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Market Power and Efficiency Analysis in Bi-level Energy Transmission Market

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Abstract

We study a bi-level energy market and a transmission system that sends the energy supply generated in the upstream market to consumers in the downstream market. The players including manufacturers and wholesalers in this market can exercise the market power to affect the productive efficiency of the energy supply. To analyze the impact of market power, we formulate the mixed complementarity problems to identify the Nash equilibrium and use a directional distance function to estimate the efficiency by the directional vector toward the Nash equilibrium. We investigate two cases of market power in the energy market: wholesalers as leaders, or bilateral bargaining power between manufacturers and wholesalers. The numerical study of a power generation market is conducted and the results show that the Nash equilibriums in the two cases provide the insights to drive the productivity and support environmental policies. Particularly, a sensitivity analysis shows that price volatility in the upstream transmission market significantly influences the equilibrium solution.

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Data Availability

The data that support the findings of this study are available from China Electric Power Yearbook (2014).

Notes

  1. Note that a bi-level market defined in the current study does not directly imply a leader–follower game, and it describes an upstream–downstream relation in an energy transmission market.

  2. This study focuses on model generalization and productivity, and does not consider the wholesalers’ geographical constraints related to energy transmission.

  3. Some assumptions in this numerical study is not be totally valid as real settings in China, such as imperfectly competitive electricity market or one aggregated demand formulated by inverse demand function. However, these assumptions can also be regarded as a scenario analysis due to deregulation in the power market [11, 10, 17]. This assumption can provide some interesting factors and insights.

  4. The numerical study was performed on a personal computer (PC) with an Intel Core i7-4770 CPU @ 3.40 GHz and 8 GB RAM. The optimization problems were solved in GAMS 23.3 using the PATH solver [8]. The typical CPU computation time ranged from 5 to 20 s.

  5. For one specific power plant, we average the contract price by all TSOs, and for one specific TSO, we average the contract price by all power plants.

  6. Note that for efficiency estimation of TSOs, Eq. (2) is corrected by the production possibility set \(T\) as the typical variable-returns-to-scale (VRS) DEA model without undesirable outputs.

  7. We calculate the normalized efficiency score by \(1 - \theta^{*} /\theta^{\max }\), where \(\theta^{*}\) is the DDF and \(\theta^{\max }\) is the largest DDF used for normalization. A nonnegative DDF \(\theta^{*} = 0\) means a firm is efficient; otherwise, it is inefficient. After normalization, the efficiency scores are comparable among four models.

  8. In Case 1 and Case 2, it is possible to have no DMU with efficiency score equal to 1; all DDFs estimated by Nash direction are larger than 0 (e.g., although Plant 10 is located on the efficient frontier according to Chung and Färe, it can still move toward the Nash equilibrium located on the efficient frontier for improvement.

  9. We only discuss a situation where the parameters have the same proportional change for each contract price setting between a plant and a TSO, to understand the impact on the Nash equilibrium solution.

Abbreviations

\(i\) :

Index of input, \(i \in I\)

\(q\) :

Index of output, \(q \in Q\)

\(k\) :

Index of manufacturer, \(k \in K\)

\(r\) :

Index for one specific manufacturer and an alias of index \(k\), \(r \in K\)

\(h\) :

Index of wholesaler, \(h \in H\)

\(n\) :

Index for one specific wholesaler and an alias of index \(h\), \(n \in H\)

\(T\) :

Production possibility set

\(\lambda_{k}\) :

The intensity weights of the convex combination between players

\(\mu_{k}\) :

The variable for the weak disposability property of Podinovski’s convex technology

\(\theta\) :

The decision variable representing the estimate of efficiency

\(p_{kh}^{C}\) :

Contract price negotiated between manufacturer and wholesaler in the upstream market

\(p_{{}}^{S}\) :

Wholesale price as an inverse demand function with a single aggregated consumer

\(I_{kh}^{T}\) :

The purchased quantity transmitted from manufacturer \(k\) into wholesaler \(h\)

\(\lambda_{rk}^{C}\) :

The intensity weights of the convex combination between manufacturers

\(\mu_{rk}^{C}\) :

The weak disposability of Podinovski’s convex technology between manufacturers

\(O_{rh}^{C}\) :

The amount of energy supply transmitted to wholesaler \(h\) from manufacturer \(r\)

\(O_{h}^{T}\) :

The amount of energy supply transmitted from wholesaler \(h\) to customers

\(\lambda_{nh}^{T}\) :

The intensity weights of the convex combination between wholesalers

\(\tau_{ri}^{C} ,\varphi_{r}^{C} ,\gamma_{rq}^{C} ,\delta_{r}^{C} ,\varepsilon_{r}^{C}\) :

Lagrange multipliers corresponding to Eqs. (3.2)–(3.6)

\(\tau_{nl}^{T} ,\varphi_{n}^{T} ,\gamma_{n}^{T} ,\delta_{n}^{T} ,\varepsilon_{nl}^{T} ,\theta_{n}^{T} ,\rho^{T} ,\omega_{n}^{T} ,\sigma_{kn}^{T}\) :

Lagrange multipliers related to Eqs. (3.9)–(3.17)

\(X_{ki}^{{}}\) :

The \(i\)th input of firm \(k\)

\(Y_{k}^{{}}\) :

The single desirable output of firm \(k\)

\(B_{kq}^{{}}\) :

The \(q\)th undesirable output of firm \(k\)

\({\varvec{g}}\) :

The direction vector \({\varvec{g}} = \left( {{\varvec{g}}_{x} ,g_{y} ,{\varvec{g}}_{b} } \right)\) used in directional distance function

\(X_{ki}^{C}\) :

The \(i\)th input of manufacturer \(k\)

\(Y_{k}^{C}\) :

The single desirable output of manufacturer \(k\)

\(B_{kq}^{C}\) :

The \(q\)th undesirable output level of manufacturer \(k\)

\(\alpha_{kh}^{C}\) :

Positive intercept of contract price equation \(p_{kh}^{C} = \alpha_{kh}^{C} - \beta_{kh}^{C} I_{kh}^{T}\)

\(\beta_{kh}^{C}\) :

Discount parameter of contract price equation \(p_{kh}^{C} = \alpha_{kh}^{C} - \beta_{kh}^{C} I_{kh}^{T}\)

\(w_{i}\) :

The price of input \(i\)

\(\alpha^{S}\) :

Positive intercept of inverse demand function \(p^{S} = \alpha^{S} - \beta^{S} \hat{O}^{T}\)

\(\beta^{S}\) :

Price-sensitive coefficient of the energy supply in the inverse demand function

\(\overline{O}^{T}\) :

The least and fixed amounts generated by the wholesalers without market power

\(X_{hl}^{T}\) :

The \(l\)th input level of wholesaler \(h\)

\(Y_{h}^{T}\) :

The single output (energy supply) level of wholesaler \(h\)

\(T_{n}\) :

The upper bound of the transmission line’s capacity from wholesaler \(n\) to customers

\(D\) :

The customer demand

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Acknowledgements

This study was funded by the Ministry of Science and Technology of Taiwan (MOST108-2221-E-006 -223-MY3).

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Authors and Affiliations

  1. Department of Information Management, National Taiwan University, Taipei, 106, Taiwan

    Chia-Yen Lee

  2. Institute of Manufacturing Information and Systems, National Cheng Kung University, Tainan City, 701, Taiwan

    Chin-Yi Tseng

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Correspondence to Chia-Yen Lee.

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Communicated by Arvind Raghunathan.

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Lee, CY., Tseng, CY. Market Power and Efficiency Analysis in Bi-level Energy Transmission Market. J Optim Theory Appl 196, 544–569 (2023). https://doi.org/10.1007/s10957-022-02147-3

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