Renewable Portfolio Standards in the Presence of Green Consumers and Emissions Trading

Abstract

Greenhouse gas (GHG) emissions trading, green pricing programs and renewable portfolio standards (RPS) are three concurrent policies implemented in the United States to reduce reliance on fossil fuel and GHG emissions. Despite their differences in policy targets, they are closely related and integrated with competitive electric markets. This paper examines the interactions among these three policies by considering two aspects of the RPS policy design: double-counting and bundling. Whereas the former grants utilities using the same MWh of renewable energy to meet RPS and to sell as green power, the latter allows them to bundle the renewable energy credits/certificates (RECs) with non-renewable electricity and sell as green power. This paper studies the policy designs by formulating each policy combination as a market model, which treats electricity as a differentiated product. We derive the conditions under which the REC price serves as the upper bound of the green premium or vice versa. The theoretical analysis shows that the bundling could be redundant in the presence of double counting. The policies that allow for double-counting appear to be a better choice, since they result in a higher social surplus. Most surplus gains are due to consumers surplus from green power sales. The framework we develop in this paper is capable of incorporating other detailed policy designs in the analysis such as strategic reserve and offset.

Appendices

Appendix A: Equilibrium market models

Table 7 summarizes the corresponding sets of equilibrium conditions under each case. Notice that these are the concatenated conditions for all market participants, including consumers, LSEs, producers and the ISO. In the remainder of this section, we present the KKT conditions for each market participants: LSE, producers, and ISO.

Table 7 Summary of equilibrium market models

KKT for LSEs

Case 1

both double-counting and bundling are allowed.

$$\forall f,i,h \in H^{\rm O}_{\mathrm if}, j \leq z_{\mathrm fihj} \perp p_{\mathrm fihj} - \omega^{\rm O}_j \geq 0 \label{eq11}\\ $$

(25)

$$\forall f,i,h \in H^{\rm G}_{\mathrm if}, j \leq z_{\mathrm fihj} \perp p_{\mathrm fihj} - \omega^{\rm G}_j \geq 0 \label{eq12}\\ $$

(26)

$$\forall j \leq z_j^{\rm O} \perp \theta_j + R \phi_j + R \tau_j \geq 0 \label{eq11z}\\ $$

(27)

$$\forall j \leq z_j^{\rm G} \perp \theta_j + (R-1) \phi_j + (R -2) \tau_j - \delta_j \geq 0 \label{eq12z}\\ $$

(28)

$$\forall j \leq s_j^{\rm O} \perp - P^0_j + \frac{P^0_j}{Q^0_j}{\left(s_j^{\rm O}+s_j^{\rm G}\right)} - \theta_j \geq 0 \label{eq13}\\ $$

(29)

$$\begin{array}{lll}\forall j & 0 \leq s_j^{\rm G} \perp - P^0_j + \frac{P^0_j}{Q^0_j}{\left(s_j^{\rm O}+s_j^{\rm G}\right)} - P^{\rm G}_j + \frac{P^{\rm G}_j}{Q^{\rm G}_j}{s_j^{\rm G}}\\ & \qquad \qquad \qquad \qquad - \theta_j + \tau_j + \delta_j \geq 0 \label{eq14}\end{array} $$

(30)

$$\forall j \omega^{\rm O}_j \mathrm{ free} \perp z_j^{\rm O} = \sum_{f,i,h\in H^{\rm O}_{\mathrm if}}z_{\mathrm fihj} \label{eqzO} \\ $$

(31)

$$\forall j \omega^{\rm G}_j \mathrm{ free} \perp z_j^{\rm G} = \sum_{f,i,h\in H^{\rm G}_{\mathrm if}}z_{\mathrm fihj} \label{eqzG} \\ $$

(32)

$$\forall j s^{\rm {REC}}_j \mathrm{ free} \perp - p^{\rm {REC}} + \phi_j + \tau_j + \delta_j = 0 \label{eq15}\\ $$

(33)

$$\forall j \theta_j \mathrm{ free} \perp z_j^{\rm O} + z_j^{\rm G} = s_j^{\rm O} + s_j^{\rm G} \label{eq16} $$

(34)

$$ \forall j \leq \phi_j \perp z_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm {REC}}_j \geq 0 \label{eq17}\\ $$

(35)

$$\forall j \leq \tau_j \perp 2z_j^{\rm G} - s^{\rm {REC}}_j - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm G}_j \geq 0 \label{eq18}\\[2pt] $$

(36)

$$\forall j \leq \delta_j \perp z_j^{\rm G} - s^{\rm {REC}}_j - s^{\rm G}_j \geq 0. \label{eq19} $$

(37)

Case 2

double-counting is allowed but bundling is not.

$$ \rm{Constraints\; (25)-(27),\; (29),\; (31)-(37) \label{eq20}} $$

(38)

$$\forall j \leq z_j^{\rm G} \perp \theta_j + (R-1) \phi_j + (R -2) \tau_j - \delta_j - \kappa_j \geq 0 \label{eq22}\\[2pt] $$

(39)

$$\forall j \leq s_j^{\rm G} p^0_j \frac{P^0_j}{Q^0_j}{\left(s_j^{\rm O} s_j^{\rm G}\right)} P^{\rm G}_j \frac{P^{\rm G}_j}{Q^{\rm G}_j}{s_j^{\rm G}} \theta_j \tau_j \delta_j \kappa_j \geq$$

(40)

$$\forall j \leq \kappa_j \perp z_j^{\rm G} - s^{\rm G}_j \geq 0. \label{eq210} $$

(41)

Case 3

bundling is allowed but double-counting is not.

$$ \rm{Constraints\; (25), \;(26), \;(29), \;(31),\; (32),\; (34),\; and\; (35) \label{eq30}} $$

(42)

$$\forall j \leq z_j^{\rm O} \perp \theta_j + R \phi_j + R \gamma_j \geq 0 \label{eq31}\\[2pt] $$

(43)

$$\forall j \leq z_j^{\rm G} \perp \theta_j + (R-1) \phi_j + (R -2) \gamma_j \geq 0 \label{eq32}\\[2pt] $$

(44)

$$\forall j \leq s_j^{\rm G} \perp - P^0_j + \frac{P^0_j}{Q^0_j}{\left(s_j^{\rm O}+s_j^{\rm G}\right)} - P^{\rm G}_j + \frac{P^{\rm G}_j}{Q^{\rm G}_j}s^{\rm G}_j - \theta_j + \gamma_j \geq 0 \label{eq34}\\[2pt] $$

(45)

$$\forall j s^{\rm {REC}}_j \mathrm{ free} \perp - p^{\rm {REC}} + \phi_j + \gamma_j = 0 \label{eq35}\\[2pt] $$

(46)

$$\forall j \leq \gamma_j \perp z_j^{\rm G} - s^{\rm {REC}}_j - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm G}_j \geq 0. \label{eq38} $$

(47)

Case 4

neither double-counting nor bundling is allowed.

$$\rm{Constraints\; (25),\; (26), \;(29), \;(31), \;(32), \;(34), \;(35), \;(41), \;(43), \;(46),\; and \;(47) \label{eq40}} $$

(48)

$$\forall j \leq z_j^{\rm G} \perp \theta_j + (R-1) \phi_j + (R -1) \gamma_j - \kappa_j \geq 0 \label{eq42}\\ $$

(49)

$$\forall j \leq s_j^{\rm G} \perp - P^0_j + \frac{P^0_j}{Q^0_j}{\left(s_j^{\rm O}+s_j^{\rm G}\right)} - P^{\rm G}_j + \frac{P^{\rm G}_j}{Q^{\rm G}_j}s^{\rm G}_j - \theta_j + \gamma_j + \kappa_j \geq 0. \label{eq44} $$

(50)

KKT for Producers

$$\forall f,i,h \in H_{\mathrm if},j \leq x_{\mathrm fihj} \perp - p_{\mathrm fihj} + C_{\mathrm fih} + (w_j-w_i) + p^{\rm GHG}E_{\mathrm fih} + \rho_{\mathrm fih} \geq 0 \label{eq:producerfoc1}\\ $$

(51)

$$\forall f,i,h \in H_{\mathrm if},j \leq \rho_{\mathrm fih} \perp X_{\mathrm fih} - \sum\limits_jx_{\mathrm fihj} \geq 0. \label{eq:producerfoc2} $$

(52)

Condition (51) suggests that the price offered by a firm to LSE j is equal to the sum of the costs associated with fuel (C _fih), transmissions (w _j  − w _i ), emissions ($p^{\rm GHG}E_{\mathrm fih}$) and the capacity scarcity rent (ρ _fih.)

KKT for ISO

$$\forall i y_i \mathrm{ free} \perp \sum_k PTDF_{\mathrm ki} (\lambda_k^+ - \lambda_k^-) - w_i = 0 \label{eq:isofoc1}\\[3pt] $$

(53)

$$\forall k \leq \lambda^+_k \perp T_k - \sum_i PTDF_{\mathrm ki} y_i \geq 0. \label{eq:isofoc2}\\[3pt] $$

(54)

$$\forall k \leq \lambda^-_k \perp T_k + \sum_i PTDF_{\mathrm ki} y_i \geq 0. \label{eq:isofoc2} $$

(55)

Appendix B: Equivalent quadratic programming formulations

We construct four quadratic programs that are equivalent to the four cases in the sense that the KKT conditions of these quadratic programs are the same as the market equilibrium conditions outlined in Table 7. Since all four objectives are maximization of concave quadratic functions, a Nash equilibrium is a solution of the KKT conditions if and only if it is an optimal solution of the corresponding quadratic program. Notice that the objective function of the quadratic programs for all four cases is to maximize the social welfare. These formulations are useful for proving some of the propositions presented in Section 3.3.

Case 1

both double-counting and bundling are allowed.

$$\begin{array}{lll} \max \qquad &\Pi_1 = \sum_j \,\left[P^0_j \,\left(s_j^{\rm G} \,+\, s_j^{\rm O}\right)\, - \frac{P^0_j}{2Q^0_j}{\left(s_j^{\rm G} + s_j^{\rm O}\right)^2}\right]\\ &+ \sum_j \left[P^{\rm G}_j s_j^{\rm G} - \frac{P^{\rm G}_j}{2 Q^{\rm G}_j} \left(s_j^{\rm G}\right)^2 \right] & \label{qp2}\\ & - \sum_{f,i,h \in H_{\mathrm if}, j} C_{\mathrm fih} x_{\mathrm fihj} \end{array} $$

(56)

$$s.t. \forall j, z_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm {REC}}_j \geq 0 (\phi_j \geq 0) \label{qp3} $$

(57)

$$ \forall j, 2z_j^{\rm G} \,-\, R\left(z_j^{\rm O}\,+\,z_j^{\rm G}\right) \,-\, s^{\rm {REC}}_j - s^{\rm G}_j \geq 0 (\tau_j \geq 0) \label{qp4}\\ $$

(58)

$$ \forall j, z_j^{\rm G} - s^{\rm {REC}}_j - s^{\rm G}_j \geq 0 (\delta_j \geq 0) \label{qp5}\\ $$

(59)

$$ \forall f,i,h \in H_{\mathrm if}, X_{\mathrm fih} - \sum_jx_{\mathrm fihj} \geq 0 (\rho_{\mathrm fih} \geq 0) \label{qp6}\\ $$

(60)

$$ \forall k, T_k - \sum_i PTDF_{\mathrm ki}y_i \geq 0 (\lambda_k^+ \geq 0) \label{qp7}\\ $$

(61)

$$ \forall k, T_k + \sum_i PTDF_{\mathrm ki}y_i \geq 0 (\lambda_k^- \geq 0) \label{qp7-}\\ $$

(62)

$$ \sum_j s^{\rm {REC}}_j \geq 0 (p^{\rm {REC}} \geq 0) \label{qp8}\\ $$

(63)

$$ \overline{E}- \sum_{f,i,h\in H_{\rm if},j}E_{\rm {fih}} x_{\rm {fihj}} \geq 0 (p^{GHG} \geq 0) \label{qp9}\\ $$

(64)

$$ \forall f,i,h \in H_{\mathrm if},j, x_{\mathrm fihj} = z_{\rm {fihj}} (p_{\rm {fihj}} \rm{ {free}}) \label{qp10} $$

(65)

$$ \forall i, \sum_{f,h \in H_{\mathrm if},j}x_{\mathrm fihj} - s_i^{\rm O} - s_i^{\rm G} + y_i = 0 (w_i \mathrm{ free}) \label{qp11}\\ $$

(66)

$$ \forall j, s_j^{\rm O} + s_j^{\rm G} = z_j^{\rm O} + z_j^{\rm G} (\theta_j \mathrm{ free}) \label{qp12} $$

(67)

$$ \forall j, z_j^{\rm O} = \sum_{f,i,h\in H^{\rm O}_{\mathrm if}}z_{\mathrm fihj} (\omega^{\rm O}_j) \label{qpzO} \\ $$

(68)

$$\forall j, z_j^{\rm G} = \sum_{f,i,h\in H^{\rm G}_{\mathrm if}}z_{\mathrm fihj} (\omega^{\rm G}_j) \label{qpzG} \\ $$

(69)

$$\forall f,i,h,j, x_{\mathrm fihj}, z_j^{\rm O}, z_j^{\rm G}, s_j^{\rm O}, s_j^{\rm G} \geq 0; s_j^{\rm {REC}}, y_i \mathrm{ free.} \label{qp13} $$

(70)

Case 2

double-counting is allowed but bundling is not.

$$\begin{array}{lll} \max &\qquad \Pi_2 = \sum_j \left[P^0_j \left(s_j^{\rm G} + s_j^{\rm O}\right) - \frac{P^0_j}{2Q^0_j}{\left(s_j^{\rm G} + s_j^{\rm O}\right)^2}\right]\\ &+ \sum_j \left[P^{\rm G}_j s_j^{\rm G} - \frac{P^{\rm G}_j}{2 Q^{\rm G}_j} \left(s_j^{\rm G}\right)^2 \right] - \sum_{f,i,h \in H_{\mathrm if}, j} C_{fih} x_{fihj} \label{case2:0} \end{array} $$

(71)

$$s.t. \rm{Constraints\; (57)-(70)} \label{case2:1}\\ $$

(72)

$$ \forall j z_j^{\rm G} - s^{\rm G}_j \geq 0 (\kappa_j \geq 0). \label{case2:2} $$

(73)

Case 3

bundling is allowed but double-counting is not.

$$\max \Pi_3 = \sum_j \left[P^0_j \left(s_j^{\rm G} + s_j^{\rm O}\right) - \frac{P^0_j}{2Q^0_j}{\left(s_j^{\rm G} + s_j^{\rm O}\right)^2}\right] + \sum_j \left[P^{\rm G}_j s_j^{\rm G} - \frac{P^{\rm G}_j}{2 Q^{\rm G}_j} \left(s_j^{\rm G}\right)^2 \right]\\ $$

$$\begin{array}{lll}&\qquad\qquad\qquad\qquad\qquad\qquad - \sum_{f,i,h \in H_{\mathrm if}, j} C_{\mathrm fih} x_{\mathrm fihj} \\ s.t. \qquad\qquad\qquad& \rm{Constraints\; (57), (60)--(70)} \\ & \forall j z_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm {REC}}_j - s^{\rm G}_j \geq 0 (\gamma_j \geq 0). \label{case3:2} \end{array} $$

(74)

Case 4

neither double-counting nor bundling is allowed.

$$\begin{array}{lll} \max &\qquad \Pi_4 = \sum_j \left[P^0_j \left(s_j^{\rm G} + s_j^{\rm O}\right) - \frac{P^0_j}{2Q^0_j}{\left(s_j^{\rm G} + s_j^{\rm O}\right)^2}\right]\\ &+ \sum_j \left[P^{\rm G}_j s_j^{\rm G} - \frac{P^{\rm G}_j}{2 Q^{\rm G}_j} \left(s_j^{\rm G}\right)^2 \right] - \sum_{f,i,h \in H_{\mathrm if}, j} C_{\mathrm fih} x_{\mathrm fihj}\\ \end{array} $$

(75)

$$s.t. \rm{Constraints \;(57),\; (60)-(70), \;(73), \;(74).} \label{case4:2} $$

(76)

Appendix C: Proofs

1.1 Proposition 1

Proof

For Case 1. From $s^{O}_j > 0$, Eqs. (29)–(33) and (35), we have $p^{\rm {REC}} - p^{\rm G}_j \geq \phi_j \geq 0$. Suppose bundling occurs at firm j, which means $s^{\rm G}_j > z_j^{\rm G} \geq 0$. From $s^{\rm O}_j > 0$, Eqs. (30), and (33) we have $p^{\rm {REC}} - \phi_j = p^{\rm G}_j$. From Eqs. (35) and (36) we have $z_j^{\rm G} - R(z_j^{\rm O}+z_j^{\rm G}) - s^{\rm {REC}}_j = [2z_j^{\rm G} - s^{\rm {REC}}_j - R(z_j^{\rm O}+z_j^{\rm G}) - s^{\rm G}_j] + (s^{\rm G}_j- z_j^{\rm G})>0.$ Therefore, ϕ _j  = 0 and $p^{\rm {REC}} = p^{\rm G}_j$. The proof for Case 3 is similar with τ _j  + δ _j replaced by γ _j in the demonstration.□

1.2 Proposition 2

Proof

For Case 2. From $s^{REC}_j > 0$, Eqs. ( 34), ( 37) and ( 41), we have $s^{O}_j = z_j^{O} + z_j^{G} - s^{G}_j \geq z_j^{G} - s^{G}_j \geq s^{REC}_j > 0$ and κ _j  = 0. From Eqs. ( 29), ( 40), ( 33) and ( 35), we have $p^{REC} - p^{G}_j \geq \phi_j \geq 0$. Moreover, if $s^{G}_j > R(z_j^{O}+z_j^{G})$, then $p^{REC} - p^{G}_j = \phi_j = 0$. The last equation is because in Eq. ( 35) $z_j^{G} - R(z_j^{O}+z_j^{G}) - s^{REC}_j = z_j^{G} - s^{REC}_j - s^{G}_j + [s^{G}_j - R(z_j^{O}+z_j^{G})] > 0$, using Eq. ( 37). The proof for Case 4 is similar with τ _j  + δ _j replaced by γ _j in the demonstration.□

1.3 Proposition 3

Proof

We show that for any Case 1 solution $(s_j^{\rm G}, s_j^{\rm O}, x_{\mathrm fihj}, z_{\mathrm fihj}, y_i, s_j^{\rm {REC}})$, there exists a Case 2 solution $(\hat{s}_j^{\rm G}, \hat{s}_j^{\rm O}, \hat{x}_{\mathrm fihj}, \hat{z}_{\mathrm fihj}, \hat{y}_i, \hat{s}_j^{\rm {REC}})$ that satisfies the following constraints:

$$ \forall j, \hat{s}_j^{\rm G} = s_j^{\rm G} \label{new1}\\[2pt] $$

(77)

$$ \forall j, \hat{s}_j^{\rm O} = s_j^{\rm O} \label{new2}\\[2pt] $$

(78)

$$\forall i, \hat{y}_i = y_i \label{new7}\\[2pt] $$

(79)

$$\forall f,i,h\in H_{\mathrm if}, \sum_j \hat{x}_{\mathrm fihj} = \sum_j x_{\mathrm fihj} \label{new3}\\[2pt] $$

(80)

$$\forall f,i,h\in H_{\mathrm if}, \hat{x}_{\mathrm fihj} = \hat{z}_{\mathrm fihj} \label{new4}\\[2pt] $$

(81)

$$ \forall j, \hat{z}_j^{\rm G} := \sum_{f,i,h\in H^{\rm G}_{\mathrm if}} \hat{z}_{\mathrm fihj} \label{new5}\\[2pt] $$

(82)

$$ \forall j, \hat{z}_j^{\rm O} := \sum_{f,i,h\in H^{\rm O}_{\mathrm if}} \hat{z}_{\mathrm fihj} \label{new6}\\[2pt] $$

(83)

$$ \forall j, \hat{z}_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - \hat{s}^\mathrm{\mathrm REC}_j \geq 0 (\phi_j \geq 0) \label{new7}\\[2pt] $$

(84)

$$ \forall j, \hat{z}_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - \hat{s}^\mathrm{\mathrm REC}_j - s^{\rm G}_j \geq 0 (\tau_j \geq 0) \label{new8}\\[2pt] $$

(85)

$$ \forall j, \hat{z}_j^{\rm G} - \hat{s}^\mathrm{\mathrm REC}_j - s^{\rm G}_j \geq 0 (\delta_j \geq 0) \label{new9}\\[2pt] $$

(86)

$$ \sum_j \hat{s}^\mathrm{\mathrm REC}_j \geq 0 (p^\mathrm{\mathrm REC} \geq 0) \label{new10} \\[2pt]$$

(87)

$$ \forall j, \hat{z}_j^{\rm G} + \hat{z}_j^{\rm O} = s_j^{\rm G} + s_j^{\rm O} (\theta_j \mathrm{ free}) \label{new11}\\[2pt ] $$

(88)

$$ \forall j, \hat{z}_j^{\rm O} \geq 0, \hat{z}_j^{\rm G} \geq 0, \hat{s}_j^{\rm {REC}} \mathrm{ free} \label{new12}\\[2pt] $$

(89)

$$ \forall j, \hat{z}_j^{\rm G} \geq s_j^{\rm G} (\kappa_j \geq 0). \label{new13} $$

(90)

It is easy to see that, given $(x_{\mathrm fihj}, \hat{z}_j^{\rm G}, \hat{z}_j^{\rm O})$, there exists a solution (possibly among infinitely many others) $(\hat{x}_{\mathrm fihj}, \hat{z}_{\mathrm fihj})$ that satisfies Eqs. (80)–(83). For any given Case 1 solution $(s_j^{\rm G}, z_j^{\rm G}, z_j^{\rm O})$, we prove the existence of $(\hat{z}_j^{\rm O}, \hat{z}_j^{\rm G}, \hat{s}_j^{\rm {REC}})$ that satisfies Eqs. (84)–(90) by showing the non-existence of a solution to the following constraints:

$$\begin{array}{lll} &\qquad \sum_j \Bigl[R\left(z_j^{\rm O}+z_j^{\rm G}\right) (\phi_j+\tau_j)\\[4pt] & +\, s^{G}_j \left(\tau_j+\delta_j+\theta_j+\kappa_j\right) + s^{O}_j \theta_j\Bigr] > 0 & \label{dual1} \end{array} $$

(91)

$$\forall j, \phi_j + 2 \tau_j + \delta_j + \theta_j + \kappa_j \leq 0 (z_j^{\rm G} \geq 0) \label{dual2} \\[4pt] $$

(92)

$$\forall j, \theta_j \leq 0 (z_j^{\rm O} \geq 0) \label{dual3} \\[4pt] $$

(93)

$$\forall j, \phi_j - \tau_j - \delta_j + p^{\rm {REC}} \leq 0 (s_j^{\rm {REC}} \mathrm{ free}) \label{dual4} \\[4pt] $$

(94)

$$\forall j, \phi_j, \tau_j, \delta_j, \kappa_j, p^{\rm {REC}} \geq 0; \theta_j \mathrm{ free}. \label{dual5} $$

(95)

According to Farkas’ lemma (Bertsimas and Tsitsiklis 1997), for any A ∈ ℝ^{m ×n} and b ∈ ℝ^{m ×1}, exactly one of these two sets is empty: {x ∈ ℝ^{n ×1}: A x ≥ b, x ≥ 0} and {y ∈ ℝ. In this context, Farkas’ lemma means that Eqs. (84)–(90) possesses a solution if and only if the set of Eqs. (91)–(95) does not. We prove the infeasibility of Eqs. (91)–(95) by contradiction. Suppose $(\phi_j^0, \tau_j^0, \delta_j^0, \kappa_j^0, p^\mathrm{REC0}, \theta_j^0)$ satisfies Eqs. (91)–(95), then it is easy to see that $(\phi_j = \phi_j^0, \tau_j = \tau_j^0, \delta_j = \delta_j^0 + \kappa_j^0, \kappa_j = 0, p^{\rm {REC}} = p^\mathrm{REC0}, \theta_j = \theta_j^0)$ also satisfies Eqs. (91)–(95). However, the latter further implies, also by Farkas’ lemma, that Eqs. (84)–(89) is infeasible, which contradicts the fact that $(\hat{z}_j^{\rm O} = z_j^{\rm O}, \hat{z}_j^{\rm G} = z_j^{\rm G}, \hat{s}_j^{\rm {REC}} = s_j^{\rm {REC}})$ satisfies Eqs. (84)–(89). As a result, the $\hat{p}^{\rm G}_j$ and $\hat{\Pi}$ for Case 2 will be the same with those in Case 1.

We prove that $\hat{p}^{\rm {REC}} \leq p^{\rm {REC}}$ as follows:

$$\hat{p}^{\rm {REC}} - p^{\rm {REC}}\\[4pt] $$

(96)

$$= \hat{\phi}_j + \hat{\tau}_j + \hat{\delta}_j - (\phi_j + \tau_j + \delta_j), \forall j \label{p1}\\[4pt] $$

(97)

$$= \hat{\tau}_j + \hat{\delta}_j - (\tau_j + \delta_j), \forall j: s_j^{\rm G} > z_j^{\rm G} \label{p2}\\[4pt] $$

(98)

$$= - \hat{\kappa}_j \leq 0, \forall j: s_j^{\rm G} > z_j^{\rm G}, s_j^{\rm O} > 0. \label{p3} $$

(99)

Here, Eq. (97) is from Eq. (33). Equation (98) is because in Eqs. (35) and (36) we have

$$\begin{array}{lll} && z_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm {REC}}_j\\[4pt] &&{\kern12pt} = \left[2z_j^{\rm G} - s^{\rm {REC}}_j - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s_j^{\rm G}\right] + (s_j^{\rm G} - z_j^{\rm G})> 0. \end{array}$$

Therefore, ϕ _j  = 0, and similarly $\hat{\phi}_j = 0$. Equation (99) is due to Eqs. (29), (30), and (40), which yield that $\tau_j + \delta_j = \hat{\tau}_j + \hat{\delta}_j + \hat{\kappa}_j$.□

1.4 Proposition 4

Proof

The first part of the proof is similar to that of Proposition 3, except that Eqs. (85) and (86) are substituted with

$$\forall j, \hat{z}_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - \hat{s}^{\rm {REC}}_j - s^{\rm G}_j \geq 0 (\gamma_j \geq 0). \label{new894} $$

(100)

Using a similar argument, we can prove that Eqs. (87)–(90) and (100) possesses a solution if and only if the following set of Eqs. (101)–(105) does not.

$$\begin{array}{lll} &\qquad \sum_j \Big[R\left(z_j^{\rm O}+z_j^{\rm G}\right)\\ & \gamma_j + s^{\rm G}_j (\gamma_j+\delta_j+\theta_j+\kappa_j) + s^{\rm O}_j \theta_j\Big] > 0 & \label{dual14} \end{array} $$

(101)

$$\forall j, \gamma_j + \delta_j + \theta_j + \kappa_j \leq 0 (z_j^{\rm G} \geq 0) \label{dual24} \\ $$

(102)

$$\forall j, \theta_j \leq 0 (z_j^{\rm O} \geq 0) \label{dual34} \\ $$

(103)

$$\forall j, \gamma_j - \delta_j + p^{\rm {REC}} \leq 0 (s_j^{\rm {REC}} \mathrm{ free}) \label{dual44} \\ $$

(104)

$$\forall j, \delta_j, \kappa_j, p^{\rm {REC}} \geq 0; \gamma_j, \theta_j \mathrm{ free}. \label{dual54} $$

(105)

It is easy to see that the $\hat{p}^{\rm G}_j$ and $\hat{\Pi}$ for Case 4 will be the same with those in Case 3. We prove that $\hat{p}^{\rm {REC}} \leq p^{\rm {REC}}$ as follows:

$$\hat{p}^{\rm {REC}} - p^{\rm {REC}}\\ $$

(106)

$$= \hat{\phi}_j + \hat{\gamma}_j - (\phi_j + \gamma_j), \forall j \label{p14}\\ $$

(107)

$$= \hat{\gamma}_j - \gamma_j, \forall j: s_j^{\rm G} > 0 \label{p24}\\ $$

(108)

$$= -\hat{\kappa}_j \leq 0, \forall j: s_j^{\rm G} > 0, s_j^{\rm O} > 0. \label{p34} $$

(109)

Here, Eq. (107) is from Eq. (46). Equation (108) is because in Eqs. (35) and (47) we have

$$\begin{array}{lll} && z_j^{\rm G} - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s^{\rm {REC}}_j\\ &&{\kern12pt} = \left[z_j^{\rm G} - s^{\rm {REC}}_j - R\left(z_j^{\rm O}+z_j^{\rm G}\right) - s_j^{\rm G}\right] + s_j^{\rm G} > 0. \end{array}$$

Therefore, ϕ _j  = 0, and similarly $\hat{\phi}_j = 0$. Equation (109) is due to Eqs. (29), (45), and (50), which yield that $\gamma_j = \hat{\gamma}_j + \hat{\kappa}_j$.□

1.5 Proposition 5

Proof

From Propositions 3 and 4, we have that Π ₁ = Π ₂ and Π ₃ = Π ₄, respectively. Moreover, any feasible solution to Case 3 is also feasible to Case 1, since Constraint (74) is tighter than Eq. (58), which is the only difference between the two cases. Therefore, Π ₁ ≥ Π ₃.□

Appendix D: Data

Table 8 Assumptions of generation characteristics

Table 9 Derived power transfer distribution factor and transmission thermal limit

Table 10 Assumptions of the inverse demand curves