1. Introduction
Financial Transmission Rights (FTR) is an energy derivative that allows market participants to receive an annual or monthly share of congestion cost revenues collected in settled electricity prices, or locational marginal price (LMP), by Independent System Operators (ISO) [
1,
2]. The ISO is a third-party organization to ensure electric systems reliability in generation resources and transmission lines. Congestion cost at a pricing location is a price difference between the least expensive electricity available in the ISO region and more expensive options due to transmission system constraints.
FTR holders are paid a congestion cost difference (credit) settled on a transmission path when it is positive (prevailing flow FTR) and must pay the difference (charge) when negative (counterflow FTR). The above FTR is called the FTR Obligation, compared to the FTR Option where FTR Option holders do not have to pay the difference even when the settled value is negative [
1]. As of April 2020, FTR Option products are not available in ISO New England (ISO-NE) in the U.S., but exist in other electric markets, such as the Pennsylvania–New Jersey–Maryland Interconnection LLC (PJM Interconnection).
As FTR values are derived from the pairs of pricing locations in an electric market, the possible combinations of FTR paths could be very large, providing multiple FTR bidding opportunities for market participants. For example, ISO New England has an existing generating capacity of 31,200 MW from 1976 generators in its six member states, with a generation mix of natural gas (40%), nuclear (25%), net imports (19%), renewable (9%), and hydro (7%) [
3]. According to the Day-Ahead Energy Market Hourly LMP Report for 14 April 2020 published by the ISO-NE, the total number of pricing locations in the region is 1192, comprising 1125 network nodes, 32 hub nodes, 20 demand response locations, 8 load zones, 6 external nodes, and 1 hub. The total pricing locations could theoretically be translated into 1.4 million FTR paths in the ISO auction for prevailing flows, and when added by counterflows, the total number could be doubled.
There are two types of FTR market participants: hedgers and speculators. Hedging participants, with electricity supply obligations, want to hedge against congestion costs by purchasing FTRs on the paths from their supply sources to their customer load zones, while speculating participants, without any physical supply obligations, may purchase FTRs to arbitrage differences between expected and actual settled values of FTR paths [
1]. With such numerous choices of potential FTR paths available, FTR market participants need to reasonably evaluate which FTR paths to bid in the auctions, subject to their limited capital budgets. Consistent and standardized methodology is crucial in evaluating interested paths, in terms of profitability and risks associated with the paths.
Item Response Theory (IRT) is one of the most influential methods in the field of educational and psychological measurement, to understand the behaviors of individual test items or variables [
4]. IRT models provide information about item parameters and latent traits of test respondents, helping gain insights and assessments about their performance as well as the items. It is also useful for test development, item analysis, equating, item banking, and computer aided test (CAT) [
5]. As a group of statistical models with probabilistic and stochastic procedures, IRT connects the pattern of responses to a group of items to predict a latent trait/ability, and then, converts discrete item responses into the levels or locations of probability estimates which respondents possess underlying the latent trait [
6,
7].
The most basic model is the One-Parameter Logistic model (1PL), or the Rasch model named after Georg Rasch, a Danish mathematician. In the model, the probability of correct response (denoted as
Xi = 1) to each item (labeled
i) is a function of the item’s difficulty level (labeled
bi) and the respondent’s trait level (labeled
θ), with a mathematical expression as in Equation (1) [
7]:
In the Equation (1),
Xi = 1 indicates that a respondent endorsed an item i or provided a correct response. A horizontal line at
Pi(
θ) = 0.5 on the y-axis in
Figure 1 denotes a mid-probability of the correct response to item
i being correct. That is, it indicates that the respondent has 50% chance of providing a correct response to the item. The difficulty coefficient (
bi) of an item are the value of a latent trait level (
θ) on the x-axis which is an intersection point between the mid-probability, shown on the horizontal line, and an individual characteristic curve (ICC) of the item.
Figure 1 illustrates three items in Rasch 1PL model, with the values of difficulty coefficients with
bi = −1, 0, and 1. Both the latent trait level (
θ) and difficulty coefficient (
bi) are on the same z-score metric, with the latent trait level (
θ) typically in the range of [−2, 2] [
6].
IRT models are logistic regression models to predict dichotomous, or binary, outcomes, with a monotonically increasing S-shaped curve, called the Item Characteristic Curves (ICCs) [
8]. ICCs display the relationship between a latent trait level and the probability of correct response.
Figure 1 illustrates three ICCs of probability of correct response (
Pi (
θ)), with assumptions of three items’ difficulty parameters of −1, 0, and 1, respectively, given a range of the latent trait levels [−4, 4]. This paper will use the term latent trait and ability interchangeably in describing the IRT and its application to FTR path evaluation in the U.S. electricity market.
The ICCs in
Figure 1 may be interpreted such that an item is more difficult to the right of the chart, and easier to the left of the chart, and that low
Pi(
θ) of the item implies highly unlikeliness of providing correct responses at a given latent trait level [
7]. For example, in
Figure 1 at the level of −1 for the latent trait (
θ), the probability of correct response varies with each of the three items, 50%, 27%, and 12% as indicated on a vertical line on the
Figure 1.
While IRT is popular in the psychometrics discipline, there have been several studies on the applications of the IRT theory in the fields of health behavior research [
6,
7], as well as in financial literacy measurement [
9]. The Two-Parameter Logistic model (2PL) is an extension of the Rasch 1PL model, with additional parameter of an item’s discrimination parameter. Item discrimination represents how well an item differentiates among respondents on the latent trait continuum, e.g., differentiating respondents of different ability levels. The IRT 2PL model and ICCs will be discussed in greater detail later in this study.
In summary, the IRT 2PL model provides analytical advantages in terms of parameter parsimony, easier parameter interpretation, distinguishability among multiple items, and visual presentation. The parameters, difficulty (bi) and differentiation (ai), may be derived from historical data observation, and the magnitude of the parameters may be used to easily interpret how difficult or different each item is relative to other items. The parameters also provide a foundation to build the probability function of a right response (Pi (θ) or Pi(Xi = 1|θ) in visual presentation of ICCs on a level of latent trait variable (θ). When there are multiple opportunities available in the marketplace, essential decision-making factors involve estimation and comparison of their return and risk profiles. With such capability, IRT may be applied to FTR markets, where the astonishing number of paths are available, 1.4 million paths in U.S. New England ISO alone, and a consistent and standardized evaluation model is required for FTR participants to understand return and risk profiles of path they are interested in.
This paper is the first experiment to apply IRT, particularly the IRT 2PL model, to the U.S. energy market, in evaluating and selecting the FTR paths to bid in market auctions. This paper is organized as follows:
Section 2. Literature Review;
Section 3. Data and Methodology;
Section 4. Results and Discussion; and
Section 5. Conclusion and Implications.
3. Data and Methodology
3.1. Data
Historical electricity price data is collected from the pricing reports of ISO New England Inc. (
https://www.iso-ne.com/isoexpress/web/reports/pricing/-/tree/zone-info), where historical Standard Market Design (SMD) hourly data files are available for LMP by month and year. This paper used a file of 2019 SMD Hourly Data in the reports for analysis.
Each spreadsheet in the file has price information for eight zones and one hub, including hourly LMPs, the components of energy, congestion cost, and marginal loss, as well as hourly demand and weather. This research will use the DAM congestion costs because an FTR value is settled at the DAM. Besides, hourly prices will be broken down to two categories, onpeak and offpeak. Analysis focus will be on daily onpeak prices only, with onpeak hours being from hours ending 0800 to 2300 during weekdays, while offpeak hours include hours ending 2400 to 0700 during weekdays and all 24 h during weekends and NERC holidays.
3.2. Methodology
ISO New England has one internal hub and eight customer load zones in its six member states. The trading hub, called Massachusetts Hub (MassHub), is actively traded at electricity futures markets. Market hedging participants are interested in the FTRs between the MassHub, a liquid pricing point, and a customer zone that they are obligated to serve, while speculators look to capture any arbitrage opportunities associated with active pricing locations. In this context, this creates eight FTR paths, with Massachusetts Hub as a source pricing point and each of the eight customer zones as a sink pricing point. The number of Massachusetts Hub-related FTRs awarded in 2020 annual auction was significant at 276 (as a source), or 8% of total paths, and at 165 (as a sink), while the number for April 2020 auction was 332 (as a source), or 2% of total, and 313 (as a sink) (
https://www.iso-ne.com/isoexpress/web/reports/auctions/-/tree/auction-results-ftr).
The eight customer load zones for analysis designated as sink points, are: northeastern Massachusetts (NEMA), Vermont (VT), New Hampshire (NH), Maine (ME), Rhode Island (RI), Southeast Massachusetts (SEMA), Connecticut (CT), and West/Central Massachusetts (WCMA), with Mass Hub being a source point.
Figure 4 presents a map of the eight load zones, and hourly day-ahead LMP prices for the zones, Mass Hub, and import interface locations. (Sources:
https://www.iso-ne.com/about/key-stats/maps-and-diagrams/ and
https://www.iso-ne.com/isoexpress/web/charts).
The R-software package was used to build the IRT model, and the analysis package is “ltm” package, or Latent Trait Models under IRT Analyses, dated 17 April 2018. The detail and functions are available on
https://cran.r-project.org/web/packages/ltm/ltm.pdf.
In this study, an item (i), or a variable in IRT, is an FTR path for analysis to evaluate the profitability and risk levels of the FTR path relative to other candidate paths. Hourly FTR path values for each path during each of onpeak hours are first calculated as the difference between congestion costs at Massachusetts Hub and each of customer delivery zones, and then summarized as average FTR values for each of the onpeak days in 2019. When the average daily FTR values on each path is positive, it is coded as 1, else 0, to build a dichotomous variable as an item response (Xi).
The binary FTR value (Xi) of 1 in this analysis represents positive profitability of an FTR path on an onpeak day. Analysis will begin with data summary statistics of the eight FTR paths in binary format (0, 1), or from Mass Hub to each customer load zone. Each onpeak date is treated as an individual respondent identifier (ID) in this analysis. In 2019, there were 255 onpeak days, resulting in 255 IDs, and daily binary items (FTR profitability) of all the eight FTR paths may be used to derive a latent trait variable (ability, θ). A latent trait (ability) will be labeled as congestion ability.
The IRT 2PL model, as specified in Equation (2), is used to estimate difficulty coefficient (bi), discrimination coefficient (ai), as well as to build the ICCs. The ICC, that is, Pi (θ) or Pi(Xi = 1|θ) derived from the Equation, will provide visual comparisons of path profitability and risk levels, based on the parameters (bi, ai) and congestion ability level (θ) in the range of [−4, 4].
Item difficulty coefficient (bi) will represent profitability in this paper for each FTR path, and be interpreted that the greater the coefficient, the lower profitability, or vice versa. The item discrimination coefficient (ai) will represent the risk level of an FTR path, translating into the level of differentiating among candidate FTR paths on different congestion levels. The greater the discrimination coefficient for an FTR path, the wider the FTR value distribution is, indicating riskiness itself of an FTR path. The two parameters will be compared among the paths, to evaluate and select bidding paths in FTR auctions. Congestion ability (θ) refers to an underlying latent trait measured by daily responses to each FTR path, and it may be interpreted as a daily congestion ability. The parameter would indicate how often congested transmission situation took place in daily electricity market, potentially resulting in creating more congestion cost difference or values on FTR paths.
5. Conclusion and Implications
Financial Transmission Rights (FTR) is an energy derivative and financial instrument in electricity markets. Transmission system constraints are one of the risks for market participants, called congestion cost risk as part of LMP, particularly for those with supply obligations to serve customer loads from a different pricing location. The U.S. New England ISO, with six members states, has about 1200 pricing nodes, including load zones, hubs, and generating plant nodes. The number of the pricing nodes may be translated into 1.4 million potential FTR paths on prevailing flows only, which created tremendous opportunities for FTR bidders, as well as challenges of making decisions which paths to bid in FTR auctions. It is essential that FTR participants have a standardized and consistent model to evaluate those paths, given the complexity and magnitude of FTR path choices available to them.
Item response theory (IRT), a popular statistical model in psychometrics, measures information of an item (e.g., item difficulty and item discrimination) with a latent variable estimated by all the item responses. This paper examined a way to apply the IRT 2PL model to evaluate and select the FTR paths to bid in market auctions, with a historical price data for eight candidate paths in 2019 in U.S. New England. The parameters in IRT were defined in a way that an FTR path is an item, the FTR value (binary) for an item response, the difficulty parameter for path profitability, and the discrimination for its risk level.
This study selected eight FTR paths, with Massachusetts Hub as a source and eight load zones in six states as a sink, with several steps of evaluation. Balancing FTR profitability and risk level was prudently considered in the whole process of applying the IRT model. For each of the eight paths, the IRT 2PL model produced difficulty parameter (bi) for FTR path profitability and discrimination (ai) for risk level. As a first step, significance of the parameters for the candidate paths was calculated based on a p-value hurdle of 0.05. Two paths of MassHub_SEMA and MassHub_RI were removed for further consideration because their p-values of both profitability (bi) and discrimination (ai) were greater than 0.05. As a result, six remaining FTR paths were selected for second evaluation process, that is, ranking the paths based on two criteria of profitability (bi) and discrimination (ai). The two rankings of each path were finally evaluated with weighting factor of 60% and 40% each on profitability and discrimination, resulting in priority order among the paths. Results show that FTR paths of Mass_Maine and Mass_NEMA took the top two spots, followed by Mass_NH, Mass_WCMA, Mass_VT, and Mass_CT.
This experiment shows that the IRT model may provide a standardized analytical framework, with three parameters, in the evaluation of FTR paths, and may be implemented to address the path choice challenges for FTR participants. The model could also be useful and applicable in other energy markets, with proper definitions of terms for analysis and interpretation of estimated parameters. In this study, the lowest item difficulty coefficient of an FTR path was interpreted as a greatest profitability path, and the greatest item discrimination coefficients, or the steepest slope on the ICCs, as the riskiest path.
We note that there are a couple of limitations in this research. One limitation is that it has not developed rigorous interpretation and utilization of the latent trait variable (θ), or congestion ability, measured by the FTR path item (binary) responses (Xi), in the path evaluation process. Another limitation is that the paper focused on evaluation and selection aspect of FTR paths to bid under the analytical framework of item response theory (IRT). Future research may include topics of how to utilize the latent trait level (congestion ability) in the FTR path evaluation, and how to determine FTR bid prices for the auction under uncertainty of electric prices in future. Another extension of IRT applications may involve using more FTR paths for analysis, longer time horizons across multi-years, and testing results with other future time periods.