*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).

**Figure 4.** Eight load zonal map (**left**); and hourly day-ahead LMP Prices for zones, hub, and import interface locations (**right**). Reprinted with permission from ISO New England.

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 (*<sup>a</sup>i*), 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, <sup>a</sup>i*) 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 (*<sup>a</sup>i*) 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.

### **4. Results and Discussion**

### *4.1. Summary Statistics for Eight FTR Paths*

Table 1 summarizes proportions of positive profits, or FTR profitability days, for eight FTR paths, from the Mass Hub to each customer load zone, during 255 onpeak days in 2019. Among the eight FTR paths, two FTR paths of MassHub\_RI (0.2392 and MassHub\_SEMA (0.2235) took the lead in FTR profitability days, with FTR paths of MassHub\_VT (0.0392) and MassHub\_WCMA (0.0431) at the bottom. Notably, standard deviations were also the greatest on the top two FTR paths with about 0.42, while the rest of FTR paths stayed in the range of 0.20 to 0.28. In other words, sink points in Rhode Island and Southeast Massachusetts, sourcing electricity from Massachusetts Hub, recorded profits at 23.9% and 22.4% of onpeak days in 2019, while the profitability distribution is wider than other zones, as indicated by their higher standard deviations.


**Table 1.** Summary statistics of profitable onpeak days for FTR path in 2019.

### *4.2. IRT 2PL Model Results*

Table 2 presents a summary of estimated coefficients of FTR profitability (difficulty, *bi*) and risk level (discrimination, *<sup>a</sup>i*), estimated by the IRT 2PL model. As a smaller difficulty coefficient on an FTR path represents the greater probability of profitability, the FTR paths, MassHub\_RI (0.56) and MassHub\_SEMA (0.61), showed the highest profitability among the eight candidate paths, with the least profitable paths of MassHub\_VT (3.67) and MassHub\_WCMA (3.32). These profitability results are consistent with the results of proportions of positive profits, as shown in Table 1. Item discrimination coefficient, a measure of risk level in this paper, is also the greatest on the two FTR paths, MassHub\_RI (26.80) and MassHub\_SEMA (26.25), implying higher risks than other paths. The two paths with the smallest discrimination coefficients were MassHub\_VT (0.92) and MassHub\_WCMA (1.00), implying the least risks among all the candidate paths.


**Table 2.** 2PL model—path difficulty (profitability) and discrimination (risk) coefficients.

\* Statistically not significant at the 0.05 level. \*

The two zones of Rhode Island and Southeast Massachusetts, as sink points from Massachusetts Hub, recorded the highest profitability, and at the same time, the highest risk profiles, representing high-return, high-risk opportunities. On the other hand, the least risky sink zones, Vermont and West/Central Massachusetts, from Massachusetts Hub, did not necessarily display the highest profitability.

When parameters of difficulty (profitability, b*i*) and discrimination (risk level, <sup>a</sup>*i*) are estimated from IRT 2PL model, the estimates need to be tested if they are different from 0. This research designed the priority ranking system to evaluate FTR paths, after accounting for statistical significance of the two parameter estimates, FTR profitability (b*i*), and FTR path risk level (a*i*).

First, the p-value criteria of statistical significance at the 0.05 level, for difficulty and discrimination coefficients are compared. Two FTR paths, MassHub\_RI and MassHub\_SEMA, are not statistically significant at the 0.05 level, meaning that the difficult and discrimination estimates could be unreliably zero, and be excluded for further evaluation. Table 3 presents a summary of evaluation processes, after exclusion of two insignificant paths, to determine bidding priority among FTR paths.


**Table 3.** Evaluation processes to determine bidding priority among FTR paths.

 Excluded for further evaluation, since the coefficients are not statistically significant at the 0.05 level.

The second step is ranking the remaining six FTR paths with the difficulty and discrimination coefficients. The FTR paths are ranked by two individual categories, in the ascending order of the two individual coefficients, since the lower difficulty of an FTR path stands for higher profitability and lower discrimination for less risk level, as shown columns of "rank by (b*i*) and rank by (a*i*)" in Table 3. As an FTR auction bidder, more profitable and less risky paths are favorable target paths to bid on.

The third and last step is to obtain weighted scores of the two ranks of each path, in this example, 60% for profitability (difficulty, *bi*) and 40% for risk level (discrimination, *<sup>a</sup>i*), laying the foundation to determine FTR bidding priority for the candidate paths. As a result, the FTR bidding priority is set up in the ranking order of the paths: MassHub\_ME, MassHub\_NEMA, MassHub\_NH, MassHub\_WCMA, MassHub\_VT, and MassHub\_CT.

### *4.3. Item (Path) Characteristic Curves (ICCs)*

The FTR path profitability (item difficulty, *bi*) and risk level (discrimination, *<sup>a</sup>i*) in Tables 2 and 3 may be translated and illustrated on graphical forms (ICCs), representing the probability of profitability (*Pi* (θ)) on the y-axis on a given congestion ability (θ) on the x-axis. Figure 5 presents ICCs for each of the eight FTR paths for discussion purpose, where FTR paths, MassHub\_RI (*<sup>a</sup>i* = 26.80) and MassHub\_SEMA (*<sup>a</sup>i* = 26.25), show steep slopes, indicating high risk levels, due to their higher discrimination coefficients. FTR paths, MassHub\_VT (*<sup>a</sup>i* = 0.92) and MassHub\_WCMA (*<sup>a</sup>i* = 1.00), show the smallest slopes, indicating the least discrimination, or the least risks among all the FTR paths.

As the item (path) difficulty (b*i*) coefficients represent a scale on the x-axis of a latent trait variable (congestion ability θ) at a mid-probability (0.50) on the y-axis, their implication is that the further right an FTR path is, the less profitable (more difficult) it is. For example, the two FTR paths with the greatest difficulty coefficients are MassHub\_VT (3.67) and MassHub\_WCMA (3.32), as shown in Tables 2 and 3, and Figure 5.

**Figure 5.** Individual item characteristic curves: eight 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 di fficulty 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 di fficulty 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 di fficulty parameter (b*i*) for FTR path profitability and discrimination (a*i*) 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 (b*i*) and discrimination (a*i*) 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 (b*i*) and discrimination (a*i*). 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 di fficulty coe fficient of an FTR path was interpreted as a greatest profitability path, and the greatest item discrimination coe fficients, 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.

**Author Contributions:** Conceptualization, P.Y.J.; Data curation, P.Y.J.; Formal analysis, P.Y.J.; Investigation, P.Y.J.; Methodology, P.Y.J. and K.J.; Project administration, K.J. and M.G.B.; Resources, P.Y.J.; Software, P.Y.J.; Supervision, K.J. and M.G.B.; Validation, P.Y.J., K.J. and M.G.B.; Visualization, P.Y.J.; Writing – original draft, P.Y.J.; Writing – review & editing, P.Y.J., K.J. and M.G.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
