenever the incremental PTDF is large, with the exception of the eleven PTDFs for the line that joins the reference bus to the rest of the system.

Figures 1 to 3 show results for the first approach to PTDF calculation described in section III with each point representing the average across several busses in an injection group. Potentially, this averaging across the busses in an injection group could mask some discrepancy between the DC and incremental PTDFs.

To investigate the possibility that the averaging across an injection group masked some variation in PTDFs, we considered PTDFs calculated from individual busses in an injection group. We considered 10 generator busses in the LCRA injection group and calculated DC and incremental PTDFs for injection at each generator bus, withdrawal at the reference bus, and for each line. The generators were selected to be in different parts of the LCRA area.

Figure 4 shows the results. Again, the DC and incremental PTDFs are essentially identical, with the exception of the PTDFs for the line that joins the reference bus to the rest of the system. In particular, Figure 4 suggests that the use of injection groups in figures 1 to 3 did not mask any significant deviation between the DC and incremental PTDFs. In the rest of this paper, for analysis of PTDFs to all lines, we will use injection groups rather than individual busses so as to be able to cover a large number of combinations of busses and lines.

Figure 4. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for generator busses in LCRA injection group for all lines in ERCOT. The PTDFs for each generator are drawn with a different symbol.

There are several main corridors of lines in ERCOT: 1. Graham-Parker and Graham-Benbrook, 2. the Sandow-Temple double circuit, and 3. the South Texas Project-Dow double circuit.

As a further test of the relationship between DC and incremental PTDFs, we calculated the PTDFs for each of these lines and all generator busses in ERCOT. There are 394 such generator busses. Figures 5-7 show the results. (Figure 5 shows the case for Graham-Parker, but the results for Graham-Benbrook were essentially the same. The results for each line in a double circuit are the same.)

Figure 5. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for all generator busses to Graham-Parker.

Figure 6. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for all generator busses to Sandow-Temple.

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Figure 7. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for all generator busses to South Texas Project-Dow.

In almost all cases, the DC and incremental PTDFs are essentially the same. However, there are a few exceptions. In particular, for the Graham-Parker line there are several DC PTDFs around 5% for which the incremental PTDFs are somewhat larger. For the South Texas Project-Dow there are several DC PTDFs around 13% to 14% for which the incremental PTDF is 17% to 19%.

To determine whether these exceptions are typical or atypical, we selected two of the generator busses, AMS #2 and EPLVT, for which the DC and incremental PTDFs differed significantly and calculated the PTDFs for these busses to all lines in the ERCOT system. Figures 8 and 9 show the results. These figures show that for these two busses, the DC and incremental PTDFs are mostly very similar, although there is a systematic deviation from equality that is at its most pronounced for PTDFs around 15% to 30%.

Figure 8. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for AMS #2 to all lines in ERCOT system.

Figure 9. DC PTDFs versus incremental PTDFs for 2003 Summer peak conditions for EPLVT to all lines in ERCOT system.

These two busses, and the others with significant deviation in figures 5-7, involve generators with relatively small capacity, on the order of a few tens of MW. Consequently, errors between the DC and incremental PTDFs would produce only small errors in the predicted effect on the flow on most lines. Significant errors between the DC and incremental PTDFs occur for just a relatively small number of combinations of such busses and lines.

2) Line outage cases Outages of lines on the three main corridors in ERCOT described in the previous section constitute the most important transmission contingencies for ERCOT and determine transfer capability between zones in the ERCOT congestion management system [17]. Outage transfer distribution factors (OTDFs) are necessary to properly represent the effect of these constraints on operation of the system [18]. We investigate whether the DC OTDFs are good approximations to the incremental OTDFs.

The three outage cases we consider are: 1. Outage of Graham-Parker and Graham-Benbrook, 2. Outage of the Sandow-Temple double circuit, and 3. Outage of the South Texas Project-Dow double

circuit. These are the three sets of contingencies that determine inter-zonal transfer capability in ERCOT.

Figures 10-12 show the DC OTDFs versus the incremental OTDFs for the three outage cases, respectively. In essentially all cases, and with the exception of the line that joins the reference bus to the rest of the system, the DC OTDFs and the incremental OTDFs are essentially the same.

Figure 10. DC OTDFs versus incremental OTDFs for outage of Graham-Parker and Graham-Benbrook.

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Figure 11. DC OTDFs versus incremental OTDFs for outage of Sandow-Temple double circuit.

Figure 12. DC OTDFs versus incremental OTDFs for outage of South Texas Project-Dow double circuit.

3) Capacitor outage cases In [2], the conditions for small error between DC and incremental PTDFs include the assumption that bus voltages are maintained constant. To investigate the significance of this assumption, we considered reducing the available reactive support from transmission system capacitors. In the base case system, there is approximately 11.6 GVAr of reactive support from capacitors and about 14.7 GVAr of reactive support from generators. We ordered the amounts of reactive support from the capacitors from most to least and then outaged, in order, each capacitor until the modified case would no longer solve. We then replaced the last capacitor and re-solved the case. We will call this the capacitor outage case. It represents a system that is close to being infeasible because of a lack of reactive support from capacitors.

The reactive support from capacitors in the capacitor outage case is decreased from the base case to approximately 8.9 GVAr. Correspondingly, the reactive support from generators is increased compared to the base case to approximately 18.0 GVAr, with increased reactive losses in the capacitor outage system and lowered voltages at many busses, reflecting the need to import reactive power from relatively remote generators.

Figure 13 shows the per unit voltages for the busses in the system. In figure 13, the 5165 ERCOT busses have been sorted in order of voltage and shown for the base case and the capacitor outage case. Voltages have been lowered at most busses but are still mostly within normal operating limits.

Figure 14 shows DC PTDFs versus the incremental PTDFs calculated for the capacitor outage case for eleven different groups of injection busses distributed across ERCOT and for the 5989 transmission lines in ERCOT. 3 As in previous graphs, the DC and incremental PTDFs are very similar, even though the reactive support from capacitors in this system has been reduced almost to the point where there is no solution. This means that the DC PTDFs are a good approximation to the incremental PTDFs even when voltage support from capacitors is relatively weak in the system.

Figure 13. Per unit voltages in base case and capacitor outage case.

Figure 14. DC PTDFs versus incremental PTDFs for capacitor outage case.

4) Summary For almost every generator bus and line combination in the ERCOT system, the DC PTDFs and incremental PTDFs are

3 The DC PTDFs are the same for the base case as for the capacitor outage

case since the DC PTDFs only depend on line impedance.

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nearly the same. Significant deviations between DC and incremental PTDFs occur for a small number of combinations associated with:

the line connecting the reference bus to the rest of the system, and

some generator busses having small capacity generators.

For most bus and line combinations, the DC and incremental PTDFs remain close together in a variety of line and capacitor outage conditions.

B. Western Interconnection In this section, we compare the DC and incremental PTDFs for the Western Interconnection. We consider PTDFs for a 2004 Summer peak case having 13614 busses and 46013 transmission lines. We refer to this as the Western base case.

Figure 15(a)-15(d) show DC PTDFs versus the incremental PTDFs calculated for the Western base case for twenty-two different groups of injection busses distributed across the Western Interconnection and for the 46013 transmission lines.4 As with the ERCOT case study, for the twenty-two PTDFs calculated for the line that joins the reference bus to the rest of the system, the DC and incremental PTDFs differ somewhat. Moreover, for some injection groups there is a consistent discrepancy between the DC and incremental PTDFs. Nevertheless, the DC and incremental PTDFs are in relatively close correspondence, although not as close as in the ERCOT system.

4 The injection groups are shown in separate graphs because our graphing

software was unable to plot all of the approximately 100,000 points on a single graph.

Figure 15(a)-(d). DC PTDFs versus incremental PTDFs for Western base case. PTDFs to each line in the Western Interconnection and for each of twenty-two injection groups are shown.

C. Eastern Interconnection In this section, we compare the DC and incremental PTDFs for the Eastern Interconnection. We consider PTDFs for a 2000 Summer peak case having 33538 busses and 53552 transmission lines. We refer to this as the Eastern base case.

We calculated DC and incremental PTDFs for 130 injection groups distributed across the Eastern Interconnection and for the 53552 transmission lines. Figure 16(a)-(b) shows typical

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results for the DC PTDFs versus the incremental PTDFs.5 As in the ERCOT system, there is a close correspondence between the DC and incremental PTDFs.

Figure 16(a)-(b). DC PTDFs versus incremental PTDFs for Eastern base case. PTDFs to each line in the Eastern Interconnection and for each of several injection groups are shown.

D. Comparison of results for ERCOT, Western, and Eastern Interconnection For both the ERCOT and Eastern Interconnections, the correspondence between DC and incremental PTDFs is very close for a very large fraction of all combinations of busses and lines. This is also true for the ERCOT system under various line and capacitor outage situations.

5 As in the case of the Western Interconnection, we were only able to show a

small number of injection groups on each graph because of limitations in the graphing software.

In the Western Interconnection, there was a larger discrepancy between DC and incremental PTDFs. In [2], it is noted that the difference between DC and incremental PTDF increases with the angle across a line. The Western Interconnection has many long, stability limited lines, suggesting that the angles across lines in the Western Interconnection tend to be larger than in the ERCOT or Eastern Interconnection. In turn, this accounts for the greater discrepancy between DC and incremental PTDFs in the Western Interconnection.

V. CONCLUSION In this paper, we have demonstrated empirically that values of DC PTDFs are very close to incremental PTDFs for practical operating points of realistic power systems, with the Western Interconnection showing greater discrepancies between DC and incremental PTDFs than either the ERCOT or Esatern Interconnections. We have also shown that DC OTDFs are close to the values of incremental OTDFs for the ERCOT system and that DC and incremental PTDFs in ERCOT remain close even under outage of significant amounts of reactive support from capacitors.

VI. ACKNOWLEDGMENT This work was supported, in part, by the Federal Energy Regulatory Commission Office of Markets, Tariffs and Rates. We would like to thank the Lower Colorado River Authority for help with the operation of PowerWorld Simulator.

VII. REFERENCES [1] William Hogan, Flowgate rights and wrongs, John F. Kennedy School of Government, Harvard University, August 2000. [2] Ross Baldick, Variation of distribution factors with loading, IEEE Transactions on Power Systems, 18(4), November 2003. [3] Santiago Grijalva, Complex Flow-based non-linear ATC Screening, Ph.D. thesis, University of Illinois, Urbana, July 2002. [4] Minghai Liu and George Gross, Effectiveness of the distribution factor approximations used in congestion modeling, in Proceedings of the 14th Power Systems Computation Conference, Seville, 2428 June 2002, 2002. [5] Federal Energy Regulatory Commission, Form No. 715 Overview, http://www.ferc.gov/electric/f715/f715overview.htm. Accessed August 3, 2003. [6] Hung-po Chao and Stephen Peck, A market mechanism for electric power transmission, Journal of Regulatory Economics, vol. 10, no. 1, pp. 2559, July 1996. [7] Richard P. ONeill, Benjamin F. Hobbs, Jr. William R. Stewart, and Michael H. Rothkopf, The joint energy and transmission rights auction: A general framework for RTO market designs, Unpublished Manuscript, 2001. [8] Minghai Liu and Georgre Gross, Framework for the Design and Analysis of Congestion Revenue Rights, To appear in IEEE Transactions on Power Systems, 2004. [9] Thomas J. Overbye, Xu Cheng, and Yan Sun, A Comparison of the AC and DC Power Flow Models for LMP Calculations, Proceedings of the 37th Hawaii International Conference on System Sciences, January 2004, Big Island, Hawaii. [10] Martin L. Baughman and Fred C. Schweppe, Contingency evaluation: Real power flows from a linear model, Presented at the IEEE PES Summer Meeting, paper CP 689-PWR, 1970. [11] P. W. Sauer, On the formulation of power distribution factors for linear load flow methods, IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 2, pp. 764770, February 1981. [12] Wai Y. Ng, Generalized generation distribution factors for power system security evaluation, IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 3, pp. 10011005, March 1981.

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[13] Allen J. Wood and Bruce F. Wollenberg, Power Generation, Operation, and Control, Wiley, New York, second edition, 1996. [14] Peter W. Sauer, Karl E. Reinhard, and Thomas J. Overbye, Extended factors for linear contingency analysis, in Proceedings of the 34th Hawaii International Conference on System Sciences, 2001, pp. 697703. [15] PowerWorld Corporation, Features and Benefits, http://www.powerworld.com/features.html. Accessed August 3, 2003. [16] Lawrence T. Pillage, Ronald A. Rohrer, and Chandramouli Visweswariah, Electronic Circuit and System Simulation Methods, McGraw-Hill, Inc., New York, 1995. [17] Electric Reliability Council of Texas, Zonal CSC TTC, http://tcr.ercot.com/documents/P39_D235.xls. Accessed August 3, 2003. [18] Ross Baldick, Shift factors in ERCOT Congestion Pricing, www.ksg.harvard.edu/hepg/Papers/baldick_shift.factors.ercot.cong_3-5-03.pdf. Accessed August 3, 2003.

Ross Baldick is a Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin. He received his M.S and Ph.D. in Electrical Engineering and Computer Sciences from the University of California, Berkeley.

Krishnan Dixit is a graduate student in the Department of Electrical and Computer Engineering at The University of Texas at Austin

Thomas J. Overbye (S87-M92-SM96) received the B.S., M.S. and Ph.D. degrees in electrical engineering from the University of Wisconsin-Madison. He is currently a Professor of electrical and computer engineering at the University of Illinois at Urbana-Champaign. He was with Madison Gas and Electric Company, Madison, WI, from 1983-1991. His research interests include power system operations and visualization.