Risk-averse and strategic prosumers: A distributionally robust chance-constrained MPEC approach

Abstract

Under the ruling of FERC Order 2222, prosumers who own distributed energy resources are expected to play an essential role that can significantly affect the market outcomes. This paper assesses the market power potential of a risk-averse prosumer by designating it as a Stackelberg leader in the market. We formulate the problem as a mathematical program with equilibrium constraints with a distributionally robust chance-constrained framework to account for the renewable generation uncertainty. The numerical results demonstrate that the prosumer’s strategy depends on the magnitude of renewable generation uncertainty and the degree of risk aversion, which jointly affect the perceived quantity of its renewable resources. The risk-averse Stackelberg case always yields a higher payoff for the prosumer compared to the Cournot and perfect competition cases. Moreover, it is more potent for the prosumer to exercise buyer’s market power rather than seller’s market power. The situation worsens when the prosumer is less risk averse. This highlights the importance of understanding the role of the prosumer acting either as a consumer or as a producer and the risk faced by the prosumer when evaluating its interaction with the wholesale market.

Appendices

Appendix A: Wolfe duality of lower-level problem

Since the lower-level problem is a concave program, we can obtain the optimal solution by solving its dual. Particularly, if we have a general concave program $\textrm{max}_{x} \{f(x):g(x)\le 0\}$, the corresponding Wolfe dual is $\textrm{min}_{x,\lambda }\{{\mathcal {L}}(x,\lambda ): \nabla _x {\mathcal {L}}(x,\lambda ) = 0, \lambda \ge 0\}$, where $\nabla _x {\mathcal {L}}(x,\lambda )$ are the gradients of the Lagrangian ${\mathcal {L}}(x,\lambda ) = f(x) - \lambda ^{\top } g(x)$. For a concave (or convex) programming problem, strong duality holds between the primal and Wolfe dual problems. Consequently, the Wolfe dual of the social welfare maximization problem in the lower level is:

$$\begin{aligned}&\min _{\Omega \cup \Lambda } \qquad \sum _i B_i(d_i) - \sum _{f,i,h \in H_{fi}}C_{fih}(x_{fih}) \end{aligned}$$

(A.1a)

$$\begin{aligned}&- \sum _{f,i,h \in H_{fi}}\beta _{fih}(x_{fih} - X_{fih}) -\sum _k \lambda _k^+\left(\sum _i PTDF_{ki}-T_k\right) \nonumber \\&-\sum _k\lambda _k^-\left(\sum _i -PTDF_{ki}y_i -T_k\right) - \sum _i \eta _i d_i + \sum _{i} \eta _i z_{i} \nonumber \\&+ \sum _{f,i,h \in H_{fi}} \eta _i x_{fih} + \sum _i \eta _i y_i - \theta \sum _i y_i + \sum _{f,i,h \in H_{fi}}\varepsilon _{fih}x_{fih} \nonumber \\&+ \sum _i \xi _i d_i \nonumber \\&\text {subject to} \nonumber \\&-C'_{fih}(x_{fih}) - \beta _{fih} + \eta _i + \varepsilon _{fih} = 0 \quad \forall f,i,h \in H_{fi} \end{aligned}$$

(A.1b)

$$\begin{aligned}&B_i'(d_i) - \eta _i + \xi _i = 0 \quad \forall i \end{aligned}$$

(A.1c)

$$\begin{aligned}&-\sum _k (\lambda _k^+ -\lambda _k^-)PTDF_{ki} + \eta _i - \theta = 0 \quad \forall i \end{aligned}$$

(A.1d)

$$\begin{aligned}&\beta _{fih}, \varepsilon _{fih} \ge 0 \quad \forall f,i,h \in H_{fi} \end{aligned}$$

(A.1e)

$$\begin{aligned}&\lambda _k^+, \lambda _k^- \ge 0 \qquad \forall k \end{aligned}$$

(A.1f)

$$\begin{aligned}&\xi _i \ge 0 \qquad \forall i \end{aligned}$$

(A.1g)

Note that non-negativity constraints are imposed on $\{\beta _{fih},\lambda ^+_k,\lambda ^-_k,\varepsilon _{fih},\xi _i\}$, which are associated with the inequality constraints in the primal problem. Otherwise, variables are unrestricted.

Given the concavity of the social welfare maximization problem and that strong duality holds, the original objective functions and (A.1a) have the same value. Thus:

$$\begin{aligned}&- \sum _{f,i,h \in H_{fi}} \beta _{fih} (x_{fig} - X_{fih}) + \sum _{f,i,h \in H_{fi}}\varepsilon _{fih} x_{fih} \nonumber \\&- \sum _k \lambda ^{+}_k \left(\sum _i PTDF_{ki}y_i - T_k\right) + \sum _i \xi _i d_i \nonumber \\&- \sum _k \lambda ^{-}_k \left(\sum _i - PTDF_{ki}y_i - T_k\right) - \theta \sum _i y_i \nonumber \\&-\sum _i \eta _i d_i + \sum _{f,i,h \in H_{fi}}\eta _i x_{fih} + \sum _{i}\eta _i z_{i} + \sum _i \eta _i y_i = 0 \end{aligned}$$

(A.2)

Using constraints (A.1b)–(A.1d), we further simplify (A.2). In particular, from (A.1b) we have:

$$\begin{aligned} \begin{aligned}&\bigl ( -C'_{fih}(x_{fih}) - \beta _{fih} + \eta _i + \varepsilon _{fih} \bigr ) x_{fih} = 0 \\&\sum _{f,i,h \in H_{fi}} ( - \beta _{fih} + \eta _i + \varepsilon _{fih} ) x_{fih} = \sum _{f,i,h \in H_{fi}} C'_{fih}(x_{fih}) x_{fih} \end{aligned} \end{aligned}$$

(A.3)

From (A.1c), we derive:

$$\begin{aligned} \begin{aligned}&\bigl ( B'_{i}(d_i) -\eta _i + \xi _i \bigr ) d_i = 0 \\&\sum _i \bigl ( -\eta _i + \xi _i \bigr ) d_i = -\sum _i B'_{i}(d_i) d_i \end{aligned} \end{aligned}$$

(A.4)

Fom (A.1d), we obtain:

$$\begin{aligned} \begin{aligned}&\Bigl ( -\sum _k (\lambda ^{+}_k - \lambda ^{-}_k)PTDF_{k} + \eta _i - \theta \Bigr ) y_i = 0 \\&\sum _i \bigl ( \eta _i - \theta \bigr ) y_i = \sum _{k,i} (\lambda ^{+}_k - \lambda ^{-}_k)PTDF_{k}y_i \end{aligned} \end{aligned}$$

(A.5)

Substituting (A.3), (A.4), and (A.5) into (A.2), we can rewrite the bilinear term $\sum _{i}\eta _i z_{i}$ as follows:

$$\begin{aligned}&\sum _{i}\eta _i z_{i} = \sum _i B'_{i}(d_i) d_i - \sum _k(\lambda ^{+}_k + \lambda ^{-}_k)T_k\nonumber \\&- \sum _{f,i,h \in H_{fi}} \bigl (C'_{fih}(x_{fih})x_{fih} + \beta _{fih}X_{fih} \bigr ) \end{aligned}$$

(A.6)

Appendix B: Concavification of objective function in MPEC

We can then substitute (A.6) into the bilinear term $\sum _{i}\eta _i z_{i}$ in the original objective function of the MPEC. Along with other constraints, the objective function of MPEC, or the Stackelberg leader formulation for the prosumer, is then rewritten as follows:

$$\begin{aligned}&\max _{\Phi \cup \Omega \cup \Lambda } \sum _i B'_{i}(d_i) d_i - \sum _k(\lambda ^{+}_k + \lambda ^{-}_k)T_k \nonumber \\&- \sum _{f,i,h \in H_{fi}} \bigl (C'_{fih}(x_{fih})x_{fih} + \beta _{fih}X_{fih} \bigr ) \nonumber \\&+ \sum _i \bigl ( B^l_i(l_i) - C^g_i(g_i) \bigr ) + {\sum _i P_i^c \Bigl (K_i-z_{i}-l_i+g_i \Bigr ) } \end{aligned}$$

(B.1)

Appendix C: MIQP reformulation

As the final step, we further remove non-convex terms caused by the complementarity conditions from the lower-level problem by utilizing disjunctive constraints. With binary variables Φ = $\{{\bar{r}}_{fih}, r^+_k, r^-_k, r_{fih},{\hat{r}}_i\}$ and positive big constants $\{M_1,M_2,M_3,M_4,M_5\}$, we reformulate the MPEC into an MIQP as follows:

$$\begin{aligned}&\max _{\Phi \cup \Omega \cup \Lambda \cup \Psi } \sum _i B'_{i}(d_i) d_i - \sum _k(\lambda ^{+}_k + \lambda ^{-}_k)T_k \end{aligned}$$

(C. 1a)

$$\begin{aligned}&- \sum _{f,i,h \in H_{fi}} \bigl (C'_{fih}(x_{fih})x_{fih} + \beta _{fih}X_{fih} \bigr ) \nonumber \\&+ \sum _i \bigl ( B^l_i(l_i) - C^g_i(g_i) \bigr ) + {\sum _i P_i^c \Bigl (K_i- z_{i}-l_i+g_i \Bigr ) } \nonumber \\&\text {subject to} \nonumber \\&\sigma _i \sqrt{\frac{1-R_i}{R_i}} +z_{i} + l_i - g_i - K_i \le 0 \qquad \forall i \end{aligned}$$

(C. 1b)

$$\begin{aligned}&g_i \le G_i \qquad \forall i \end{aligned}$$

(C. 1c)

$$\begin{aligned}&l_i,g_i \ge 0 \qquad \forall i \end{aligned}$$

(C. 1d)

$$\begin{aligned}&-C'_{fih}(x_{fih}) - \beta _{fih} + \eta _i + \varepsilon _{fih} =0 \quad \forall f, i, h \in H_{fi} \end{aligned}$$

(C. 1e)

$$\begin{aligned}&B'_{i}(d_i) -\eta _i + \xi _i =0 \qquad \forall i \end{aligned}$$

(C. 1f)

$$\begin{aligned}&-\sum _k (\lambda ^{+}_k - \lambda ^{-}_k)PTDF_{ki} + \eta _i - \theta =0 \qquad \forall i \end{aligned}$$

(C. 1g)

$$\begin{aligned}&0\le -(x_{fih}-X_{fih}) \le M_1 {\bar{r}}_{fih} \qquad \forall f, i, h \in H_{fi} \end{aligned}$$

(C. 1h)

$$\begin{aligned}&0\le \beta _{fih} \le M_1 (1-{\bar{r}}_{fih}) \qquad \forall f, i, h \in H_{fi} \end{aligned}$$

(C. 1i)

$$\begin{aligned}&d_i - \sum _{f,h \in H_{fi}}x_{fih} - z_{i} - y_i =0 \qquad \forall i \end{aligned}$$

(C. 1j)

$$\begin{aligned}&0 \le - \Bigl (\sum _i PTDF_{ki}y_{i} - T_{k} \Bigr ) \le M_2 r^{+}_{k} \qquad \forall k \end{aligned}$$

(C. 1k)

$$\begin{aligned}&0 \le \lambda _{k}^+ \le M_2 (1-r^{+}_{k}) \qquad \forall k \end{aligned}$$

(C. 1l)

$$\begin{aligned}&0 \le - \Bigl (-\sum _i PTDF_{ki}y_{i} - T_{k} \Bigr ) \le M_3 r^{-}_{k} \qquad \forall k \end{aligned}$$

(C. 1m)

$$\begin{aligned}&0 \le \lambda _{k}^- \le M_3 (1-r^{-}_{k}) \qquad \forall k \end{aligned}$$

(C. 1n)

$$\begin{aligned}&\sum _i y_i =0 \end{aligned}$$

(C. 1o)

$$\begin{aligned}&0 \le x_{fih} \le M_4 r_{fih} \qquad \forall f, i, h \in H_{fi} \end{aligned}$$

(C. 1p)

$$\begin{aligned}&0 \le \varepsilon _{fih} \le M_4 (1 - r_{fih}) \qquad \forall f, i, h \in H_{fi} \end{aligned}$$

(C. 1q)

$$\begin{aligned}&0 \le d_i \le M_5 {\hat{r}}_{i} \qquad \forall i \end{aligned}$$

(C. 1r)

$$\begin{aligned}&0 \le \xi _{i} \le M_5 (1 - {\hat{r}}_{i}) \qquad \forall i \end{aligned}$$

(C. 1s)

$$\begin{aligned}&{\bar{r}}_{fih},r^{+}_{k},r^{-}_{k},r_{fih},{\hat{r}}_i \in \{0,1\} \end{aligned}$$

(C. 1t)