# Time-of-use pricing in the electricity markets: mathematical modelling using non-linear market demand

• Theoretical Article
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## Abstract

We determine the gains in efficiency accruing to a monopolist producer facing a non-linear market demand under a time-of-use (TOU) pricing structure as opposed to a flat rate pricing (FRP) structure. In particular, we consider the constant elasticity of demand function and the exponential demand function for this analysis. We estimate the price and quantity demanded for these two types of functions and optimize the profit earned by the producer. A comparison of the linear, exponential, and constant elasticity of demand functions shows that in cases of linear and exponential demand, the TOU pricing works to reduce the peak demand below the installed capacity and saves on additional investment and operation costs, while no such reduction takes place in the case of constant elasticity of demand. However, profit accruing to the monopolist under the TOU pricing structure exceeds that under FRP, irrespective of the form of the demand function. Thus, we conclude that regardless of the shape of the demand function, a time-varying pricing structure is better than the traditional FRP. Finally, we study some implications for the policy maker if such a pricing structure is implemented.

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## Notes

1. Detailed solution available on request.

2. Detailed Proof available on request.

3. The value of p obtained after running the C program is 8.15 in contrast to the optimal value of 8.99 obtained through AMPL. Thus, the N-R method does not provide an optimal solution to the NLP problem but for the sake of theoretical completeness, it is applied to solve the non-linear equation in (8) that cannot otherwise be solved by usual mathematical methods. When the problem is fed into AMPL, the optimal solution of the non-linear programming problem is obtained, nevertheless.

4. The data on power generation and operating costs have been obtained by modelling a 500 MW generator in a power plant in central India. Each day has been divided into 9 h of off-peak, 8 h of shoulder, and 7 h of peak period, as assumed by Cellibi and Fuller (2001)..

## References

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Correspondence to Nidhi Kaicker.

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Kaicker, N., Dutta, G. & Mishra, A. Time-of-use pricing in the electricity markets: mathematical modelling using non-linear market demand. OPSEARCH 59, 1178–1213 (2022). https://doi.org/10.1007/s12597-021-00564-y