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Estimating utilities of price-responsive electricity consumers

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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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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),

$$\begin{aligned} -u_1 T + u_2\sum _{t=1}^{T}f_d^t - u_2\sum _{t=1}^{T}{z^*}_d^t \le 0, \forall d, \end{aligned}$$

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


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:

$$\begin{aligned} \min \quad&\sum _{t=1}^{T}{(\rho ^t - u_1)y_t+\frac{1}{2}u_2{y_t}^2}\nonumber \\ \text {s.t.}\quad&\sum _{t=1}^{T} y_t \ge 0 , \nonumber \\ \quad&y_t \le f^t, \ \forall t. \end{aligned} $$

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:

$$\begin{aligned} \sum _{t=1}^{T} y^*_t \ge 0&\end{aligned}$$
$$\begin{aligned} y^*_t \le f^t, \ \forall t&\end{aligned}$$
$$\begin{aligned} (\rho ^t - u_1) + u_2y^*_t - \delta ^* + \lambda _t^* = 0, \quad&\forall t \end{aligned}$$
$$\begin{aligned} \delta ^*\sum _{t=1}^{T}y^*_t = 0 \quad&\end{aligned}$$
$$\begin{aligned} \lambda ^*_t(y^*_t - f^t)=0, \quad&\forall t \quad&\end{aligned}$$
$$\begin{aligned} \delta ^* \ge 0, \quad&\end{aligned}$$
$$\begin{aligned} \lambda _t^* \ge 0, \quad&\forall t\quad&\end{aligned}$$

Adding all T (16c) equations, one obtains

$$\begin{aligned} Tu_1 - u_2\sum _{t=1}^Ty^*_t = \sum _{t=1}^T\rho ^t + T\left(\sum _{t=1}^T\lambda ^*_t - \delta ^*\right). \end{aligned}$$

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

$$\begin{aligned} \sum _{t=1}^T\lambda ^*_t - \delta ^* \ge 0. \end{aligned}$$

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).

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