Emissions trading, point-of-regulation and facility siting choices in the electric markets

Abstract

This paper examines the effects of emissions trading on firms’ capacity siting choices in the electric sector, considering three programs that are different by their point-of-regulation: source-, load and first-jurisdictional-deliverer approaches. We model each program by assuming electricity sales are through bilateral contracts, and analyze the solutions properties and economic and emissions implications. We have three central findings in this paper. First, when electricity sales in both directions (i.e., in-state to out-of-state and vice verse) are subject to an emissions cap, three programs will produce identical siting distribution. Second, the emissions leakage in the long-run could be lower than in the short-run case in which capacity expansion is not allowed. This is so because in the long-run power producers may opt for clean technologies in fear of the emission costs under a cap-and-trade program. Third, if compared to pure source-based program in which the import sales are not subject to an emissions cap, more pollution-intensive facilities are expected to build in uncapped states, where the regulation is less stringent. Our findings are therefore consistent with some empirical evidence of pollution haven hypothesis.

Appendices

Appendix A notation summary

A summary of notation is provided here for the ease of reading the modeling sections. In general we use script capital letters for sets, capital letters for parameters, lower-case letters for variables and indices and greek lower-case letters for dual variables. Also we use bold letters to represent vectors of appropriate dimensions.

Sets, Indices and Dimensions
$i, j \in \mathcal{N }$	Set of zones in a network, including both power producers and LSEs; $|\mathcal{N }| = N$
$\mathcal{N }^C \subset \mathcal{N }$	Set of zones that are under the emissions cap
$\mathcal{N }^{NC} \subset \mathcal{N }$	Set of zones that are not under the emissions cap; $\mathcal{N }^{NC} = \mathcal{N }\backslash \mathcal{N }^C$
$l \in \mathcal{A }$	Set of (directional) transmission links in the network $\mathcal{N }$; $|\mathcal{A }| = A$
$f\in \mathcal{F }$	Set of power producers; $|\mathcal{F }| = F$.
Parameters
$P_j^0$	Price intercept of the inverse demand function in zone $j$ ($/MW(h))
$Q_j^0$	Quantity intercept of the inverse demand function in zone $j$ ($/MW(h))
${\overline{E}}^{^P}_f/{\overline{E}}^{^{LSE}}_j$	Initial free allocation of $\mathrm {CO}_2$ allowances to power producer $f$ or LSE $j$ (tons)
$E_{fi}$	Emission rate of firm $f$ in zone $i$ (tons/MW(h))
${\overline{E}}$	System-wide $\mathrm {CO}_2$ cap (tons)
$\alpha _{fi}$	Per unit cost of capacity expansion ($/MW)
$K_{fi}$	Firm $f$’s existing capacity at location $i$ (MW); $K_{fi} \ge 0,\ \forall f\in \mathcal{F },\ i\in \mathcal{N }$
$C_{fi}(\cdot )$	Firm $f$’s production cost function in zone $i$ ($/MW(h))
$PTDF_{li}$	The $(l, i)$-th element of the power transmission-distribution factor matrix
$T_l$	Transmission capacity bound on line $l$; $T_l > 0 \ \forall l\in \mathcal{A }$
${{1}\!\!\mathrm{l}}_{\left[ {\cdot } \right] }$	An indicator function—if the condition in the brackets is true, ${{1}\!\!\mathrm{l}}_{\left[ {\cdot } \right] } = 1$; else, ${{1}\!\!\mathrm{l}}_{\left[ {\cdot } \right] } = 0$.

Variables
$d_{j}$	Electricity demand in zone $j$ (MW(h))
$s_{fij}$	LSE $j$’s purchase of power from firm $f$ in zone $i$ (MW(h)). (Note that there is
	assumed to be just one LSE per zone; more general assumptions are possible,
	but would not change the conclusions of our analysis.)
$s_{fj}$	Power firm $f$’s sale of electricity to LSE in zone $j$ (MW(h))
$g_{fij}$	Electricity sold to LSE $j$, produced by firm $f$ in zone $i$ (MW(h))
$x_{fi}$	New capacity addition by firm $f$ at location $i$ (MW)
$y_j$	MWs transmitted from the hub to zone $j$ (MW)
$w_j$	Fees charged for transmitting electricity from the hub to zone $j$ ($/MW)
${\tilde{p}}_{j}$	The price of electricity paid by consumers to an LSE in zone $j$ ($/MW(h))
$p_{fij}$	The price of electricity produced by generator $(f,i)$ sold to LSE $j$ ($/MW(h))
$p^{CO_2}$	$\mathrm {CO}_2$ allowance price ($/ton)

Appendix B Theoretical properties of the open-loop models

In this Appendix we provide theoretical results for the three equilibrium models introduced in Sect. 3—Model Mo-SB, Model Mo-LB and FJD). More specifically, we show equilibrium existence and uniqueness.

Proposition 3

Given the inverse demand functions in (12), the investment cost functions in (13) and the production cost functions in (14), there exists an equilibrium to Model Mo-SB, Model Mo-LB and Model FJD, respectively.

Proof

If consumers’ highest willingness-to-pay for electricity is less than (or equal to) the lowest-cost generation unit’s marginal cost, (that is, $\max _{f\in {\mathcal{F }},\ i, j\in {\mathcal{N }}}\{P^0_j - a_{fi}\} \le 0$), then it is straightforward to verify that $\mathbf 0$ (a vector of all 0’s with a proper dimension) is an equilibrium to all the three models. If $\max _{f\in {\mathcal{F }},\ i, j\in {\mathcal{N }}}\{P^0_j - a_{fi}\} > 0$, then let every variable be 0 except for the capacity expansion variables $x_{fi}$ and the Lagrange multipliers associated with the generation capacity constraints (denoted as $\rho _{fi}$). For the nonzero variables, we set that $\rho ^{\circ }_{fi} = \max _{j\in {\mathcal{N }}}\{P^0_j - a_{fi}, 0\}$, and $x^{\circ }_{fi} = \rho _{fi}/\alpha _{fi}$, $\forall \ f\in {\mathcal{F }}$ and $i \in {\mathcal{N }}$. Then the tuple $(\mathbf 0 ,\mathbf x ^{\circ }, \varvec{\rho }^{\circ })$ is feasible to each of the mixed linear complementarity problem resulted from Model Mo-SB, Mo-LB and FJD. The rest of the proof follows exactly as that in Chen et al. (2011), Theorem 1. $\square$

The following proposition is a straightforward extension of Theorem 2 in Chen et al. (2011), and the proof is hence omitted.

Proposition 4

Let a vector $\mathbf z$ represent the variables ($\mathbf{x,d, s,g,w,y, } \ {\tilde{\mathbf{p}}},\mathbf{p, p^{CO_2}}$) in one of the equilibrium models and $\varvec{\zeta }$ represent the (vector of) dual variables corresponding to the constraints. Assume that the system-wide $\mathrm {CO}_2$ cap among the three models is the same; namely,

$$\begin{aligned} \displaystyle {\overline{E}}^{SB} = {\overline{E}}^{LB} = {\overline{E}}^{FJD}. \end{aligned}$$

(24)

Further assume that the assumptions in Proposition 3 hold. Then if $(\mathbf z , \varvec{\zeta })$ solves the mixed complementarity problem (MiCP) derived from Model Mo-SB, Mo-LB or FJD, it solves the MiCP derived from the other two models.

Appendix C Proof of Lemma 1

Lemma 1

The inequalities (20)–(22) are equivalent.

Proof

Since we consider an electricity market with bilateral contracts; that is, $s_{fij} = g_{fij}, \forall f, i, j$, we can replace the $s_{f ij}$’s in (21) and (22) with $g_{fij}$’s. Let ${\mathcal{N }} \times {\mathcal{N }}$ denote the Cartesian product of the set of node indices. It can be partitioned into the following four sets.

$$\begin{aligned} \mathcal{I }_1 := {\mathcal{N }}^C \times {\mathcal{N }}^C,\ \mathcal{I }_2 := {\mathcal{N }}^C \times {\mathcal{N }}^{NC},\ \mathcal{I }_3 := {\mathcal{N }}^{NC} \times {\mathcal{N }}^C,\ \mathcal{I }_4 := {\mathcal{N }}^{NC} \times {\mathcal{N }}^{NC}.\nonumber \\ \end{aligned}$$

(25)

For any $(i, j)\in {\mathcal{N }}\times {\mathcal{N }}$, it must be in one and only one of the above four sets. Using the partitioned sets, we can rewrite (20)–(22) as follows.

$$\begin{aligned} \text{ Modified } \text{ SB: } \displaystyle \sum \limits _{f\in {\mathcal{F }}}\left( \underset{(i, j) \in \mathcal{I }_1\cup \mathcal{I }_2\cup \mathcal{I }_3}{\sum \limits }E_{fi}g_{fij}\right) \le {\overline{E}} \\ \text{ Modified } \text{ LB: } \displaystyle \sum \limits _{f\in {\mathcal{F }}}\left( \sum \limits _{(i, j) \in \mathcal{I }_1\cup \mathcal{I }_3\cup \mathcal{I }_3}E_{fi}g_{fij}\right) \le {\overline{E}} \\ \text{ FJD: } \displaystyle \sum \limits _{f\in {\mathcal{F }}}\left( \sum \limits _{(i, j) \in \mathcal{I }_3\cup \mathcal{I }_1\cup \mathcal{I }_2}E_{fi}g_{fij}\right) \le {\overline{E}}. \end{aligned}$$

Apparently the above three inequalities are equivalent. $\square$

Appendix D Proof of Proposition 2

To prove Proposition 2, we first discuss solution existence of the two-stage problem (16). Let $Z(x)$ denote the feasible region of the 2nd-stage problem (i.e., Problem (19)) with respect to a given $x\in X$, where $X$ is the Cartesian product of all firms’ feasible capacity expansion sets $X_f$ (that is, $X = \varPi _{f\in {\mathcal{F }}} X_f$). To rule out uninteresting cases, we make the blanket assumption that $X\ne \emptyset$. Even with this assumption, $Z(x)$ may still be empty with a given $x \in X$. In this case, by convention we assign $\mathcal{Q }(x)$ to be $-\infty$. For $X$, note that throughout the paper we only consider the case where $X\subset \mathfrak R ^{FN}_+$ (namely, all the $x_{fi}$’s are nonnegative).²² Then it is easy to check that $\mathbf 0$ (the vector of zeros of a proper dimension) is always in $Z(x)$ for any $x\in X$. Hence, $\mathcal{Q }(x) > -\infty$, $\forall x\in X$. With the $P_j(\cdot )$ and $C_{fi}(\cdot )$ functions defined in (12) and (14), respectively, the objective function $Q(\mathbf z )$ is convex quadratic and can be easily seen to be coercive over $Z(x)$.²³ Since $Z(x)$ is always nonempty and closed (as all the functions in the constraint set of (19) are continuous) for each $x\in X$, together with the coerciveness $Q(\mathbf z )$, an optimal solution of the 2nd-stage problem (19) (denoted as $\mathbf z ^*$) exists for each $x\in X$. (See the existence result in Bertsekas (1999), Proposition A.8.) Consequently, $\mathcal{Q }(x)$ is finite for each $x\in X$. Then with the investment cost functions given in (13), $\varPi (x)$ is also coercive over $X$. With a further assumption of $X$ being a closed set, we have the following existence result regarding to the two-stage problem (16).

Lemma 2

Let the inverse demand functions, the investment cost functions, and the production cost functions be given as in (12)–(14), respectively. Let $X\subset \mathfrak R ^{FN}_+$ be nonempty and closed. Then an optimal solution of (16) exists. $\square$

Note that in showing the solution existence, we do not require the set $X$ to be bounded. Though capacity expansions are bounded by engineering or other constraints in reality, modeling-wise not requiring explicit bounds on the expansion variables is nonetheless more convenient.

We then present several results regarding to the properties of the optimal value function $\mathcal{Q }(x)$ in the two-stage problem (16).

Lemma 3

Given the conditions in Lemma 2 and that $X$ is a convex set, $\mathcal{Q }(x)$ is concave on $X$.

Proof

With the given inverse demand functions and cost functions, $Q(z)$ is easily seen to be concave. In addition, $Z(x)$, the parameterized feasible region that can be viewed as a set-valued mapping, is apparently a convex mapping as all the constraints defining the region are linear. Then the concavity of $\mathcal{Q }(x)$ on $X$ follows Proposition 2.1 in Fiacco and Kyparisis (1986). $\square$

The optimal value function $\mathcal{Q }(x)$ is in general not differentiable. With the concavity of $\mathcal{Q }(x)$, however, it has subgradients on $X$, and the collection of the subgradients is referred to as the subdifferential, denoted as $\partial \mathcal{Q }(x)$. Given an $x\in X$, by the discussion for Lemma 2, an optimal solution to Problem (19) exists. Let it be denoted as ${\tilde{z}}(x)$ (or simply ${\tilde{z}}$). The optimal solution may not be unique though, such as in the case of linear production cost functions. Let $Z^*(x)$ denote the set of optimal solutions to Problem (19) with a given $x$. Since the constraints in (19) are all linear, the Linear Constraint Qualification (LCQ) holds for the entire set $Z(x)$. Then for each ${\tilde{z}}\in Z^*(x)$, there exist corresponding Lagrange multipliers. Let $\mathcal P ({\tilde{z}}; x)$ denote the set of Lagrange multipliers corresponding to ${\tilde{z}}$ and associated with the capacity constraints in (19). To establish the relationship between the equilibrium model and the optimization model, it is key to show the relationship between $\partial \mathcal{Q }(x)$ and $\mathcal P ({\tilde{z}}; x)$, which is formally stated in the following lemma.

Lemma 4

Given the conditions in Lemma 2, for an $x\in X$, let ${\tilde{z}} \in Z^*(x)$. $\forall \rho \in \mathcal{P }({\tilde{z}}; x)$, $\rho \in \partial \mathcal{Q }(x)$. If in addition that the Magasarian-Fromovitz Constraint Qualification (MFCQ) holds at each ${\tilde{z}} \in Z^*(x)$, then $\displaystyle \partial \mathcal{Q }(x) = \text{ conv }\{\cup _{{\tilde{z}}\in Z^*(x)}\mathcal{P }({\tilde{z}}; x)\}$, where ‘$\mathrm {conv}$’ denotes the convex hull of a set.

Proof

Let ${\bar{x}}$ be an arbitrary point in $X$, and let ${\bar{z}}$ be the corresponding optimal solution. Since the constraints in (19) are all linear, the Linear Constraint Qualification (LCQ) holds for any $z\in Z({\bar{x}})$. Hence there exist Lagrange multipliers associated with ${\bar{z}}$. For the ease of argument, we use $h(z)=0$, a vector of functions with proper dimension, to represent all equality constraints in (19), and use $l(z)\le 0$ to represent all inequality constraints in (19) except for the generation capacity constraint. Let $\mathbf g$ denote the vector of $(\sum _{j\in {\mathcal{N }}}g_{fij})_{f\in {\mathcal{F }}, i\in {\mathcal{N }}}$. Further define $\bar{v}$ and $\bar{u}$ to be the vectors of Lagrange multipliers (with appropriate dimensions) associated with $h({\bar{z}})$ and $l({\bar{z}})$. (19) is easily seen to be a convex optimization problem given an $x\in X$. Hence the strong duality of nonlinear programming holds. With the Lagrangian dual function of (19) (considering minimizing the negative of $Q(z)$) defined as follows

$$\begin{aligned} \theta (\rho (x), u, v) \equiv \displaystyle \inf _{z\in \mathfrak R ^{N+FN+2A}}\left\{ -Q(z) + \rho ^T(x)(\mathbf g - K - x) + u^Tl(z) + v^Th(z)\right\} \nonumber \\ \end{aligned}$$

(26)

we have the following

$$\begin{aligned} -\mathcal{Q }({\bar{x}})&= \displaystyle \theta (\rho ({\bar{x}}), {\bar{u}}, \bar{v})\nonumber \\&= \inf _{z\in \mathfrak R ^{N+FN+2A}}\left\{ -Q(z) \!+\! \rho ^T({\bar{x}})(\mathbf g \!-\! K \!-\! {\bar{x}})\!+\! {\bar{u}}^Tl(z) \!+\! \bar{v}^Th(z)\right\} \nonumber \\&= \rho ^T({\bar{x}})(x - {\bar{x}}) + \displaystyle \inf _{z\in \mathfrak R ^{N+FN+2A}}\left\{ -Q(z) + \rho ^T({\bar{x}})(\mathbf g - K - x)\right. \nonumber \\&\left. + {\bar{u}}^Tl(z) + \bar{v}^Th(z)\right\} \nonumber \\&\le \rho ({\bar{x}})^T(x - {\bar{x}}) - \mathcal{Q }(x), \end{aligned}$$

(27)

where the last inequality follows from the weak duality of nonlinear programming. By the definition of subgradients, we obtain that $-\rho (x)\in \partial \left( - \mathcal{Q }(x)\right)$ for $x\in X$, and hence, $\rho (x) \in \partial \mathcal{Q }(x)$ for $x \in X$.

To show that $\partial \mathcal{Q }(x) = \text{ conv }\{\cup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x)\}$, we apply the result in Rockafellar (1982) (Corollary 2 of Theorem 2), which states that if $\mathcal{Q }(x)$ is tame and that if the MFCQ holds at each ${\tilde{z}}\in Z^*(x)$, then

$$\begin{aligned} \partial \mathcal{Q }(x) = \text{ cl } \text{ conv }\left\{ \displaystyle \left[ \bigcup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x)\right] \bigcap \partial \mathcal{Q }(x) \right\} . \end{aligned}$$

(28)

As stated in Rockafellar (1982), ‘tameness’ is a local boundedness assumption on how a parameterized feasible set varies with the parameter. Let $x^{\prime }\in X$ be a small perturbation of $x$; i.e., for a $\delta > 0$, $|| x^{\prime } - x || \le \delta$. In the discussion of Lemma 2, the feasible set of Problem 16, $Z(x)$, is always nonempty for each $x\in X$. In addition, an optimal solution of the 2nd-stage problem (19) always exists with a given $x \in X$. Hence, there exist finite $\omega$ and $\omega ^{\prime }$ such that $\mathcal{Q }(x) = \omega$ and $\mathcal{Q }(x^{\prime }) = \omega ^{\prime }$. Let $\omega _0 = \min \{\omega , \omega ^{\prime }\} - \epsilon$ with $\epsilon > 0$. Then by the coerciveness of $Q(z)$ shown in the discussion of Lemma 2, we obtain that the level set $\{z^{\prime }\in Z(x^{\prime }): Q(z^{\prime }) \ge \omega _0 \}$ is bounded. Hence, by Proposition 9 in Rockafellar (1982), $\mathcal{Q }(x)$ is tame. Together with the MFCQ assumption in the lemma, the characterization of the subdifferential of $\mathcal{Q }(x)$ in (28) holds true. By the first part of this proof, we know that $\cup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x) \subset \partial \mathcal{Q }(x)$. Hence,

$$\begin{aligned} \partial \mathcal{Q }(x) = \text{ cl } \text{ conv }\left\{ \displaystyle \left[ \bigcup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x)\right] \right\} . \end{aligned}$$

(29)

Since the functions in the optimization problem (19) are all continuously differentiable, the Lagrange multiplier set is closed, and so is $\mathcal{P }({\tilde{z}}; x)$ for each ${\tilde{z}}\in Z^*(x)$. Further by the MFCQ assumption, $\mathcal{P }({\tilde{z}}; x)$ is also bounded with each ${\tilde{z}}\in Z^*(x)$. Then $\cup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x)$ is a closed bounded set, and the convex hull of which is closed. Hence,

$$\begin{aligned} \partial \mathcal{Q }(x) = \text{ conv }\left\{ \displaystyle \left[ \bigcup _{{\tilde{z}}\in Z^*(x)} \mathcal{P }({\tilde{z}}; x)\right] \right\} . \end{aligned}$$

(30)

$\square$

We need one more technical result before proving Proposition 2, which also has intuitive economic meaning.

Lemma 5

Let $X$ be given as $\{x\in \mathfrak R ^{FN}: x\ge 0\}$, and $x^*$ be an optimal solution of the Problem (16). Then under the conditions of Lemma 2, for each ${\tilde{z}} \in Z^*(x^*)$ and ${\tilde{\rho }} \in \mathcal{P }({\tilde{z}}; x^*)$, $\alpha _{fi} x_{fi}^*\ge {\tilde{\rho }}_{fi}$, $\forall f\in {\mathcal{F }},\ i\in {\mathcal{N }}$, where $\alpha _{fi}$’s are the coefficients of the capacity expansion variables $x_{fi}$’s in (13).

Proof

By Lemma 2, an optimal solution exists to Problem (16). Let $x^*$ be such a solution. Suppose that there exists a ${\tilde{z}}\in Z^*(x^*)$ and a ${\tilde{\rho }} \in \mathcal{P }({\tilde{z}}; x^*)$ such that for an $h\in {\mathcal{F }}$ and $j\in {\mathcal{N }}$, ${\tilde{\rho }}_{hj} > \alpha _{hj}x_{hj}^*$. Define a vector ${\bar{x}}\in \mathfrak R ^{FN}$ such that ${\bar{x}}_{hj} = x^*_{hj} + \epsilon$ and ${\bar{x}}_{fi} = x^*_{fi}$ for all $h\ne f\in {\mathcal{F }}$ and $j\ne i\in {\mathcal{N }}$, where $\epsilon$ is an arbitrarily small positive number. By the way that the set $X$ is defined, ${\bar{x}}$ is also in $X$. By Lemma 4, ${\tilde{\rho }} \in \partial \mathcal{Q }{x}$. Then we have the following.

$$\begin{aligned} \varPi ({\bar{x}})&= \displaystyle - \sum \limits _{f\in {\mathcal{F }}, i \in {\mathcal{N }}} \frac{1}{2}\alpha _{fi} {\bar{x}}_{fi} + \mathcal{Q }({\bar{x}}) \nonumber \\&\ge \displaystyle - \sum \limits _{f\in {\mathcal{F }}, i \in {\mathcal{N }}} \frac{1}{2}\alpha _{fi} {\bar{x}}_{fi} + \mathcal{Q }(x^*) - {\tilde{\rho }}^T(x^* - {\bar{x}}) \nonumber \\&= \displaystyle - \sum \limits _{f\in {\mathcal{F }}, i \in {\mathcal{N }}} \frac{1}{2}\alpha _{fi} x^*_{fi} - \alpha _{hj} \left( \epsilon x^*_{hj} + \frac{\epsilon ^2}{2}\right) + \mathcal{Q }(x^*) + {\tilde{\rho }}_{hj} \epsilon \nonumber \\&= \displaystyle \varPi (x^*) + \epsilon \left( {\tilde{\rho }}_{hj} - \alpha _{hj}x_{hj}^* - \frac{1}{2}\alpha _{hj}\epsilon \right) , \end{aligned}$$

(31)

where the first inequality is implied by the definition of subgradients. Since it is assumed that ${\tilde{\rho }}_{hj} > \alpha _{hj}x_{hj}^*$, then there must exist an $\epsilon > 0$ such that (${\tilde{\rho }}_{hj} - \alpha _{hj}x_{hj}^* - \frac{1}{2}\alpha _{hj}\epsilon ) > 0$, which implies that $\varPi ({\bar{x}}) > \varPi (x^*)$. This contradicts with the assumption that $x^*$ maximizes $\varPi (x)$ over $X$. Hence, the claim of the lemma holds true. $\square$

The interpretation of Lemma 5 is that for any optimal capacity expansion decision $x_{fi}$, its corresponding (per unit) capital cost must be greater than or equal to the marginal value of expanding that capacity.

Now we are ready to prove Proposition 2, which is restated below (in more mathematically precise statements) to ease the presentation.

Proposition 2

Assume that the set $X$ is given as $\{x\in \mathfrak R ^{FN}: x\ge 0\}$. Under the conditions in Proposition 1, if a tuple $(x^*_,\ d^*_,\ s^*_,\ g^*_,\ y^*_,\ \zeta ^*)$ solves Model FJD, where $\zeta ^*\equiv (\ w^*_,\ {\tilde{p}}^*_,\ p^*_,\ p^{{CO_2}^*})$, then $x^*$ solves the two-stage optimization problem (16) with $(d^*_,\ s^*_,\ g^*_,\ y^*) \in {\arg \max }\{Q(z): {z\in Z(x^*)}\}$. Conversely, if $x^*$ solves (16) with $(d^*, s^*, g^*, y^*) \in {\arg \max }\{Q(z): {z\in Z(x^*)}\}$, then there exists a vector $\zeta ^*$ such that $(x^*, d^*, g^*, y^*, \zeta ^*)$ solves Model FJD. In addition, the two-stage optimization problem (16) has a unique solution in capacity expansion decisions.

Proof

By Lemma 3, the optimal value function $\mathcal{Q }(x)$ is concave over $X$. Together with the convex quadratic investment functions (13) and the assumption on the set $X$, the two-stage optimization problem (16) is a convex problem. By the general result of optimality conditions of convex programming (see Theorem 28.3 in Rockafellar 1997), $x$ is an optimal solution to (16) if and only if there exist $\gamma \in \mathfrak R ^{FN}$ such that

$$\begin{aligned} 0&\in \varLambda ^T x - \partial \mathcal{Q }(x) - \gamma , \nonumber \\ 0&\le x \perp \gamma \ge 0, \end{aligned}$$

(32)

where $\varLambda \in \mathfrak R ^{FN\times FN}$ is a diagonal matrix with the diagonal entries being the investment cost parameters $\alpha _{fi}$’s in (13). The optimality condition (32) can be equivalently stated as that for some $\xi \in \partial \mathcal{Q }(x^*)$,

$$\begin{aligned} 0\le x \perp \varLambda ^Tx - \xi \ge 0. \end{aligned}$$

(33)

Now we first assume that $(x^*_,\ d^*_,\ s^*_,\ g^*_,\ y^*_,\ \zeta ^*)$ solves Model FJD. Since the optimization problems in Model FJD are all convex with linear constraints, the first-order conditions are both necessary and sufficient conditions of optimality. By grouping the KKT systems of problems (1), (5), (9), (10), together with the market clearing conditions (11), it is straightforward to check that the resulting mixed complementarity problem (minus the first-order condition corresponding to the variable $x^*_f$ of the generator’s problem (9)) is exactly the KKT system of the 2nd-stage optimization problem (19), with $x$ set at $x^*$. Since (19) is a convex program as well with a fixed $x$, then satisfying the KKT system is a sufficient optimality condition; and hence, $z^* \equiv (d^*_,\ s^*_,\ g^*_,\ y^*) \in {\arg \max }\{Q(z): {z\in Z(x^*)}\}$. Let $\rho ^*$ denote the vector of Lagrange multipliers associated with the capacity constraints in the generating firms’ optimization problem (9). Then $\rho ^* \in \mathcal{P }(z^*; x^*)$, the set of Lagrange multipliers corresponding to $z^*$ and associated with the capacity constraints in (19). By Lemma 4, $\rho ^* \in \partial \mathcal{Q }(x^*)$. Then the first-order condition corresponding to the variable $x^*_f$ of the generator’s problem (9) is exactly the same as (33), and hence, $x^*$ is an optimal solution of (16).

Conversely, if $x^*$ is an optimal solution to the optimization problem (16), then (33) holds for some $\xi \in \partial \mathcal{Q }(x^*)$. By Lemma 4, we know that $\xi \in \text{ conv }\{\cup _{{\tilde{z}}\in Z^*(x^*)} \mathcal{P }({\tilde{z}}; x^*)\}$. Let $\mathcal M ({\tilde{z}}; x^*)$ denote the set of Lagrange multipliers associated with a ${\tilde{z}}\in Z^*(x^*)$, the optimal solution set of the 2nd-stage problem (19) with the given $x^*$. $\mathcal M ({\tilde{z}}; x^*)$ is convex by the well-known fact about Lagrange multiplier sets (see, for example, Bertsekas 1999). Then the set $\mathcal{P }({\tilde{z}}; x^*)$ as a projection of $\mathcal M ({\tilde{z}}; x^*)$ onto the $\rho$-space is also a convex set. By a corollary to the Carathéodory’s Theorem (Corollary 17.1.1 in Rockafellar 1997), which states that the convex hull of a collection of convex sets can be expressed by a convex combination of affinely independent points, each belonging to a different convex set in the collection, we can write $\xi$ as follows.

$$\begin{aligned} \displaystyle \xi = \sum \limits _{m=0}^{FN} \tau _m {\tilde{\rho }}^m,\ \text{ with }\ {\tilde{\rho }}^m\in \mathcal{P }^m({\tilde{z}};x^*),\ \sum \limits _{m=0}^{FN} \tau _m = 1,\ \tau _m \ge 0,\ \forall m = 0,\ldots ,\! FN.\nonumber \\ \end{aligned}$$

(34)

By (33), with the same ${\tilde{\rho }}^m$’s and $\tau _m$’s as in (34), we have that

$$\begin{aligned} 0&\le x^* \perp \varLambda ^Tx^* - \sum \limits _{m=0}^{FN} \tau _m {\tilde{\rho }}^m \ge 0 \Longrightarrow 0\\&\le x^*_{fi} \perp \sum \limits _{m=0}^{FN}\tau _m(\alpha _{fi}x_{fi}^* - {\tilde{\rho }}^m_{fi}) \ge 0,\quad \forall f \in {\mathcal{F }}, i\in {\mathcal{N }}. \end{aligned}$$

Let $\mathcal{I }$ be an index set such that $\mathcal{I } := \{(f, i) \in {\mathcal{F }} \times {\mathcal{N }}:\ x_{fi}^* > 0 \}$. By complementarity, for each $(h, j) \in \mathcal{I }$, $\displaystyle \sum \nolimits _{m=0}^{FN}\tau _m(\alpha _{hj}x_{hj}^* - {\tilde{\rho }}^m_{hj}) = 0$. By Lemma 5, $\alpha _{hj}x_{hj}^*- {\tilde{\rho }}^m_{hj} \ge 0$ for each $m$. Hence, $\alpha _{hj}x_{hj}^* - {\tilde{\rho }}^m_{hj} = 0$ for each $(h, j) \in \mathcal{I }$ and $m = 0, \ldots , FN$. Then pick an arbitrary $m$ from $0, \ldots , FN$, we have that

$$\begin{aligned} 0\le x^* \perp \varLambda x^* - {\tilde{\rho }}^m \ge 0. \end{aligned}$$

(35)

Let $z^*\equiv (d^*_,\ s^*_,\ g^*_,\ y^*)$ be the optimal solution of (19) corresponding to the multiplier ${\tilde{\rho }}^m$ with the given $x^*$, and let $\mathcal M ^m(z^*; x^*)$ be the corresponding multiplier set. Then for any $\zeta ^* \in \mathcal M ^m(z^*; x^*)$, it is straightforward to check that the KKT system of (19), together with (35), is exactly the same as the mixed complementarity problem of Model FJD, which consists of the KKT systems of (1), (5), (9) and (10), together with the market clearing conditions (11). Hence, the tuple $(x^*_,\ z^*_,\ \zeta ^*)$ solves Model FJD.

The uniqueness of solutions to the optimization model (16) follows directly from the above equivalence result and Proposition 1 (part (b)). $\square$