## Abstract

Utility companies are interested in understanding their consumers’ behavior in response to dynamic pricing to help them devise effective demand response (DR) programs in a smart grid. In this paper, we propose a bi-level optimization model to estimate the coefficients of utility functions associated with price-responsive electricity consumers. One coefficient represents the utility of consuming energy while the other presents the price-responsiveness. The upper level problem, in some form of inverse optimization, determines the optimal utility coefficients for individual homes by minimizing the error between estimated and observed electricity consumption. The lower level problem describes individual homes’ utility maximization behavior in electricity consumption when faced with dynamic pricing in DR programs. We develop a trust region algorithm to solve the bi-level program after introducing a cut for the mixed integer program reformulation of the problem. Numerical experiments with real-world data collected from a field demonstration DR project in a midwestern US municipality demonstrate the effectiveness of the proposed bi-level model and the efficiency of the re-formulation and the proposed trust region algorithm.

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to check access.## References

Altman, N.S.: An introduction to kernel and nearest-neighbor nonparametric regression. Am. Stat.

**46**(3), 175–185 (1992)Boßmann, T., Eser, E.J.: Model-based assessment of demand-response measures—a comprehensive literature review. Renew. Sustain. Energy Rev.

**57**, 1637–1656 (2016)Breiman, L.: Random forests. Mach. Learn.

**45**(1), 5–32 (2001)Breiman, L.: Classification and Regression Trees. Routledge, London (2017)

Colson, B., Marcotte, P., Savard, G.: A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput. Optim. Appl.

**30**(3), 211–227 (2005)Conejo, A.J., Morales, J.M., Baringo, L.: Real-time demand response model. IEEE Trans. Smart Grid

**1**(3), 236–242 (2010)Edmunds, T.A., Bard, J.F.: Algorithms for nonlinear bilevel mathematical programs. IEEE Trans. Syst. Man Cybern.

**21**(1), 83–89 (1991)Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput.

**13**(5), 1194–1217 (1992)Li, N., Chen, L., Low, S.H.: Optimal demand response based on utility maximization in power networks. In: Power and Energy Society General Meeting, pp. 1–8. IEEE (2011)

Liu, G., Han, J., Wang, S.: A trust region algorithm for bilevel programing problems. Chin. Sci. Bull.

**43**(10), 820–824 (1998)Mohsenian-Rad, A.H., Leon-Garcia, A.: Optimal residential load control with price prediction in real-time electricity pricing environments. IEEE Trans. Smart Grid

**1**(2), 120–133 (2010)Ratliff, L.J., Dong, R., Ohlsson, H., Sastry, S.S.: Incentive design and utility learning via energy disaggregation. IFAC Proc. Vol.

**47**(3), 3158–3163 (2014)Saez-Gallego, J., Morales, J.M., Zugno, M., Madsen, H.: A data-driven bidding model for a cluster of price-responsive consumers of electricity. IEEE Trans. Power Syst.

**31**(6), 5001–5011 (2016)Samadi, P., Mohsenian-Rad, A.H., Schober, R., Wong, V.W., Jatskevich, J. Optimal real-time pricing algorithm based on utility maximization for smart grid. In: Smart Grid Communications (SmartGridComm), 2010 First IEEE International Conference on, IEEE, pp. 415–420 (2010)

Savard, G., Gauvin, J.: The steepest descent direction for the nonlinear bilevel programming problem. Oper. Res. Lett.

**15**(5), 265–272 (1994)Schweppe, F.C., Caramanis, M.C., Tabors, R.D., Bohn, R.E.: Spot Pricing of Electricity. Springer Science & Business Media, Berlin (2013)

Thoai, N., Yamamoto, Y., Yoshise, A.: Global optimization method for solving mathematical programs with linear complementarity constraints. J. Optim. Theory Appl.

**124**(2), 467–490 (2005)Vicente, L., Savard, G., Judice, J.: Discrete linear bilevel programming problem. J. Optim. Theory Appl.

**89**(3), 597–614 (1996)Zugno, M., Morales, J.M., Pinson, P., Madsen, H.: A bilevel model for electricity retailers’ participation in a demand response market environment. Energy Econ.

**36**, 182–197 (2013)

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## Appendix A: Proof of Theorem 1

### Appendix A: Proof of Theorem 1

### Theorem 2

Let \(z^* = \left( z_d^1,\ldots ,z_d^T \right) ^T\) be the optimum curtailed load for a given consumer for problem \(LL_d\), then \(z^*\) must satisfy equation (14),

where \(f_d^t\) is the predicted consumption at time *t* on day *d*.

### Proof

Without loss of generality, consider the lower-level problem \(LL_d\) for a single day i.e. \(d = 1\) and hence omit *d* in the rest of this proof. Let \(y_t=f^t - z^t\), the lower-level problem can be rewritten as follows:

Letting \(\delta\) and \(\lambda _t\) be the lagrangian multipliers for the two constraints in (15), respectively, yields the following Krush-Kuhn-Tucker (KKT) conditions at optimality:

Adding all *T* (16c) equations, one obtains

Note that by subtracting equation (16d) from (16e), we have

Therefore, using (18) and (17), we obtain \(T u^*_1 - u^*_2\sum _{t=1}^{T} y^*_t\ge 0\), or, \(-T u_1^* + u_2^*\sum _{t=1}^{T}f^t-u_2\sum _{t=1}^{T}{z_t^*}\le 0.\) \(\square\)

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Roy, A., Bai, L. Estimating utilities of price-responsive electricity consumers.
*Energy Syst* **14**, 893–912 (2023). https://doi.org/10.1007/s12667-021-00496-y

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DOI: https://doi.org/10.1007/s12667-021-00496-y