A New Method To Find The Nash Equilibrium Point in Financial Transmission Rights Bidding Problem

Financial transmission right (FTR) is an important tool and an especially feature for stopping congestion charges in restructured electricity markets. Participants in the transmission market as players are assumed to be a generation company (Gencos) which also take part in an energy market and able to buy their require FTRs. In this regard, there are two types of FTR: obligation or option. There are three main questions which immediately arise for each player who is placed in the market. First, which type of FTR is the best choice second, how much power is needed to generate by each player and third, how bid prices should be offered. Deciding on these trade-offs is difficult and requires definition of special matrices to measure risk in each possible condition in the transmission market. These matrices include: possibility of flow direction alteration, probable forward and reverse power flow on each line, maximum and minimum offering FTRs and the worst condition of load variation which influence on each players decision. Based on these matrices, players try to maximize their expected payoffs by taking into account the associated risks. Supposing these matrices are known to respective players, the FTR bidding problem is modeled as a bi-level optimization based on the Nash equilibrium game theory with the upper sub-problem representing player profit maximization and the lower sub-problem representing the optimal solution to the market clearing. An eight-bus system with six players is simulated to verify the proposed method and the obtained results are illustrated the complex interaction between FTR obligation and FTR option bidding strategies. Furthermore, the results are demonstrated to be consistent between the impacts of FTR type, forecast bid offer of the other players and players preferred risk levels on FTR bidding strategies.

💡 Research Summary

The paper addresses the problem of how generation companies (GenCos) participating simultaneously in energy and transmission markets should decide on three intertwined decisions: (i) whether to acquire financial transmission rights (FTRs) as obligations or options, (ii) how much power to secure through those rights, and (iii) what bid price to submit. Recognizing that FTRs are used to hedge congestion charges, the authors note that obligations expose the holder to both positive and negative cash flows depending on the direction of the LMP (locational marginal price) spread, whereas options guarantee non‑negative cash flow at the cost of a higher premium.

To capture the risk inherent in these decisions, the authors introduce three novel matrices: (1) the probability of flow‑direction alteration, (2) forward‑potential flow and reverse‑potential flow coefficients that quantify expected power movement on each line under load uncertainty, and (3) a worst‑case load‑variation probability distribution. These matrices are derived from sensitivity of line flows to generator dispatch, load probability distributions, and the interaction between load changes and transmission constraints. The forward‑potential flow (α⁺) reflects the expected increase in the original flow direction, while the reverse‑potential flow (α⁻) captures the expected magnitude of a possible reversal. Using these coefficients, the paper defines two “chance” parameters, γ⁺ (probability that flow direction is maintained) and γ⁻ (probability of reversal).

With these risk metrics, the expected profit for an obligation‑type FTR is expressed as γ⁺·ΔLMP·q – γ⁻·ΔLMP·q, where ΔLMP is the price difference between source and sink nodes and q is the quantity of FTR purchased. For an option‑type FTR, the expected profit simplifies to γ⁺·ΔLMP·q – price·q because the reverse‑flow loss term disappears; the only loss is the upfront payment for the option. Each GenCo also incorporates a risk‑aversion coefficient λ into its objective function, allowing the model to reflect different attitudes toward uncertainty.

The strategic interaction among GenCos is modeled as a bilevel (bi‑level) optimization problem grounded in Nash equilibrium game theory. The upper level (leader) represents each GenCo’s profit‑maximization problem, taking the lower‑level market‑clearing outcome as given. The lower level (follower) represents the Independent System Operator’s (ISO) transmission‑rights clearing problem, which must satisfy power‑flow equations, line‑capacity limits, and the simultaneous feasibility test (SFT) that ensures collected congestion revenues cover payments to FTR holders. The inclusion of option‑type FTRs makes the SFT non‑convex, as the ISO must consider the worst‑case combination of constraints.

To solve for the Nash equilibrium, the authors derive the Karush‑Kuhn‑Tucker (KKT) conditions for both levels, introduce Lagrange multipliers to link the two problems, and employ an iterative algorithm that alternates between updating GenCo strategies and re‑solving the market‑clearing sub‑problem until convergence.

The methodology is validated on an 8‑bus test system with six GenCos. Different risk‑aversion levels, forecasted LMP spreads, and load‑uncertainty scenarios are assigned to each player. Simulation results reveal several key insights:

1. When the probability of flow reversal on a line is high (γ⁻ large), GenCos tend to favor option‑type FTRs despite their higher price, because the risk of negative cash flow is mitigated.
2. Risk‑averse players (high λ) systematically select options, while risk‑neutral or risk‑seeking players prefer obligations to reduce upfront costs.
3. The optimal bid quantity and price of a player are highly sensitive to the anticipated bids of competitors; a change in another player’s forecasted offer can shift the equilibrium dramatically.
4. The proposed risk matrices provide a richer characterization of uncertainty than traditional expected‑profit models, leading to more realistic bidding strategies and improved overall market efficiency.

In conclusion, the paper contributes a comprehensive framework that integrates detailed risk quantification with game‑theoretic equilibrium analysis for FTR bidding. By explicitly modeling the trade‑off between expected profit and risk, and by solving the resulting bilevel Nash equilibrium, the authors offer a practical decision‑support tool for market participants and regulators seeking to design more robust transmission‑rights markets.