Information about this Project
Project information can be obtained at http://www.pserc.cornell.edu/tcc/info.html or by contacting:
Ian Dobson  Rajesh Rajaraman 
Professor  Senior Engineer 
Electrical & Computer Engineering  Laurits R. Christensen Associates 
University of WisconsinMadison  4610 University Avenue, Suite 700 
1415 Engineering Drive  Madison WI 53705 
Madison WI 53706 USA  phone : (608) 2311398 ext 191 
phone: 6082622661  email rajesh@lrca.com 
fax: 6082621267  
email: dobson@engr.wisc.edu 
Additional Copies of the Report
This report appears electronically at http://www.pserc.cornell.edu/tcc/ and is also available from the Power Systems Engineering Research Center website, http://www.pserc.wisc.edu. The PSERC publication number is 0134. For additional information, contact:
Notice Concerning Copyright Material
Permission to copy without fee all or part of this publication is granted if appropriate attribution is given to this document as the source material.
© 2001 Board of Regents of the University of Wisconsin. All rights reserved.
This report was produced by collaboration between the Power Systems Engineering Research Center and Laurits R. Christensen Associates as part of a project funded by the National Science Foundation Research Centers  Small Firms Collaborative R&D Initiative under NSF Award numbers 9815325 and 9815678. We gratefully acknowledge this support from the National Science Foundation. We also thank the Power Systems Engineering Research Center for its support in making this report available.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Transfer of bulk electrical power over long distances is routine in North America in order to have a reliable and economical electrical supply. For example, hydroelectric power generated in Canada can be transferred to consumers and industry in Los Angeles using the high voltage transmission system. But the transmission system has a limited capability to transfer power. The maximum power that can be transferred is called the transfer capability. To operate the power system safely and to gain the benefits of the bulk power transfers, the transfer capabilities must be calculated and the power system planned and operated so that the power transfers do not exceed the transfer capability.
The purpose of this document is to explain concepts and calculations of transfer capability and describe applications of transfer capability. The document aims to give a tutorial introduction to some standard transfer capability concepts and introduce some new methods in transfer capability sensitivity and accounting for uncertainty.
Some highlights of the document are:
Long distance bulk power transfers are essential for an economic and secure supply of electric power in North America.
Power system transfer capability indicates how much interarea power transfers can be increased without compromising system security. Accurate identification of this capability provides vital information for both planning and operation of the bulk power market. (Planners need to know the system bottlenecks and system operators must not implement transfers which exceed the calculated transfer capability.) Repeated estimates of transfer capabilities are needed to ensure that the combined effects of power transfers do not cause an undue risk of system overloads, equipment damage, or blackouts. However, an overly conservative estimate of transfer capability unnecessarily limits the power transfers and is a costly and inefficient use of the network. Power transfers are increasing both in amount and in variety as deregulation proceeds. Indeed, such power transfers are necessary for a competitive market for electric power. There is a very strong economic incentive to improve the accuracy and effectiveness of transfer capability computations for use by system operators, planners and power marketers.
The practical computations of transfer capability are evolving. The computations presently being implemented are usually oversimplified and in many cases do not take sufficient account of effects such as interactions between power transfers, loop flows, nonlinearities, operating policies, and voltage collapse blackouts. The goal of the methods described here is to improve the accuracy and realism of transfer capability computations. The power system must be operated with some conservatism to account for the effects of uncertainty in power system data. This uncertainty can be analyzed and quantified to provide a defensible basis for the conservatism.
The limitations on power system performance that we consider in this document are transmission line flow limits, voltage magnitudes and voltage collapse. All these limits can be handled in an AC load flow power system model. We do not address limits due to transient stability or oscillations; in our framework, these limits have to be crudely approximated by flow limits.
One way to explain transfer capability is to work topdown from the purpose of transfer capability by discussing definitions and meanings of various components of transfer capability and then explaining the various methods for calculating each component. This useful approach is followed, for example, by NERC in defining transfer capability [89], [90]. In this tutorial, we take an alternative approach of explaining how to calculate transfer capability in a simplified case and then generalizing the calculation. We think that a focus on what one typically calculates tends to reveal the meanings and limitations of transfer capability concepts.
This section introduces and explains transfer capability in a simplified context. The transmission network is assumed to be fixed so that, if any contingencies such as a line outage are considered, they are incorporated into the network and do not change during the discussion. Only one transfer capability calculation is explained; in practice the calculation is repeated with a range of assumptions. The modeling is assumed to be deterministic; probabilistic effects are neglected. A more detailed explanation of transfer capability is in Chapter 2.
There are many assumptions or choices made during a transfer capability calculation which can greatly influence the answer. In addition to choices made in modeling the power system, the base case, the transfer itself, and the limiting case must all be chosen. These choices are now described.
Base Case: The base case is an assumed power system operating condition to which the transfer is applied. That is, the base case is the assumed power system operating condition when no power has been transferred. A power system operating condition is generally obtained by specifying the powers generated or consumed at each bus and the control settings and then solving power system equations to calculate the other power system quantities such as power flows on the transmission lines. The base case is assumed to be an operating condition solved in this way and also a secure condition so that all quantities such as line flows and bus voltage magnitudes lie within their operating limits.
Specifying the transfer: A transfer is specified by changes in power injections at buses in the network. For example, a point to point transfer from bus A to bus B is specified by increasing power at bus A and reducing power at bus B. In particular, if 100 MW are to be transferred from A to B, then power at bus B is reduced by 100 MW and power at bus A is increased by 100 MW plus an amount to cover the change in losses. In this case bus A is called a source of power and bus B is called a sink of power.
Limiting Case: A solved transfer limited case is established at which the transfer has been increased to such a value that there is a binding security limit. The binding security limit can be a limit on line flow, voltage magnitude, voltage collapse or other operating constraint. Further transfer in the specified direction would cause a violation of the binding limit and compromise system security.
Summary and transfer margin: Now we can summarize the choices made in the calculation and define the transfer margin:
One way to compute transfer capability with a software model is called continuation. First, the base case is solved. The amount of the transfer is a scalar parameter which can be varied in the model. (Note that the transfer must be properly defined in terms of bus power injections to implement the amount of transfer as a scalar parameter in the model.) Then the amount of transfer is gradually increased from the base case until a binding limit is encountered. This continuation process requires a series of power system solutions to be solved and tested for limits. The transfer capability is then the change in the amount of transfer from the base case transfer.
Continuation can be simply done as a series of load flow calculations for increasing amounts of transfers or by more elaborate high performance numerical methods. Some continuation software can accurately take account of power system nonlinearity, operator actions, controls such as tap changes, and generator limits as the transfer is increased. On the other hand, useful transfer capabilities can also be obtained with simpler power system models such as the DC load flow approximation.
An interactive transfer capability calculator is available on the web at
This section assumes that the reader is viewing the calculator on the web.
The calculator allows the power system, the base case, and the transfer to be changed. Many of the transfer capability terms used in the calculator are explained by links on the calculator page.
It is useful to know the transfer capability when changes are made in the base case parameters. The changes in the base case parameters of interest can represent variation in simultaneous transfers, assumed data, and system controls.
One way to investigate this is simply to change the base case parameter and rerun the calculator to get the new transfer capability. The disadvantage of this approach is the considerable computation required every time the AC continuation load flow is executed. If, as common in power systems, there are many parameter changes to be considered, then this approach becomes computationally burdensome or impractical. (On the small, 6 bus system, the calculator is quite quick and this problem is less apparent. Running the calculator on the 3357 bus system gives a better appreciation of this point.)
Quick estimates of the transfer capability when parameters are changed can be evaluated by specifying the desired parameter change with the three popup menus as indicated. The calculator then displays the estimated transfer capability in text and graphics. The parameter that was changed appears as the vertical axis of the graph.
The estimates are quick because they are obtained from formulas for the transfer capability sensitivity (the transfer capability is NOT calculated again from scratch with the parameter changed). These formulas essentially specify the slope of the gray line in the plot and then the gray line is used to estimate the transfer capability. This procedure is a linear approximation which is accurate for very small parameter changes and less accurate for large parameter changes.
The quick estimates are derived from transfer capability sensitivity formulas. Chapter 3 shows how to compute these sensitivities very quickly for a wide range of parameters. The calculator illustrates the use of these sensitivities to estimate the effect on the transfer capability of changes in parameters.
To find out how accurate these quick estimates are for the selected parameter change, use the button to "verify the estimated transfer capability". The calculator will then rerun the transfer capability from scratch with the selected parameter change to obtain an "actual" transfer capability which can then be compared to the "estimated" transfer capability.
The following subsections explain some of the concepts that can be demonstrated with the calculator.
This apparent but important fact can be easily demonstrated and quantified with the calculator by varying the base case. Redispatch of generation or outage of a transmission line can have a particularly marked effect on the transfer capability.
The transfer capability calculation assumes a base case in which other power transfers are happening. It is of interest how the value of these other transfers affects the calculated transfer capability. In particular, the calculator offers a choice of redispatches from a list that are equivalent to transfers and it is of interest to vary these redispatches to determine their impact on the selected transfer capability.
Consider the lossless case for simplicity. Then a transfer of 10 MW from bus 1 to bus 2 increases generation at bus 1 by 10 MW and decreases generation at bus 2 by 10 MW whereas a redispatch of 10 MW from bus 1 to bus 2 decreases generation at bus 1 by 10 MW and increases generation at bus 2 by 10 MW. Redispatch of power has opposite sign to a transfer of power: a redispatch of power from bus 1 to bus 2 is equivalent to a transfer of power from bus 2 to bus 1.
Changes in transfer capability due to redispatch may also be quickly estimated from a solved limiting case using the calculator.
The 6bus system is a simple power system network loosely based on tutorial cases from the NERC Web site and the text [94]. The 6bus system is intended to illustrate in a simple context notions of transfer capability and the impact that various actions have on the given transfer capability.
Buses 1, 2, 3 are generators and buses 4, 5, 6 are loads. The general flow of power is, of course, from the generators to the loads but there is also a tendency for power to flow mainly from the lower numbered buses to higher numbered buses. Reactive demand by the loads is large. The network has 11 branches and each branch represents a transmission line.
Now we consider the transfer from generator bus 2 to generator bus 3, (that is, generator 2 increases its supply and generator 3 decreases its supply). The calculator is run with this transfer selected and all other parameters at their base case values. The transfer capability is 208.7 MW and the limiting event is the flow limit joining bus 2 to bus 3.
Now we study the impact of changing parameters on the transfer capability of 208.7 MW.
The 39bus system is a standard system for testing new methods. It represents a greatly reduced model of the power system in New England. It has been used by numerous researchers to study both static and dynamic problems in power systems.
The 39bus system has 10 generators, 19 loads, 36 transmission lines and 12 transformers. The 39bus system is organized into three areas. (Area 3 contains two portions of the network which are not directly connected.)
The following are some comments and observations about 39bus system cases that the reader can verify by running the calculator:
The NYISO system is a 3357 bus model of a portion of the North American eastern interconnect. The model contains a detailed representation of the network operated by the New York independent system operator and an equivalent representation of more distant portions of the network.
The calculator includes the NYISO system to demonstrate that the continuation and sensitivity methods apply to systems of a more practical size.
The NYISO base case is intended to be an illustrative example of a severely stressed power system. The base case is produced by artificially stressing the NYISO system near to its operating limits. The base case is motivated by a scenario identified as problematic in the New York Power Pool summer 1999 operating study and includes two 345 kV lines out of service.
This case is derived from the general data available within FERC 715 filings of the New York ISO for the year 1999. It does not necessarily represent actual operating conditions and should not be relied upon to draw conclusions about the system itself. It is intended only to illustrate the performance and capabilities of the method for estimation of transfer capability and the impact of changes in conditions to changes in transfer capability. In particular, no tap adjustments or shunt reactor adjustments are performed as part of these simulations.
Transfer capabilities are fundamental to appropriate operation of the system. In the past it has been considered that transfer capabilities calculations using AC load flow models are difficult because they require the use of nonlinear tools such as continuation power flows. The calculator demonstrates the use of AC continuation load flow to accurately compute transfer capabilities. Moreover, the calculator demonstrates that the impact on transfer limits of system controls, data and other transactions can be quickly estimated without rerunning the continuation power flow. There are many applications for these quick estimates or the sensitivities used to generate them in the rest of this document.
The Web site allows users to readily experiment with transfer capability calculations and develop intuition as to the effect of data, controls and other transfers. Hopefully, hard to explain concepts such as the impact of voltages on flow limits, or the impact of reactive power injections on remote voltage limits, or even the reasons why 1 MVAR can be worth several MW of transfer capability under some conditions will become clearer to users of this site.
A DC load flow model represents real power flows and injections and voltage angles. The limits considered are line flow limits (sometimes other system limits are approximated as equivalent line flow limits). The DC load flow is a linear model; the line flows are a linear function of the powers injected (the DC load flow equations are obtained by linearizing the power flow equations about an operating point). The DC load flow model is lossless and changes in bus injections must always sum to zero.
Implementing the transfer capability computation of section 1.3 simplifies in some ways when the DC load flow approximation is made. The security of the base case can be easily checked by ensuring that all line flows are within their limits. Since the model is linear, the behavior of the line flows as the transfer increases can be linearly extrapolated to find the limiting line flow and the "continuation" process is much easier. Indeed this calculation can be done in terms of generation shift distribution factors as illustrated below.
Consider the DC load flow approximation of the 6 bus example from the text by Wood and Wollenberg [94]. (This 6 bus example has different parameters than the 6 bus example used in the calculator.) The 6 bus example has 3 generators and any transfer can be specified by the 3 power injections at each of the 3 generators. However, since these 3 power injections must sum to zero (the DC load flow model is lossless), any transfer can be specified by the power injections at the generators at buses 2 and 3. For example, a transfer of 10 MW from bus 2 to bus 3 is specified by (10,10) and a transfer of 5 MW from bus 1 to bus 2 is specified by (5,0)
The complete transfer capability information can be evaluated from the generation shift factors and the margin remaining on each line. These data are shown in Table 1.1 (c.f. Fig. 11.8 in [94]).
line  gen. shift factor  margin remaining  
bus 2  bus 3  (MW)  
12  0.47  0.40  10 
14  0.31  0.29  10 
15  0.21  0.30  10 
23  0.05  0.34  10 
24  0.31  0.22  10 
25  0.10  0.03  10 
26  0.06  0.24  10 
35  0.06  0.29  10 
36  0.01  0.37  10 
45  0.00  0.08  10 
56  0.06  0.13  10 
The meaning of the generation shift factor of 0.47 for bus 2 and line 12 is that an injection of 1 MW at bus 2 and 1 MW at bus 1 will decrease the flow on line 12 by 0.47 MW. Similarly, the generation shift factor of 0.22 for bus 3 and line 24 implies that an injection of 1 MW at bus 3 and 1 MW at bus 1 will increase the flow on line 24 by 0.22 MW. For simplicity, the margins remaining on each line are all chosen to be 10 MW.
Let us compute a transfer capability using Table 1.1. We specify the first transfer as being from bus 1 to bus 2. If the amount of transfer is T MW, then it is specified by injections (T,0) at buses 2 and 3. Since in this example, all lines have equal margin remaining, the line that will reach its limit first is the line with the largest shift factor for bus 2, which is line 24. The transfer capability is given by 0.31 T = 10, or T = 32.3 MW.
For another example, specify a second transfer as being from bus 2 to bus 3. If the amount of transfer is T MW, then it is specified by injections (T,T) at buses 2 and 3. If T MW are transferred, then the changes in the flow on a given line are T (gen shift factor bus 3  gen shift factor bus 2). Since in this example, all lines have equal margin remaining, the line that will reach its limit first is the line with the largest difference between the generation shift factor for bus 3 and bus 2, which is line 36. The transfer capability is given by (0.37 (0.01)) T = 0.38 T = 10, or T = 26.3 MW.
The transfer capabilities computed so far assume that only one of the transfers was implemented. If the two transfers are done at the same time or one after another, then they definitely do interact. For example, if both transfers occur simultaneously and at the same rate, then if T is the amount transferred in one of the transfers, then the injections at buses 2 and 3 are (0,T). (The simultaneous transfer is equivalent to a transfer from bus 1 to bus 3.) Then the limiting line is the line that will reach its limit first is the line with the largest shift factor for bus 3, which is line 36. The transfer capability is then given by 0.37 T = 10, or T = 27.0 MW.
For further description of generator shift factors and their relation to power transfer distribution factors see section 2.7.
Generally speaking, the term "transfer capability" refers to the amount of electric power that can be passed through a transmission network from one place to another. The concept of transfer capability is useful for several reasons.
A system which can accommodate large interarea transfers is generally more robust and flexible than a system with limited ability to accommodate interarea transfers. Thus, transfer capability can be used as a rough indicator of relative system security.
Transfer capability is also useful for comparing the relative merits of planned transmission improvements. A transmission expansion that increases transfer capability between two areas of the grid might be more beneficial for increasing both reliability and economic efficiency than an alternate improvement that provides a lesser increase in transfer capability.
Along similar lines, transfer capability can be used as a surrogate for more specific circuit modeling to capture the gross effects of multiarea commerce and provide an indication of the amount of inexpensive power likely to be available to generation deficient or highcost regions.
Transfer capability computations facilitate energy markets by providing a quantitative basis for assessing transmission reservations.
Transfer capability computations play a role in both the planning and operation of the power system with regard to system security.
One benefit of interconnected power systems is the potential for increased reliability. In an interconnected system, the loss of generation in one area can be replaced by generation from other areas. Thus, several systems interconnected can survive contingencies that the individual systems could not. Transfer capability computations are useful for evaluating the ability of the interconnected system to remain secure following generation and transmission outages.
Determining the adequacy of the transmission system in allowing external generation to replace internal generation is a typical application for transfer capability computations.
For this purpose, a model of the network reflecting the anticipated conditions is assumed. Several generators within one area are selected as sinks. The power injected to the network at these locations is systematically reduced or eliminated to reflect the planned or unplanned loss of the units. For each generation outage scenario, several external generators are selected as potential sources. The choice of sources and the participation of each source depends upon the assumption concerning the time frame of the response.
The purpose of the transfer capability computation is to determine the quantity of lost generation that can be replaced by the potential reserves and the limiting constraints in each circumstance. In addition to varying the assumptions regarding the generation sources and sinks to reflect different outages and reserve locations, the computations are often repeated assuming different loading conditions or increasing loads and coincident branch element outages.
The previous example illustrates a typical planning or operational study. A very similar application of transfer capability computations is useful for near real time security analysis.
During the course of any day, events or circumstances are encountered that require operators to determine prevention or mitigation actions, or verify that the existing operating guides will perform effectively. Consider a situation in which large transfer across a system is observed during exceptional weather. Although the circumstances are within what might have been considered in planning or had been previously observed, the operators have specific information concerning the actual system state and must anticipate the future state of the system in the coming hours. The operators would like to know how much additional transfer the system could maintain before security is compromised  the security margin.
Based on the energy schedules or typical patterns, the transfer capability for a feasible transfer can be determined. In this situation, the transfer capability represents a security margin. The distance between the present state and a state violating a security criteria is the amount of the transfer that initiates a security violation.
The ability of the model to reflect the actual system depends in part on obtaining reasonable estimates of the capacities of the "sticks" as well as the proper distribution factors for each zone to zone transfer. Both these quantities are reasonably obtained from transfer capability computations.
A first approximation for the capacity of the sticks would be the sum of the ratings of the transmission elements that make up a zone to zone interface. However, since constraints away from the interface can limit interface flows before any of the interface elements are fully loaded, a method that also considers limitation due to noninterface elements is preferred.
By performing a series of transfer capability computations for many combinations of zone to zone transfers, both the distribution factors and capacities can be reasonably approximated. Starting from a reasonable base case with a full network model, one zone is selected as a reference zone. The transfer capability between that zone and every other zone is computed. The distribution factors are obtained from comparison of the base interface flows to those corresponding to a given transfer level. By identifying the transfer capability for many zone to zone transfers, limiting elements that are not interface elements can be identified and incorporated as "sticks" in the ball and stick model. The fundamental step in building the model is the transfer capability computation.
The preceding examples described the use of transfer capability for modeling market behavior. A similar application of transfer capability computations influences market operation.
Bilateral and pooled markets are fundamentally different. However, there is some overlap since pooled markets accommodate bilateral trading and the proposed RTOs designed to facilitate pure bilateral markets also plan to offer imbalance services that resemble the location based marginal pricing characteristic of pooled markets. The following examples illustrate uses of transfer capability in the operation of electricity markets.
Energy contracts can span hours or months. Thus, the quantities of transmission available for reservation must be suitable for the range of conditions that can be encountered over those time frames. A portion of the computed total transfer capability (TTC) is reserved and available only on a nonfirm basis.
The transfer capability across a transmission system is often used as a basis to determine the quantity of firm transmission service available to schedule energy delivery. For instance, transmission service might be requested a year in advance for a duration of several months. On yet a shorter time frame, the transfer capability across a system is needed to determine the quantity of nonfirm transmission service available for an hour or so starting in the next hour or the next day or week. Transfer capability computations are thus integral to the determination of TTC, ATC, and the quantities of transmission reserved for generation and transmission contingencies.
In pooled markets such as PJM, suppliers offer quantities and prices to a pool and load serving entities (LSEs) submit schedules for demand. A central authority determines prices and dispatch based on a constrained optimization program so that all load is served and all network security constraints are observed.
In time intervals when no network constraints actively affect the dispatch, the prices everywhere in the network are nearly the same, subject to differences only due to losses. However, when network constraints are active, such as in the case of a transmission line being fully loaded, prices everywhere can vary significantly depending upon how each location can influence the constraint. Also, by a property of the optimization problem governing the dispatch, during congested conditions the central authority receives more payment from demands than it pays to generators. This quantity of funds is called congestion rent.
Market participants exposed to the variation in prices caused by the discrete nature of constraints desired a mechanism to provide price consistency and hedge against the effects of congestion.
A solution involved the distribution of financial contracts that allowed the holder to receive funds in proportion to the price differences between locations in every hour. These contracts are referred to as "FTRs" for Firm Transmission Rights or Financial Transmission Rights or TCCs, for Transmission Congestion Contracts. Essentially, the congestion rents received by the central authority are distributed to the holders of FTRs.
An interesting problem arises. How can the central authority be sure that they will not sell FTRs that require payments exceeding the congestion rents? The answer [47] involves an optimization program called the simultaneous feasibility test. Each participant wishing to purchase FTRs submits bids identifying a price they are willing to pay for an FTR and the quantity in MW of the FTR and the source and sink locations for which it applies. Note that specification of an FTR abstractly resembles a specification for a physical transfer with the addition of a price per MW. An optimization program maximizes the revenue from the auction subject to the constraint that the total of all FTRs awarded, if implemented as bilateral transfers on the network model, would not violate any network constraints. In other words, the financial contracts are modeled as physical transfers to determine a feasible set of contracts. This is the exact same problem one would solve to achieve an optimum curtailment of physical bilateral transfers to maintain network security.
In these examples, the range of conditions that must be accounted for are substantial and different for each application. The transfer capability can change for different assumptions about demand, the location of transfer sources and sinks, configuration of the network, seasonal and daily changes in facility ratings, variation in dispatch in neighboring regions, and many other factors.
The need to quantify transmission capability requires computations and assumptions. Discussion of the computation is nearly inseparable from discussion of assumptions.
For example, the selection of sources and sinks between the same two areas will be different for a transfer that models the use of reserves to replace a forced generator outage than for a transfer that models the economic activity of market participants. The purpose of the transfer capability analysis determines the assumptions and approximations used for the computations. The challenge and limitation of transfer capability computations is selecting appropriate assumptions.
The determination of transfer capability for every application typically requires that the transfer capability for a fixed, specific set of assumptions be computed, and then recomputed for some prescribed changes in those assumptions. The purpose of changing the assumptions is to determine the transfer capability most appropriate for the application at hand.
For example, one might desire knowing the most limiting contingency for a particular area to area transfer. In this instance, one would compute the minimum over a set of contingencies of the maximum transfers. Alternatively, one might be interested in the maximum area to area transfer under a fixed network, but varying the point of delivery and point of receipt locations, such as in evaluating the effect the location of reserves has on system security.
In general, analysis of longer and future time frames requires that a greater variety of conditions and assumptions be considered, such as changing load patterns, generator commitments, and network configurations. Analysis of short time frames in near real time has the luxury of more limited conditions but the curse of greater urgency and required accuracy. Computation of transfer capability for one hour ahead might require only analysis of varying branch contingencies subject to a limited set of source and sink assumptions. However, computation of transfer capability over a one week period may require analysis for many different load patterns, generation commitments, and source and sink assumptions. Calculations of Available Transfer Capability (ATC), Capacity Benefit Margin (CBM), and Transfer Reliability Margin (TRM) typically require that the transfer margin computation be repeated for multiple combinations of transfer directions, base case conditions, and contingencies [90], [75].
The basic process involves these steps::
There are many ways of implementing this process. Some of the assumptions that must be specified include:
The transfer margin computation can be implemented with a range of power system models and computational techniques. One convenient and standard practice is to use a DC power flow model to establish transfer capability limited by thermal limits. The limiting cases are then checked with further AC load flow analysis to detect possibly more limiting voltage constraints.
Alternatively, a detailed AC power system model can be used throughout and the transfer margin determined by successive AC load flow calculations [38] or continuation methods [14,22,3,92]. A related approach [e.g., EPRI's TRACE] uses an optimal power flow where the optimization adjusts controls such as tap and switching variables to maximize the specified transfer subject to the power flow equilibrium and limit constraints. The formulations in [38] and [95] show the close connection between optimization and continuation or successive load flow computation for transfer capability determination.
Methods based on AC power system models are slower than methods using DC load flow models but do allow for consideration of additional system limits and more accurate accounting of the operation guides and control actions that accompany the increasing transfers. Under highly stressed conditions the effects of tap changing, capacitor switching, and generator reactive power limits become significant. A combination of DC and AC methods may be needed to achieve the correct tradeoff between speed and accuracy. The methods in this tutorial account directly for any limits which can be deduced from equilibrium equations such as DC or AC load flow equations or enhanced AC equilibrium models.
For the purposes of understanding the many methods of transfer capability, it is beneficial to consider the simple case of computing the transfer capability for a situation with a limited set of variable assumptions.
A single transfer capability computation yields:
Typically, we think of the base case and the transfer specification as inputs to the process and the identification of the limiting case and transfer capability as the outputs. However, circumstances exist for which the transfer specification is also an output. The four components of the computation are now explained in greater detail.
Base Case: A time horizon for the transfer capability is selected and a base case consistent with the time horizon is selected. The time horizon refers to the period in the future for which the transfer capability must be found. For example, to schedule firm transmission service to serve native base load, one is commonly interested in knowing a particular transfer capability for both peak and nonpeak periods over an entire month, often three to six months or more in the future.
Identification of a suitable base case is a formidable and important task. In the case of very short time frame analysis, an example of a suitable base case would be the most recent case available from an EMS state estimator updated to include load forecast and schedules for the period of question. Even in this case, several base cases may need to be developed to reflect possible contingencies. Selection of base cases for longer term analysis can require extensive studies. However, such base cases are normally useful for many planning purposes.
Regardless of the time horizon in question, several important considerations are inherent in selection of one or more base cases. The base case assumes a particular set of facility ratings, demands, unit commitment and generator dispatch and thus implies loading on all facilities. Changes in any of these assumptions lead to a different transfer capability.
For the purposes of this example, we assume that a suitable base case has been identified. The base case is assumed to be a secure and solved case.
Specifying the transfer: A transfer is specified by changes in power injections at buses in the network. For example, a point to point transfer from generator A to generator B is specified by increasing power at generator A, reducing power at generator B and making some assumption about where additional power will be generated to cover the losses associated with the transfer. In particular, if 100 MW are to be transferred from A to B, then power at generator B is reduced by 100 MW and power at generator A is increased by 100 MW plus an amount to cover the change in losses. An alternative assumption is that the power at generator B is reduced by 100 MW, power at generator A is increased by 100 MW, and generation at other locations supplies the additional power required by losses. Another way to think about transfers is in terms of changes in exports from areas or between areas in the power system. Although these changes in area exports can summarize the effect of a transfer, they do not completely define a transfer unless the generation dispatch within the areas is also specified (there are many ways to dispatch generation within an area to change the area exports). That is, the participation of each generator in the area in the transfer has to be specified. A complete specification of changes in area exports and generation dispatch within each area is equivalent to specifying the power injections at all generators.
Limiting Case: A solved transfer limited case is established at which the system transfers have been changed and there is a binding security limit. The binding security limit can be a limit on line flow, voltage magnitude, voltage collapse or other operating constraint. Further transfer in the specified direction would cause the violation of the binding limit and compromise system security.
One way to compute transfer capability with a software model is called continuation. From the solved base case, power flow solutions are sought for increasing amounts of transfer in the specified direction. The quantity of the transfer is a scalar parameter which can be varied in the model. The amount of transfer is gradually increased from the base case until a binding limit is encountered. This continuation process requires a series of power system solutions to be solved and tested for limits. The transfer capability is the change in the amount of transfer from the base case transfer at the limiting point. Continuation can be simply done as a series of load flow calculations for increasing amounts of transfers. However, when convergence could be poor, such as the case for transfers approaching voltage instability, methods that allow the transfer parameter to become a dependent variable of the model are the most successful.
Some continuation software accounts for power system nonlinearity, operator actions, controls such as tap changes, and application of generator limits as the transfer is increased. These programs are most valuable for transfer computations aimed at assessing system security margins or the effects of very near term (next hour) transactions. The appropriate use of these programs depends on the application.
Thousands of simultaneous transactions make up the energy schedule that the system dispatchers observe. The state of the power system evolves from hour to hour more or less by following the schedule, not by implementing sequences of individual transactions. The manner in which a transfer is applied during the computation might have no relation to how the transfer is implemented in the actual system.
Useful estimates of transfer capabilities can be obtained with simpler power system models such as the DC load flow approximation. A DC model may be preferable to an AC model particularly in circumstances where the extra data for an AC model is unavailable or very uncertain, such as the case of very long time frame analysis. The added detail of an AC model is not useful if it contributes greater uncertainty.
The DC approximation is good for these reasons:
The DC approximation is poor for these reasons:
In the transition to a more competitive electric power market place, the past several years have seen considerable interest in tools to determine available transmission capability, to manage congestion when limits are approached or exceeded, and to characterize the marginal value access to capability on a transmission corridor through transmission rights. In the traditional regulated utility environment, the so called Optimal Power Flow (OPF) was the tool of choice for a range of optimization problems relating to real time and near real time operation [62], [63]. However, new objectives in a competitive environment [46], [53], the associated focus on congestion management to facilitate competition [78], and the increasing volatility of dispatch and operating conditions raise significant new challenges in OPF [65], [44].
A number of authors have recognized the possibility of extending OPF tools to the problems of ATC calculation, congestion management, and transmission access valuation. Indeed, the period since 1999 has seen a flurry of activity, as represented by such works as [25], [66], [72], [93], [24], [64], [51], [98], [71]. All of these works share the common theme that they formulate an optimization problem in which the dominant elements are the equality constraints arising from the power flow. To the extent that the term ``optimal power flow" has a single definition, the use of network power flow constraints qualifies these treatments as OPF problems. Variations in these works relate to the nature of the inequality constraints represented, ranging from relatively basic operational equipment limits [71] to more detailed formulations that attempt to approximate transient stability security requirements through tractable algebraic inequalities [72]. The objective functions selected in these works also vary widely, though treatment of a range of objective functions has always been a hallmark of the OPF literature (again, the complexity of OPF problems lies in their high dimension network constraints).
Our goal here is to provide a brief representative example of how an OPF problem may be formulated to identify the available transfer capability associated with a transmission corridor. To this end, it is useful to begin from a simple representative example of a security constrained ATC maximization to allow concrete illustration of the form of objective function and constraints. This formulation will be extended to treat uncertainty in Section 6.9.
Let V represent the complex voltages at each node of the transmission network. These are the basic decision variables of the optimization framework. The key constraints relate to the electrical behavior of the network. Recognizing that the voltage/current behavior is well approximated as linear, the most fundamental fixed parameters of the problem are those that describe the electrical characteristics of the transmission network. In particular, each branch k is parameterized by its complex admittance parameter y_{k}. Note that realistic problems contain on the order of 10^{4} nodes, with an average interconnection density of 3 to 4 branches per node.
For our illustration here, we will assume that a transmission corridor is defined by a set of transmission lines that form a cutset, separating one region of the network from another. Let G denote the set of indices corresponding to transmission line branches forming the transmission corridor. Moreover, without loss of generality, we will assume that we have a specified direction of desired power flow across the corridor, and that all lines have their arbitrary reference directions assigned to agree with this direction of flow (in circuit theoretic terms, this reference direction is simply the convention used for measuring branch current, positive flows agree with the reference direction, while negative flows run counter to it). It will prove convenient to characterize the flow through the transmission corridor in terms of the sum of currents over all the lines comprising the corridor. While one may alternatively select active power flow through the corridor as the quantity to maximize, this choice introduces a significantly greater degree of nonlinearity into the objective function, without a commensurate improvement in the practical usefulness of the result.
Inequality constraints to be satisfied in the formulation below include limits on the magnitude of current flow in each line of the network, upper and lower limits on bus voltage magnitude at each bus, and upper and lower limits on active and reactive power at each bus. For simplicity of notation and algorithmic description, it is convenient to represent active and reactive power equality constraints as inequalities in which upper and lower limits are selected to be equal (or, as is often useful for improved algorithmic performance, nearly equal within a small tolerance commensurate with the desired power flow solution accuracy). Likewise, buses at which voltage magnitude is regulated to a setpoint have upper and lower limits on voltage magnitude set equal to this setpoint.
With this notation, our representative optimization formulation of the ATC calculation may be summarized as:
 (2.1) 






In this formulation, one has the advantage that the objective appears as a purely linear function of the underlying decision variables, which are the complex bus voltage magnitudes. Nonlinearity is introduced by the quadratic dependence of complex bus powers, S , on these complex bus voltages. The advantages of treating the complex bus voltage quantities in rectangular coordinates, in order to exploit the resulting quadratic form of the power flow constraints, has been described in many previous works; for example, see [37] and [9].
The use of linear approximations for updating and estimating transfer capability is widespread. This section introduces common terminology and the fundamental concepts.
Power Transfer Distribution Factors, commonly referred to as PTDFs, express the percentage of a power transfer that flows on a transmission facility. For example, if the component corresponding to the Z transformer of the PTDF for a Bus A to Bus B transfer was 0.5, then a transfer from A to B of 400 MW would result in an increase flow of 200 MW on the Z transformer. The transmission facilities, commonly referred to as flowgates, can be transformers, lines, or sets of transformers and lines. PTDFs are most useful for estimating the change in flows that result from a particular transfer and identifying which flowgates are most affected by that transfer. The sources and sinks for the power transfer must be specified for the PTDFs to be computed. In addition to specification of sources and sinks and definition of flowgates, the computation of PTDFs also depends upon the model of the power system selected. However, most commonly PTDFs are computed with the DC model assumptions.
Shift Factors, commonly referred to as GSFs (Generator Shift Factors) or Adjustment Factors, express the change in flow on a particular flowgate that results from increasing generation at a node. GSFs are meaningful only when considered in sourcesink pairs, since power injected at one location must be matched by power removed at another location. When the sink is not identified for a GSF, it can be assumed that the system slack bus has been used as the sink. GSFs are most useful for identifying which generator pairs can influence a particular flowgate. Assumptions that affect the computation and application of PTDFs also affect GSFs. An example of GSF in a DC load flow model is given in section 1.5.
GSFs are very closely related to PTDFs. For example, consider a matrix where each row corresponds to a flowgate and each column corresponds to a generator. Let each element of the matrix represent the GSF for the flowgate corresponding to that row and the generator source corresponding to that column, and a constant sink. Then each row represents the vector of GSFs for one flowgate assuming the constant sink, and each column of the matrix is the vector of PTDFs for all flowgates for the transfer between one generator and the system sink. The distinction between GSFs and PTDFs mostly concerns how one cares to view the same data. For the case of a lossless model, the PTDFs for all possible transactions can be obtained from the matrix of GSFs computed with a constant sink.
NERC provides both PTDF and GSF web based viewers and downloadable data files of PTDFs. The NERC PTDFs are used in determining which transactions are subject to curtailment when NERC Transmission Lineloading Relief (TLR) procedures are invoked on a constrained flowgate. The PTDFs and GSFs that are available from NERC are computed using the lossless, DC load flow model and some other simplifying assumptions. A new tool called the Flowgate Information Study Tool (FIST) will appear in 2001.
A variety of applications in both planning and operations require the repetitive computation of transfer capabilities. Transfer capabilities must be quickly computed for various assumptions representing possible future system conditions and then recomputed as assumptions and system conditions change. There is also uncertainty in each transfer capability calculation due to uncertainty in the assumptions and data used. Thus it is useful once a transfer capability has been computed to be able to compute the sensitivity of that transfer capability to data. In practice these sensitivities can be computed very quickly for a wide range of parameters. The sensitivities can be used to estimate the effect on the transfer capability of variation in simultaneous transfers, assumed data, and system controls.
Although sensitivity is a single concept, like most generally useful concepts, it can be thought of and applied in different ways. Sensitivity is a widely known and very widely used concept, but since it is a key concept underlying much of the material in this document, this section gives a tutorial explanation of sensitivity in several ways.
To be specific in the wording of the explanations, we explain the sensitivity of a transfer capability T with respect to a parameter p. In practice, p could be any parameter, including the amount of a real power at a load, the amount of a simultaneous transfer, a control setting, or a value of line impedance. However, for specificity in the wording of the explanations, we assume that p is the real power consumed at load bus 4.
Suppose that at the base case, the real power p consumed at load bus 4 is 70 MW and that the transfer capability T at this base case is 133 MW. We are interested in how much T varies from 133 MW when the bus 4 load p is changed from 70 MW. We can write DT = T133 and Dp = p70, so that we will be interested in how much DT varies when Dp is changed.
T is a function of p. The sensitivity of T with respect to p is the derivative of T with respect to p:
 (3.1) 
 (3.2) 
 (3.3) 
 (3.4) 
 (3.5) 
Indeed, one of the most useful approaches to sensitivity is as follows: We know that T is a nonlinear function of p. For p near 70 MW we approximate this nonlinear function by the linear function (3.5). The sensitivity is the coefficient 0.4 of p in (3.5) or the coefficient 0.4 of Dp in (3.4).
If we instead work near another value of p such as p = 60 MW, the sensitivity coefficient will change in value.
The illustrative numbers used above can be checked for the transfer from bus 1 to bus 5 at the base case of the 6 bus system on the calculator available at
Simply press CALCULATE and then compute the quick estimate of the transfer capability with the parameter change of load at bus 4 +10MW. The quick estimate for transfer capability evaluates and uses the sensitivity for the computation. In particular, the quick estimate calculates the sensitivity 0.4 of the transfer capability with respect to the load at bus 4 and then estimates the transfer capability T when the load at bus 4 is 80=70+10 MW using (3.5) to get
 (3.6) 
Moreover, the calculator shows the sensitivity graphically. In particular, the sensitivity determines the slope of the gray line (the slope of the gray line is 1/sensitivity, or 1/(0.4) = 2.5 in our illustrative numbers.)
The operation of the calculator should be thought of as follows: The AC continuation power flow is run to find the transfer capability for the selected base case. Then the quick estimate calculates the sensitivity of this transfer capability to the chosen parameter. The chosen parameter determines the vertical axis of the graph and the sensitivity determines the slope of the gray line passing through the calculated transfer capability. The gray line is a linear model for changes in the transfer capability when the chosen parameter is varied. The calculator then uses the gray line to estimate the transfer capability for the desired parameter change according to (3.5).
Suppose that a linearized (DC load flow) model for the entire power system is chosen. In this special case, the relation between the transfer capability and the load at bus 4 is linear and the equations (3.2) and (3.3) are exact. Suppose that the event limiting transfer capability was the maximum flow limit on the transmission line 15. Then for linearized (DC load flow) model, the sensitivity of the transfer capability with respect to the load at bus 4 is the negative of the generation shift factor for bus 4 and line 15. (Suppose we increase the load at bus 4 by 1 MW. Then the flow on line 15 increases by an amount given by the generation shift factor for bus 4 and line 15. Since this amount of increase in flow on line 15 will decrease the transfer capability by the same amount, this amount is also the negative of the sensitivity of the transfer capability with respect to the load at bus 4.)
Since the sensitivities are equal to the negative of the generation shift factors in the case of flow limits in a DC load flow model, we see that transfer capability sensitivities for an AC load flow model generalize the well known generation shift factors to the AC case. The transfer capabilities calculated for an AC loadflow take account of power system nonlinearity and can handle voltage magnitude and voltage collapse limits. Moreover, the AC load flow model has parameters available such as voltage magnitude and reactive power loads which are not accounted for in the DC load flow model. The variations (and hence the sensitivities) of the transfer capability with respect to voltage magnitude and reactive power loads are of interest.
One concern is how a transfer capability T varies if another transfer S is varied from its base case value. The situation can be shown graphically by a curve showing how the transfer capability T varies as the transfer S is varied. The transfer capability T at the base case of transfer S is the length of the horizontal line shown in Figure 3.1. The curve shows conceptually how the two transfers interact. With repetitive calculation many points on the curve can be exactly calculated. However, in practice, when time is short, it is useful to be able to approximate the curve by a tangent line. The tangent line approximation at the base case of transfer S is shown in Figure 3.1.
The reciprocal of the slope of this tangent line approximation is the sensitivity of the transfer capability T to transfer S. That is, the slope of the tangent line approximates how the two transfers interact. The slope of the tangent line in Figure 3.1 is 2; adding 1 MW to transfer S decreases transfer capability T by 0.5 MW. Thus the sensitivity of transfer capability T with respect to transfer S is 0.5.
For two transfers, we obtain an area of feasible transfers which is bounded by curves which are the combinations of the two transfers which correspond to transfer capabilities. Given a point on the boundary curves, we can estimate the nearby boundary using sensitivities to define the tangent lines to the boundary.
This generalizes to many transfers interacting: we obtain a hypervolume of feasible transfers which is bounded by curved hypersurfaces which are combinations of all the transfers which correspond to transfer capabilities. Given a point on the boundary hypersurfaces, we can estimate the nearby boundary using sensitivities to define the tangent hyperplanes to the boundary. An interpretation of the hyperplanes is that they represent tradeoffs or interactions between the power transfers at the limiting case.
This section is in the following paper:
Sensitivity of transfer capability margins with a fast formula
(This paper is also reference [2].)
Successful implementation of electric power deregulation requires the determination of the available transfer capability of a power system. The available transfer capability indicates the amount by which interarea bulk power transfers can be increased without compromising system security.
The value used for available transfer capability affects both system security and the profits made in bulk power transactions. Moreover, market participants can have conflicting interests in a higher or a lower available transfer capability. Thus, under deregulation, there is increasing motivation for defensible calculations of available transfer capability and its components such as transmission reliability margin.
For this section we use the main features of the NERC 1995 and 1996 definitions [89,90]: The power system is judged to be secure for the purpose of interarea transfer if "all facility loadings are within normal ratings and all voltages are within normal limits", the system "remains stable following a disturbance that results in the loss of any single element", the postcontingency system ... has all facility loadings within emergency ratings and all voltages within emergency limits" [74].
The time horizon of the calculation is established and a secure base case is chosen. A base case transfer including existing transmission commitments is chosen. Then a transfer limited case is determined. One method to determine the transfer limited case gradually increases the transfer starting at the base case transfer until the first security violation is encountered. The real power transfer at the first security violation is the total transfer capability.
The available transfer capability is then defined as

The calculation may be repeated for a short list of contingencies and the minimum of these available transfer capabilities is used.
For further summary and explanation of ATC see [74,89,90].
This section gives a brief background on the economics of power markets.
Electricity restructuring was driven in large part by the realization that new technologies were making market competition in electricity generation feasible. However, it was also recognized that the transportation of electricity would be still considered to be a natural monopoly service and continue to be regulated. The market design for electricity markets therefore involved attempting to find a model that would facilitate competition in generation while still keeping transportation services under regulatory supervision. To realize the full benefits of competition in generation services, it was recognized that full nondiscriminatory openaccess to transmission by all market participants was necessary. Thus the market design for efficient electricity markets revolved around finding the best way for the regulated transmission company ("transco'') to give access to generator companies ("gencos'') to achieve the most efficient generator dispatch possible.
Any market design for electricity must take account of the special features of electricity that differentiate it from other commodities.
An alternative to the poolco system, but that theoretically could reach the same efficient solution as the poolco model, is due to [96,20,21]. In this model, whenever flows in transmission lines violate their flow limits, the ISO makes arbitrary curtailments in trades (without necessarily looking at economical curtailments) to resolve this congestion. It can be shown that in a competitive market, the market participants would then trade among themselves to reach the same solution as the poolco solution. The obvious practical problem with this approach is that market participants would need a great deal of real time information about the transmission system to trade among themselves to reach the efficient solution (whereas in the poolco model, the ISO would do these trades based on "buy'' and "sell'' bids). With the present day information technologies this method appears to be impractical for clearing markets in real time.
Yet another alternative to the poolco system is NERC's Transmission Line Loading Relief (TLR) system (and other variants of this approach). This is the most inefficient way of resolving transmission congestion. In this method, a region's security coordinators would simply curtail all bilateral trades to relieve transmission congestion in an ad hoc manner using an administrative formula. The formula would not take any account of the willingness of market participants to pay for shipping electricity across congested transmission paths. For more details about why NERC's TLR procedures are inefficient, how they result in economic waste, and how they could potentially lead to anticompetitive gaming, see Rajaraman and Alvarado (1998) [70].
The success of the PJM model has demonstrated that the poolco approach is not only theoretically  but also practically  superior to administrative methods (such as NERC's TLR protocols) for resolving transmission congestion.
This section shows how the power transfer capability concepts presented in this document can be applied by an ISO running a poolco system.
We briefly digress to explain the role of an ISO in a poolco model. The ISO has the following main functions:
We now give examples of how the power transfer capability concepts explained in this report can be used by the ISO in a poolco setting.
This section shows how one may use the sensitivity formulas in this report to make generator and transmission investments in a market that operates under the poolco rubric.
In the past, the regulated utility company was obliged to undertake generator and transmission investments to meet the growing load forecast; the regulated utility would be guaranteed a reasonable rate of return if their network expansion proposal passed regulatory muster. Under deregulation, the incentives for generator and transmission investments are different. The problem is different for the regulated transco and the unregulated gencos.
The necessary and sufficient condition for a market participant to invest in a new generator at some location is that the return on this investment is higher than on all other alternative investments (at least as perceived by the investor). (If merchant transmission were possible, the same criteria would be used to evaluate transmission investments also.)
A regulated entity such as a transco could invest in new or old (i.e., upgrade) transmission facilities if the net "benefits'' from doing so are positive. The term "benefits'' is deliberately left nebulous here and is dependent on many parameters such as regulatory policies, the transco's performance incentives for making the investment, etc. Alternatively, the concept of transmission congestion contracts (TCC) could be expanded to induce appropriate transmission expansion by unregulated market participants [50]. Hogan shows how marketfriendly mechanisms can be developed to enable market participants (rather than the transco) to bear the costs of market transmission expansion. It must be mentioned though that the problem could be complicated by network externalities and the problem of "freeriding'' (i.e., other market participants could benefit by one market participant's investment decisions without bearing the costs of the investment). Another complication is that transmission investments (and some generator investments) tend to be "lumpy'' because of economies of scale; these can complicate the analysis of these investments.
It is beyond the scope of this document to examine what the regulatory policies ought to be to induce optimal investments for transmission or generation, or to examine what the optimum methods for making these investments should be. We will limit ourselves to showing how the methods of this document can be used, in part, in the evaluation of investment decisions in generation and transmission facilities.
As a practical matter, someone planning to make generator or transmission investments needs to know the answers to at least two important problems. One, given that such investments have a large lifetime (20 to 50 years), and that projecting revenues (and costs) over this timeframe is generally a difficult proposition because of inherent uncertainty involved, how should one evaluate the investments? Two, which locations are the optimal places for investing in new generation/transmission?
A partial answer to the first problem is that the investor must have a reasonably good idea of the revenue/cost flow in the first few years, and must rely more on qualitative analysis (or at least approximate quantitative analysis) for the later years. This is where the sensitivity formulas of our report could be useful. For example, an investor locating a new generator may estimate sensitivities of flows along key transmission paths (along which the new generator's output would tend to flow) to changes in various parameters, such as load growth in various load centers, new generator investments by competitors at other locations, changes in transmission configuration, changes in economic dispatch patterns because of changes in fuel prices, etc. These sensitivity estimates could enable the investor to develop  in a computationally efficient manner  a rank of reallife scenarios that may affect the investor's profits; the investor may then analyze some of these scenarios in more detail using more detailed tools.
For the second problem, the sensitivity formulas could aid in the following way. Consider a market participant that is trying to find an optimal location to invest in new generation. Given different candidate locations, the market participant can estimate the sensitivities of key variables such as transmission line flows, transmission congestion pattern, loading margins to operational limits, etc., to changes in MW injections at the candidate locations. The sensitivity results can then be used to rank the different locations appropriately; more detailed analysis can then be performed on a relatively smaller set of cases that have the highest sensitivity to the investment. For example, if transmission congestion on transmission paths connecting the generator to the loadcenter is increased as a result of generator investment in one location relative to other locations, then investment in that location is a less desirable because the generator may be constrained off under congested conditions.
In summary, sensitivity analysis can be a very useful computational aid in solving complex generator or transmission investment problems. The primary purpose of the sensitivities is to be used as a screening (and ranking) tool to get (even if approximate) estimates of the value of the investment under different "whatif'' scenarios. These sensitivity results can provide guidance to conduct more detailed analysis on a few targeted set of scenarios or investment possibilities.
This section is in the following paper:
Interactions among limits during maximum loadability and transfer capability determination
(This paper is also reference [1].)
Available Transfer Capability in a marketoriented context and in an environment where uncertainty plays a major role requires careful balancing of exactly what is meant by "maximum" and "available." In a recent work [25] the authors have done an excellent job in discussing and defining issues of measuring and monitoring ATC under these conditions. The transfer capability is determined by increasing the studied transfers according to an incremental step and to assumptions on power injections until congestion cannot be relieved by any form of redispatching. This transfer level is referred to as the Maximum Transfer Capability (MTC). For each level of power transferred, the paper uses a twophase approach that permits the determination of the cost upon the system imposed by congestion cost for a given state of the system. The paper provides for an explicit determination of risk for each value of the transfer capability and for each transaction. Risk is defined in this paper as the probability of curtailment of a transaction. Firmness and risk are recognized as two different concepts. Risk assessment takes into account the firmness associated with the transaction, i.e. the conditions defined for its curtailment (maximum congestion cost accepted for the transaction and priority among simultaneous transactions). Risk for a transfer is determined as the proportion of simulated situations where the desired transfer level cannot been reached. The paper illustrates the importance of considering redispatching in the determination of additional transfer limits for highly stressed systems.
This section reviews the main elements of power system security and optimization as they pertain to transfer capability.
Of greatest interest is the notion of maximum loadability and maximum transfer capability. Open access to transmission services requires continual determination of the available transfer capability (ATC) for each interface of a region or area of the power network. The following are the main limits to transfer capability:
Continuation methods for the solution of general nonlinear equations and bifurcation problems are described in [33,77], and the application to power systems is covered in [14,22]. In particular, [68,91] illustrate the combined use of continuation methods and sensitivity analysis for security margin computation.
Also of importance is recent work on sensitivity of margins. [41] presents exact analytical margin sensitivity formulas with illustrative examples including the effect of variation in interarea transfers on the loading margin to voltage collapse. [95] applies sensitivity analysis to the determination of the contingencies that most limit transfer capability, and motivates the work on contingency analysis presented in [42].
For a recent survey of the economic theory of electricity pricing, see [27]. Spot pricing of electricity was first proposed by William Vickrey in 1971 and was subsequently elaborated by the MIT Energy Laboratory and summarized in [76]. The theory of locational pricing of electricity was first proposed by R.E. Bohn, M.C. Caramanis, and F.C. Schweppe in 1984 and has subsequently been developed by W.W. Hogan in [47] and its sequels.
The background material relating to optimization and optimum power flows is now summarized. Economic operation of power systems has been a topic that has received much attention since the 1930's. In the late 1950's, Leon Kirchmeyer published his important work on economic operation of a system considering network losses. Subsequently, Carpentier presented a formulation of the Optimum Power Flow problem in 1962, and nicely summarized this work in 1979 [17]. Since this time, much significant work has taken place in this direction. Important early practical references included [29] and [73]. Subsequent, significant progress was attained with the development of quadratic Newtontype optimization methods [83,87,88]. For more recent surveys and development in the OPF area, refer to [52]. A recent reference that adds a lot of insight into the workings of the optimum power flow in practical problems is [4].
Transmission reliability margin accounts for uncertainties related to power transmission in transfer capability calculations. This chapter suggests a formula that quantifies transmission reliability margin based on transfer capability sensitivities and a probabilistic characterization of the various uncertainties. The formula is tested on 8 and 118 bus systems by comparison with Monte Carlo simulation. The formula contributes to more accurate and defensible transfer capability calculations.
The available transfer capability indicates the amount by which interarea bulk power transfers can be increased without compromising system security. Available transfer capability is described in more detail in section 4.1. The available transfer capability is the total transfer capability minus the base case transfer together with adjustments such as the Transmission Reliability Margin (TRM), see (4.1).
The transmission reliability margin accounts for uncertainties in the transmission system and safety margins. According to NERC [90], "The determination of ATC must accommodate reasonable uncertainties in system conditions and provide operating flexibility to ensure the secure operation of the interconnected network". There are two margins defined to allow for this uncertainty: The transmission reliability margin is defined in [90] as "that amount of transmission capability necessary to ensure that the interconnected transmission network is secure under a reasonable range of uncertainties in system conditions". The capacity benefit margin ensures access to generation from interconnected systems to meet generation requirements. The capacity benefit margin is calculated separately from the transmission reliability margin. Since uncertainty increases as conditions are predicted further into the future, the transmission reliability margin will generally increase when it is calculated for times further into the future.
This chapter suggests a straightforward method to quantify transmission reliability margin. The method exploits formulas for the first order sensitivity of transfer capability [43,40]. These formulas can be quickly and easily computed when the transfer capability is determined. The formulas essentially determine a linear model for changes in transfer capability in terms of changes in any of the power system parameters. This chapter supposes that the uncertainty of the parameters can be estimated or measured and shows how to estimate the corresponding uncertainty in the transfer capability. A formula for transmission reliability margin is then developed based on the uncertainty in the transfer capability and the degree of safety required.
In our framework [2], the following limits are accounted for in the transfer capability computation:
The transfer capability is a function A of many parameters p_{1},p_{2},...,p_{m}:
 (5.1) 
We assume that the nominal transfer capacity has been calculated when all the parameters are at their mean values. The uncertainty U in the available transfer capability due to the uncertainty in all the parameters is:
 (5.2) 
 (5.3) 
 (5.4) 
When the available transfer capability is limited by voltage magnitude or thermal limits, the sensitivity of the available transfer capability to parameters can be computed using the formulas of [2,43,40]. When the available transfer capability is limited by voltage collapse, the sensitivity of the available transfer capability to parameters can be computed using the formulas of [41]. (Topology changes can also be accommodated with limited accuracy using the fast contingency ranking techniques in [42].)
In each case a static, nonlinear power system model is used to evaluate the sensitivities. The computation of dA/dp_{i} is very fast and the additional computational effort to compute dA/dp_{i} for many parameters p_{i} is very small [41,2,43,40]. For example, the sensitivity of the available transfer capability to all the line admittances can be calculated in less time than one load flow in large power system models [41,2].
Since the parameters are assumed to be independent,

 (5.7) 
The central limit theorem asserts that (under suitable conditions which are discussed in the appendix) the sum of n independent random variables has an approximately normal distribution when n is large. Reference [45] states: "in practical cases, more often than not, n = 10 is a reasonably large number, while n = 25 is effectively infinite." Hence for practical power system problems with many parameters, we expect that the uncertainty U is approximately a normal random variable with mean zero and standard deviation given by (5.7). This approximation gives a basis on which to calculate the transmission reliability margin. The conditions described in the appendix are mild and require little knowledge of the distribution of the parameters.
There are cases in which the central limit theorem approximation does not work so well: As stated in [45], "the separate random variables comprising the sum should not have too disparate variances: for example, in terms of variance none of them should be comparable with the sum of the rest." This can occur in the sum (5.4) when there are a few parameters which heavily influence the available transfer capability (large dA/dp_{i}) and the other parameters have little influence on the available transfer capability (small dA/dp_{i}) and are insufficiently numerous. In these cases, accurate answers can be obtained by using the central limit theorem to estimate the combined effect of the numerous parameters of little influence as a normal random variable and then finding the distribution of U with the few influential parameters by Monte Carlo or other means (c.f. [19] in the context of probabilistic transfer capacity). This partial use of the central limit approximation dramatically reduces the dimension of the problem and the computational expense of solving it. For other methods not relying on the central limit theorem see chapter 6.
In all cases the central limit theorem approximation improves as the number of similar parameters increases and thus the approximation generally improves as the power system models become larger and more practical.
We want to define the transmission reliability margin large enough so that it accounts for the uncertainty in U with rare exceptions. More precisely, we want
 (5.8) 
 (5.9) 
Combining (5.7) and (5.9) yields a formula for transmission reliability margin:
 (5.10) 
The available transfer capability is computed from a base case constructed from system information available at a given time. There is some uncertainty or inaccuracy in this computation. There is additional uncertainty for future available transfer capabilities because the available transfer capability computed at the base case does not reflect evolving system conditions or operating actions. These two classes of uncertainty are detailed in the following two subsections.
These uncertainties generally increase when longer time frames are considered.
While some of these uncertainties may be quite hard to characterize a priori, it is important to note that it would be practical to collect empirical data on the changes in base cases as time progresses. Then variances of the uncertain parameters corresponding to various time frames could be estimated.
It is important to satisfy the statistical independence assumption when modeling the parameter uncertainty. For example, if the uncertainty of different loads has a common temperature component, then this temperature component should be a single parameter and the load variations should be modeled as a function of temperature (see section 6.1).
This section tests the transmission reliability margin formula by comparing it with Monte Carlo simulations in two examples. The first example uses the 8 bus system shown in Figure 5.1. The transfer capability from area 1 (buses 1,2,5,6) to area 2 (buses 3,4,7,8) is limited by the power flow limit on the line between bus 2 and 3. The parameters are listed in Table 5.1.
parameter  distribution 
line susceptance B25  binary; prob{B25=5.0}=0.95, 
prob{B25=0}=0.05  
line susceptance B38  binary; prob{B38=2.5}=0.95, 
prob{B38=0}=0.05  
line impedance X12  uniform; m=0.1, s=0.0029 
line impedance X23  uniform; m=0.2, s=0.0058 
line impedance X34  uniform; m=0.1, s=0.0029 
line impedance X15  uniform; m=0.1, s=0.0029 
line impedance X26  uniform; m=0.1, s=0.0029 
line impedance X37  uniform; m=0.1, s=0.0029 
line impedance X48  uniform; m=0.1, s=0.0029 
line impedance X56  uniform; m=0.1, s=0.0029 
line impedance X67  uniform; m=0.2, s=0.0058 
line impedance X78  uniform; m=0.1, s=0.0029 
system loading p13  normal; m=0.0, s=0.1 
bus 5 generation p14  normal; m=0.0, s=0.1 
line 24 flow limit  normal; m=1.5, s=0.1 
line 67 flow limit  normal; m=1.5, s=0.1 
The base case of the system assumes all parameters at their mean values. At the base system, the available transfer capability (ATC) is 2.8253 (with no contingency). Sensitivity of ATC to these parameters can be calculated with no difficulty. Given a desired high probability P, the transmission reliability margin defined in (5.8) is calculated using formula (5.10). Table 5.2 lists transmission reliability margins with respect to different probabilities P. 10,000 samples are used in the Monte Carlo simulation.
P  90%  95%  99%  99.5% 
TRM formula  0.6012  0.7750  1.0944  1.2118 
Monte Carlo  0.6027  0.7846  1.1083  1.2171 
The second example uses the IEEE 118 bus system. There are 186 lines and the real power flow limit was assumed to be 1.0 p.u. at all lines except that the real power flow limit for line 54 was assumed to be 3.0 p.u.. We consider a point to point power transfer from bus 6 to bus 45. The uncertain parameters are the power injections to all buses. The power injections are assumed to have a uniform distribution around 5% of their nominal values.
An AC power flow model was used. At the base case, the available transfer capability is 1.8821 p.u.. Given a desired probability P, transmission reliability margin defined in (5.8) is calculated using formula (5.10). Table 5.3 lists transmission reliability margins with respect to different probabilities P. 10,000 samples were used in the Monte Carlo simulation.
P  90%  95%  99%  99.5% 
TRM formula  0.0803  0.1036  0.1462  0.1619 
Monte Carlo  0.0795  0.1027  0.1427  0.1585 
In both the 8 and 118 bus examples, the Monte Carlo results confirm the TRM estimates from formula (5.10).
We observe that our approach is not limited to the determination of transmission reliability margin. Since our approach yields an approximately normal distribution of transfer capability uncertainty U and an estimate (5.7) of the standard deviation of U, this is an alternative way to find the probabilistic transfer capacity as presented in [18,19,61,97,60]. The probabilistic transfer capacity can be used for system planning, system analysis, contract design and market analysis. Reference [61] suggests promising applications of probabilistic transfer capacity in the new market environment.
This chapter suggests a way to estimate transmission reliability margin with the formula (5.10). The formula requires estimates of the uncertainty in independent parameters, the evaluation of transfer capability sensitivities, and specification of the degree of safety. The transfer capability sensitivities with respect to many parameters are easy and quick to evaluate once the transfer capability is determined [41,2]. This ability to quickly obtain sensitivities with respect to many parameters makes it practical to account for the effects of many uncertain parameters in large power system models and improves the central limit theorem approximation used to derive the formula. The formula has been confirmed by comparison with Monte Carlo runs on 8 and 118 bus systems.
The approach includes estimating the statistics of the uncertainty in the transfer capability and thus gives an alternative way to obtain a probabilistic transfer capacity.
The formula provides a defensible and transparent way to estimate transmission reliability margin; in particular, the degree of safety assumed and the sources of uncertainty are apparent in the calculation. The improved estimate of transmission reliability margin will improve the accuracy of available transfer capabilities and could be helpful in resolving the tradeoff between security and maximizing transfer capability. The sensitivities used in the calculation highlight which uncertain parameters are important. Indeed, the calculation provides one way to put a value on reducing parameter uncertainty because a given reduction in uncertainty yields a calculable reduction in transmission reliability margin and this can be related to the profit made in an increased transfer.
Let X_{1},X_{2},...,X_{m} be independent, zero mean random variables and write s^{2}_{m} = S_{k = 1}^{m} s^{2}(X_{k}) for the variance of S_{k = 1}^{m} X_{k}. The approximate normality of S_{k = 1}^{m} X_{k} requires a central limit theorem. (Note that the most straightforward version of the central limit theorem does not apply because we do not assume that X_{1},X_{2},...,X_{m} are identically distributed.) A special case of the Lindeberg theorem [10] states that if
 (5.11) 
One useful class of random variables satisfying the Lindeberg condition (5.11) is random variables which are both uniformly bounded and whose variance uniformly exceeds some positive constant. It is also possible to augment the random variables in this class with some normal random variables.
A key engineering judgement in assigning flow limits to transmission lines is determination of an acceptable margin that should be maintained between planned flow levels and the absolute operational limit. Such a margin allows for deviation of the actual line flow from predicted values under the influence of random load and power injection variations.
We start with an illustrative example in which uncertainties in regional temperature variations drive deviations in load patterns from predicted values, and ultimately, deviation in transmission line flows (or socalled "flowgate" flows, constructed as sums of individual line flows). With a set of regional temperature measurements as the fundamental random variable, we step through the necessary modeling assumptions and data requirements to translate temperature statistics (covariances) to covariances in predicted values of line/flowgate flows. These calculations establish the strong dependence on the power transfer distribution factors (PTDFs).
The chapter then continues by establishing a framework for an important optimization problem: given joint probability distributions for variations in bus power injections and loads, maximize nominal power transfers between two buses subject to the constraint that the probability of flowgate flows remaining within limits is above a user specified threshold. Then we describe practical approximation techniques for extending from the case of normally distributed random variables to more general distributions using the CornishFisher expansion.
The chapter finishes by discussing a stochastic optimal power flow formulation which can be addressed with standard power system optimization tools.
The motivation for this chapter is the need to treat transmission capability calculations in a manner that accounts for the inherently probabilistic nature of power demand, and, in some market scenarios, of power production. While there are a large number of sources of uncertainty in a power systems operating environment, the work here will focus on continuous variations in demand and production, as opposed to random changes in system structure, as might be associated with generator outages or transmission line tripping.
A key source of uncertainty in power systems operation is the response of power consumers to regional temperature variations. Despite the significant changes in operational procedures that have been brought about under restructuring, one may still assume that operation of the power system is conducted based on advanced planning, using predicted values for load, and that the accuracy of predictions is successively refined as one moves the time horizon of interest closer and closer to real time. A predicted value of load is based on many factors, and load prediction algorithms are a well developed field. However, given an advance load prediction, with a prediction horizon ranging from, say, a week down to an hour, a key source of uncertainty in this prediction will be predicted temperature.
The functional dependence of a load demand level (e.g., a distribution substation) to a set of regional temperature measurements is typically well established in most operating environments. Therefore, we will assume that the following data and models are available:

 (6.1) 


We will illustrate the interpretation of this covariance matrix in a small example to follow. However, it is useful to note the way in which some qualitative features one might intuitively expect are reflected in the computation. First, even relatively small variances in an underlying random temperature can yield large covariances for line flows. If a group of loads are all dependent on one temperature variation in a like direction, and these loads all contribute in a like direction to a line flow change, the effect of a temperature uncertainty can be greatly magnified. Second, the structure of the computation of Q makes it highly likely that Q will have significant offdiagonal components; i.e. variations within certain sets of line flows will often be highly correlated.
With this background in mind, consider the IEEE 39 bus test system diagram in Figure 6.1. Note that for convenience, each line in this network has been labelled with a number for 1 to 46, as indicated by the underscored numbers adjacent to the reference direction arrow on each line. For the data to follow, we make the simple assumption of the network operating with the generator at bus 30 as slack (i.e., for the incremental load changes examined, the corresponding generation adjustments are made exclusively at this one machine).
As noted in the preceding section, a key piece of data describing network behavior are the PTDFs for lines and load changes of interest. To illustrate the techniques described here, suppose that the line flows of interest are at lines 2, 3, 4, 6, 7, 14 and 38; all are active power flows, measured at the receiving end of lines, as indicated by reference arrows shown in Figure 6.1. In the notation introduced above, these line flow deviations would be denoted by the variables Dp_{L2} , Dp_{L3} , Dp_{L4} Dp_{L6} , Dp_{L7} , Dp_{L14} , Dp_{L38}. Suppose further that the load variations of interest are active power demands at buses 3, 4, 7, 8, 15, 16, 18, 23, and 24, denoted by the variables Dp_{D3} , Dp_{D4} , Dp_{D7} Dp_{D8} , Dp_{D15} , Dp_{D16} , Dp_{D18} , Dp_{D23} , Dp_{D24} . For the system as considered, with bus 30 as slack bus, the resulting PTDFs are given by the matrix below.

This matrix is computed from the common format power flow data describing the IEEE 39 bus test system. The authors wish to make clear that the remaining data to follow is purely hypothetical, and is introduced for purposes of illustrating the techniques described in this chapter.
Given that temperature variation (from, perhaps, a day ahead predicted value) is the underlying random element in this analysis, a fundamental piece of data is the covariance matrix for the temperature variation. Such a matrix of covariance values could be estimated from historical time series data comparing (day ahead) predicted temperature to actual observed temperature, under the assumption of a zero mean Gaussian distribution for the error. This type of construction is standard in time series analysis, and will not be treated here.
For illustration, we shall assume that temperature is measured at each of the load buses where demand is assumed to vary: buses 3, 4, 7, 8, 15 16, 18, 23 and 24. It is reasonable to assume that the temperature variations are correlated, so that the covariance matrix will have nonzero offdiagonal elements. However, assuming the buses were ordered so that greater separation in their numeric order represents increasing geographic separation, one would also expect that the magnitude of the off diagonal terms would drop off rapidly away from the diagonal (because wider geographic separation would be associated with lower correlation between temperature errors). For illustrative purposes, below is a representative matrix R_{t} of temperature errors in degrees C.

Next, as noted in Section 6.1, it is necessary to identify a matrix B which represents the relation of temperature variation to load variation. While it is certainly possible that this quantity could vary with operating conditions, we will assume that a fixed nominal value is established. Given that a given load bus may serve loads that are geographically dispersed about that bus, we will expect that temperature variations throughout the system might have an impact on any given load, but that the strongest impact will be associated with the temperature variation measured at the bus in question. This again results in a matrix with nontrivial offdiagonal elements, but typically diagonally dominant with positive entries. As above, we provide a sample numerical matrix for our example that is meant to be representative of these properties. We assume that B as given below relates degree C variation in temperature (from its predicted value) with per unit variation in load demand (from its predicted value).

As an example of the interpretation of B, suppose one wanted to compute the deviation of loads from predicted values if temperatures at every measurement point deviated from their predicted values by +1 degree C. This would be computed simply by multiplying B times a vector of all positive 1 entries.
As described in our earlier development, we are now prepared to compute the covariance matrix for line flow variations, Q , as

yielding

This example illustrates how the impact of the network characteristics, as reflected in the PTDFs, may "concentrate'' uncertainty on certain line flows. For example, note that the ( 2, 2) entry in Q is much larger than any other diagonal entry. The ( 2, 2) entry in Q corresponds to the variance of line 3, which is the line connecting buses 2 and 3. While such an interpretation is only a rough heuristic in the case of correlated Gaussian variables, as we have here, we may very roughly interpret the square root of the diagonal covariance entry as a standard deviation. In this rough interpretation, the active power variation on line number 3 is roughly 0.138 per unit. If one computes a corresponding covariance matrix for the bus power deviations, and examines the largest "standard deviation'' (square root of a diagonal covariance term), one would find that none exceed 0.038 per unit. Again, one can think of the network layout, and the relative admittances of lines, playing the role of concentrating smaller bus power variations to produce considerably larger line flow variations. This is the key observation the reader should take away from this example.
The preceding development illustrates the procedure for mapping a fundamental random variable (regional temperature variations) through to variations in load demand, and ultimately, to individual line flow variations. In this section, we will broaden our perspective, and consider a random vector of net bus power injections, which may be positive (net generation) or negative (net load).
As we no longer restrict random variation to be associated with deviation from deterministic predicted values, we will drop the prefix of D on the new random variables to be introduced below. In particular, we begin with the following
Assumption 1 There are n locations (or buses) in the power system. The net real power injection into buses are represented by a column vector p with n entries (the units of power flows are often in MW).
To allow further generality in our formulation, we also wish to extend from flows on individual lines (previously denoted as Df_{L} ), to a more general class of flows composed as weighted sums of individual line flows. Therefore, we proceed with
Assumption 2 A flowgate is a transmission line or a set of transmission lines. Each flowgate has physical limits that must be obeyed. In particular, flows on a constrained flowgate cannot increase in the congested (constraining) direction. There are m flowgates in the power system. The column vector f = [f_{1},...,f_{m}] represents the flow in each of the flowgates (associated with each flow is an arbitrarily chosen flow direction).
Assumption 3 Flowgate flows obey^{1} Kirchhoff's laws. The flowgate flows can be expressed as a function of node injections using the following reasonable approximation:
 (6.2) 
Note that the power transmission distribution factor matrix A is closely related to the matrix D introduced earlier in (6.1). In particular, A has rows formed as weighted sums of rows of D , corresponding to a flowgate flow being defined as weighted sums of individual line flows. The choice to focus on flow gates, rather than individual lines, follows the NERC practice of setting transfer limits for large scale regional transfers in terms of flowgates. With this goal in mind, we continue with
Assumption 4 Flows on flowgates must be within their physical limits,
 (6.3) 
 (6.4) 
We will find it convenient to describe A as A = [a_{1}, a_{2}, ...,a_{m}], where a_{k} is the k^{th} row of A and is of dimension n. Then the flow in flowgate k is given by
 (6.5) 
Our main focus in this chapter will be on the following question: what can we say about the uncertainty in flows given uncertainty in the power injections at each node for a given time period? In particular,
It is a wellknown result that the probability distribution of a random variable can be completely described by its cumulants (see, for example, [55]). For most practical problems of interest, the first few cumulants generally are sufficient.
Assumption 5 We will represent the jth order cumulant for a random variable Z as k_{j}(Z). When there is no ambiguity, we will suppress the functional dependence.
We note that for the normal distribution, only the mean and variance are generally nonzero; all higher order cumulants are zero. Therefore higherorder cumulants capture departures from normality for nonnormal distributions. For most practical distributions, most of the departure is captured in k_{3} and k_{4}. Therefore, in this chapter all cumulants of order higher than 4 will be assumed to be zero, i.e, k_{j} = 0, for j > 4. We note the results in this chapter are general and are independent of this approximation.
For convenience, we define the first few cumulants for a random variable Z (here the operator E[Z] describes the expected value of Z):

Cumulants have the very desirable property that they are additive for linear sums of independent random variables. More precisely, for a random variable Z = S_{i} a_{i} X_{i}, where the X_{i}'s are independent random variables and the a_{i}'s are constants,
 (6.10) 
When the X_{i}'s in (6.10) are not independent, then the cumulants will include "crosscumulant'' terms. In this chapter, we will only worry about the crosscumulants for the first two cumulants, and make the approximation that higherorder crosscumulant terms are zero. There are two reasons why this is a practical approximation.
Therefore, we will use the following generalization to (6.10), when the X_{i}'s are not independent random variables.



In the sequel, we will find it useful to "standardize'' a random variable Z by using the following transformation
 (6.20) 
For a standardized random variable, the cumulants can be related to the more familiar statistical terms

We end this section with an extremely important theorem in statistics called the CornishFisher expansion [26] (see also [54] for more detail and other references). The power in the theorem lies in its ability to relate random variables with arbitrary probability distributions to the normal distribution.
Proposition 1 [CornishFisher] Let Z be a random variable standardized according to (6.20) with probability distribution g(z). Let N(x) be the probability distribution function of a standardized normal random variable, i.e., a normal distribution with mean 0, and variance 1. Then


We stress that the CornishFisher expansion stated in (6.26) is for standardized random variables. So g(z) must represent the probability distribution function of the standardized random variable Z. Also, it is very important to verify that the higher order terms are indeed small before they can be neglected. For example, if the skewness is much larger than 1, then one should include higher order skewness terms because they cannot be discarded. See [54] for more details.
Let Y(x) represent the cumulative probability distribution of a standardized normal probability distribution. There are standard tables for Y(x). We now solve the following two practical cases of interest.
The first case can be solved from (6.25) as
 (6.27) 
For the second case, we proceed as follows. The equation
 (6.28) 
 (6.29) 
 (6.30) 
To summarize,
In this section, we will estimate the probability of transmission congestion for a given time period, given exogenous uncertain inputs. In particular, assume that we are given the cumulants for p (and the secondorder cumulant term, which is the covariance matrix S). Then what is the probability that, for a flowgate k, f_{k} < = b_{k}? Or, from (6.5),
 (6.31) 
From (6.31), we get
 (6.32) 
Let Z = (f_{k}E[f_{k}])/s_{fk}, and note that Z is now in standardized form. From (6.19), we get the following cumulants for Z.


Rewriting (6.32) in terms of Z gives
 (6.38) 
Using the result (6.27) from the previous section, we get
 (6.39) 
We note that it is difficult to generalize the result to the vector case (to find out prob{Ap < = b}), because the CornishFisher expansion is not applicable to vectors. Indeed, even for the relatively simple case when p has a normal distribution (and each element of p is independent), it is surprisingly difficult to obtain general analytical results for prob{Ap < = b} (except for special cases). Indeed, even numerical results for such integrals are hard for large dimensions. For example, see [35,36]; Genz's website also contains a wide variety of research papers and useful software.
 (6.40) 
Using the formulas in (6.39), (6.26), we get the following results.
We next tackle a more complicated problem. Suppose that probability distributions for certain node injections are known for a prospective time period. For example, since loads at buses are related to temperature, node loads would be known with a certain probability distribution. Suppose that we would like to maximize power transfers between a subset of other buses subject to the constraint that no flowgate flow should exceed its limit with a given probability a. How should we approach this problem?
To answer this question, we make the following assumptions regarding the probability distribution of node injections.
Assumption 6 Let x_{r} of dimension n_{r} represent those buses whose net power flows are assumed to have a known probability distribution. Without loss of generality, we will label the first n_{r} buses to be these buses. Then we can write p = [x_{r}^{t} x_{d}^{t} ]^{t} where x_{d} is a vector of dimension n_{d} = nn_{r}. Similarly partition the PTDF matrix A as A = [A_{r} A_{d}]. The vector x_{d} represent a vector of parameters.
We want to solve the problem of the type below.
 (6.41) 

Writing the k^{th} rows of A_{r}, A_{d} as A_{r}_{k}, A_{d}_{k} respectively, (6.43) can be rewritten as

 (6.45) 
 (6.46) 
Moreover, from (6.13), we can write
 (6.47) 
 (6.48) 
 (6.49) 
Now the optimization problem becomes
 (6.50) 
 (6.51) 
This section uses the results of section 6.7 to estimate the maximum transfer capacity between 2 busses for the 6 bus example introduced in section 1.4.5. We make the following simplifying assumptions:
a  max transfer (MW) 
0.99999  39.67 
0.9999  15.26 
0.999  12.85 
0.99  47.02 
0.9  93.74 
0.8  113.41 
0.5  151.06 
0.25  181.22 
0.1  208.37 
0.05  224.62 
0.01  255.09 
0.001  289.26 
0.0001  317.38 
0.00001  341.79 
0  infinity 
Given these assumptions, we find the maximum transfers from bus 1 to bus 2 so that the probability of no transmission line overload for each transmission line (in both directions) is above a threshold probability a. Thus a is the probability of no congestion for that maximum transfer. This involves solving the linear programming problem (6.51). The solution is described in Table 6.1.
Table 6.1 demonstrates that the maximum transfer capacity varies highly nonlinearly with the threshold probability a. For very low threshold probabilities, the maximum transfer capacity is very high; for high threshold probabilities, the maximum transfer capacity is more constrained.
In determining the optimal capability of a transmission network, and more specifically, limits on its ability to transfer power along a transmission corridor, it is natural to seek an adaptation of existing power system optimization tools to this problem. A wide class of optimization problems relating to real time and near real time operational decisions in the grid are grouped under the general framework of Optimal Power Flow (OPF). However, new objectives in a competitive environment [46], [53], the associated focus on congestion management to facilitate competition [78], and the increasing volatility of dispatch and operating conditions raise significant new challenges in OPF [65]. Adding to these challenges is the need to explicitly treat uncertainty in optimization problems relating to transmission network limits. While some efforts in this direction exist in the literature (see, for example, [79], [31], and [32] for early work), the greater levels of uncertainty introduced by market based dispatch make the need for this class of methods much more pressing.
In a competitive power systems environment, the policy decision to manage the provision of power via day ahead or hourbyhour auctions inevitably introduces stochastic behavior in the amount of power produced in any period, as the supply versus demand equilibrium will be dictated by market variations and by time varying producer and consumer preferences. Averaged over the large number of producers and consumers, this variation may be approximated as a continuous quantity. However, these uncertainties in amount of power production are further exacerbated by uncertainties in the geographic point of connection. Given a portfolio of individual generating plants that may bid into the market to meet demand, their various geographic locations correspond to different points of connection to the electrical circuit that comprises the power system. Given the limits on the power or current carrying capability of transmission lines, cables, and transformers, and given the stochastic nature of production and load, the challenge in setting limits on transmission capability so as to maintain an acceptably small probability of line overload is clear. Yet in contrast to this clear need, present day industry standards for security, through the "n1 criterion," are purely deterministic. Satisfaction of the n1 security criterion simply provides a guarantee of worst case performance in the face of those operating conditions chosen to be included in the contingency list. Therefore, consideration of relative probability of conditions which might cause overload is entirely "buried" in the choice of the contingency list to be studied. If a threatening condition is included in the contingency list, the system limits (such as ATC) are set such that this condition is survivable with probability one. If a threatening condition is judged to be of low enough probability that it is not included in the contingency list, ATC limits based on n1 criterion do not reflect the threat in any way. Likewise, threats associated with multiple events are not reflected at all. Clearly, given the growing volatility of operating conditions observed in competitive markets, this does not appear to be a viable means of setting secure ATC limits in the future electric power grid. The shortcomings of the n1 security criteria for a competitive environment are clearly described in [86].
We begin from the simple representative example of a deterministic formulation of OPF (2.1) in section 2.6 and we indicate how uncertainty is included in the problem formulation. With the notation developed in section 2.6, we can state a representative form of stochastic optimal power flow problem designed to identify the maximum power that may be shipped along a transmission corridor. Recognizing that stochastic variation in production and demand will makes these current flows random quantities, it is the expected value of flow that is to be maximized.
Inequality constraints to be satisfied in the formulation below include limits on the magnitude of current flow in each line of the network, upper and lower limits on bus voltage magnitude at each bus, and upper and lower limits on active and reactive power at each bus. All these inequality constraints are formulated probabilistically as chance constraints. That is, recognizing that in a stochastic setting there is always small but finite probability that there will be no operable solution for which all constraints are satisfied, these inequalities are required to be satisfied with some high probability. For simplicity below, all are required to meet a uniform probability of satisfaction of 1  e. e is interpreted as the probability that any operating constraint would be exceeded.
As a final observation before introducing our representative formulation, it is also important to note the class of OPF operating controls not being addressed; namely those relating to discrete decisions, which would transform the problem to a mixed integer/nonlinear program. Treatment of issues of discrete decisions presents significant challenges in deterministic OPF [58]. While optimization over discrete decisions in a stochastic environment has been treated recently in the context of traditional unit commitment [84], [85], these works do not introduce the challenging element of detailed network power flow constraints. It would appear that treatment of full power flow constraints, along with mixed discrete/continuous decision variables in a stochastic framework, challenges the current state of the art available in optimization algorithms and software.
With this background, our representative stochastic optimization formulation of the ATC calculation may be summarized as shown below.

subject to






Given the simple quadratic form of the constraint equations, with random variations entering linearly, the linearized version of this problem directly reduces to the standard form of a linear program with chance constraints. In particular, letting z represent the real and imaginary parts of the bus voltage magnitudes, linearization of the optimization above about a given operating point produces a problem with the structure

subject to

Note the great simplification inherent in the fact that the random variables s representing uncertainty in injection/demand enter linearly; the effect of the linearized power injection equations is simply one of shifting the mean of random variables formed as linear transformation on s. In the case for which s is a vector of jointly Gaussian random variables, the requirement that the inequality constraints be satisfied with given probability transforms directly to a set of deterministic constraints. A specific numeric example of this class of computation is provided in section 6.3.
^{1} Kirchhoff's laws are a reflection of the physical constraints of the system: the flows on lines are a function of the voltage differences across the lines, and the sums of flows at any node must be precisely zero.
^{2} Generally, when a flowgate flow exceeds its limit, the power injections at different buses will be "redispatched'' to bring them back to within their limits. This redispatch is usually accomplished most economically by ramping up cheaper generators at one end of the congested flowgate, and ramping down more expensive generators at the other end of the congested flowgate. In some cases, transmission overloads are resolved by removing the transmission line. In some extreme cases, redispatch of power injections are done by curtailing loads in congested pockets of the system.