General modeling assumptions and network setting
The deterministic equilibrium model discussed in this section is based on the electricity market equilibrium model given in [22], where the authors apply a well-simplified application of the study published in [30]. In the latter paper, an LCP for a Nash–Cournot market structure in bilateral or pool-type electricity markets is introduced; see [47] for a detailed version of this equilibrium model. Moreover, in [22] a stochastic version of the original equilibrium model is considered and solved with a generalized Benders decomposition approach. A robust version of this electricity market equilibrium model is presented in [39]. On the one hand, we simplify the economic setting by considering a perfectly competitive market but extend the model in [39] by incorporating generation and transmission investment decisions that affect the capacity of generators and transmission lines. Moreover, the specific setting of the robustification differs compared to the one studied in [39] since we do not bound the number of uncertainty realizations per consumer over time but bound the number of uncertainty realizations over the set of all consumers.
The basic assumptions of our generation and transmission investment planning model are as follows. We consider an equilibrium model for perfectly competitive day-ahead markets with transmission constraints. Balancing or real-time markets are not considered. As a common practice in the literature, transmission and generation investments will be done for a “target year” in the future; see, e.g., [14]. However, note that it can be extended to model a dynamic investment model for each year in the planning horizon; see, e.g., [37]. In compliance with the latter point, investment costs are discounted on an hourly basis. Potential generation investments are applicable for certain firms and buses, and they are bounded above. Similarly, transmission line investments are defined between certain buses and they are considered to have upper bounds as they are constrained by a certain available budget. Finally, for the ease of presentation, existing line capacity can be expanded without changing the line’s impedance in our market models. This simplifying assumption can be relaxed as in [49].
In our model, electricity generators can sell to all consumers in the entire system and they use the transmission system operator (TSO) as a mediator. In this structure, generating firms optimize their profits according to capacity and generation-sales constraints and the TSO optimizes its transmission service revenue according to the network constraints. The latter are modeled using lossless linear DC (direct current) load flow constraints. In addition, consumers change their amount of consumption as a reaction to price levels for optimizing their utility.
In this section, we first define each decision maker’s deterministic optimization problem separately and then form the overall equilibrium problem by concatenating each problem’s optimality conditions. Together with nodal flow balance equations, this leads to an MCP. The solutions of this MCP are market equilibria and the nodal electricity prices are, as usual, obtained as dual variables of the nodal balance equations; see, e.g., [14, 30]. Due to the fact that we consider a perfectly competitive market, all players act as price takers and we can thus state their optimization problems using exogenously given market prices.
In what follows, we model the electricity transmission network by using a connected and directed graph \(G = (I, A)\) with node (or bus) set I and arc set A. Transmission lines \(a \in A\) are usually denoted by its start and end points, e.g., \(a = (i,j)\) for start point \(i \in I\) and end point \(j \in I\). All notation used in the model is given in Table 1.
Table 1 Indices (top), variables (middle), and parameters (bottom) of the model Consumers
We start by introducing the models of the consumers that are located at the nodes \(i\in I\) of the network. The consumers decide on their demand \(d_i \ge 0\) and their willingness to pay is modeled by inverse market demand functions \(p_i = p_i(d_i)\). For the latter functions, we assume that they are continuous and strictly decreasing. Under this assumption, the gross consumer surplus
$$\begin{aligned} \int _0^{d_i} p_i(\omega ) \,\mathrm {d}\omega \end{aligned}$$
is a strictly concave function in \(d_i\) and the benefit maximization problem
$$\begin{aligned} \max _{d_i}&\quad \int _0^{d_i} p_i(\omega ) \,\mathrm {d}\omega -\pi _i d_i \end{aligned}$$
(1a)
$$\begin{aligned} \text {s.t.}&\quad d_i\ge 0 \end{aligned}$$
(1b)
of the consumer at node \(i\in I\) thus is a strictly concave maximization problem. Here and in what follows, \(\pi _i\) denotes the exogenously given market price at node \(i\in I\).
Generating firms
Every generating firm \(f \in F\) solves the problem
$$\begin{aligned} \max _{s_{f},x_{f},{\Delta K}_{f}} \quad&\sum _{i\in I} \pi _i s_{fi} - \sum _{i\in I_f} c_{fi}^\text {op}x_{fi} - \sum _{i\in I_f} c_{fi}^\text {inv}\Delta K_{fi} \end{aligned}$$
(2a)
$$\begin{aligned} \text {s.t.}\quad&\sum _{i\in I}s_{fi} - \sum _{i\in I_f} x_{fi} = 0,&[\nu _f] \end{aligned}$$
(2b)
$$\begin{aligned}&x_{fi}\le K_{fi} + \Delta K_{fi}, \quad i\in I_f,&[\mu _{fi}] \end{aligned}$$
(2c)
$$\begin{aligned}&\Delta K_{fi} \le \Delta K_{fi}^+, \quad i\in I_f,&[\delta _{fi}] \end{aligned}$$
(2d)
$$\begin{aligned}&x_{fi} \ge 0, \quad \Delta K_{fi} \ge 0, \quad i\in I_f, \end{aligned}$$
(2e)
$$\begin{aligned}&s_{fi}\ge 0,\quad i\in I, \end{aligned}$$
(2f)
where \(\Delta K_f = (\Delta K_{fi})_{i\in I_f}\) is the vector of all capacity investments of firm f, \(x_f = (x_{fi})_{i\in I_f}\) is the vector of all generations, and \(s_f = (s_{fi})_{i\in I}\) is the vector comprising all sales. The generating firm f is modeled as a price-taker, i.e., it assumes that the price at every single bus is exogenously given. The firms maximize their profits, which are given by revenue from sales less operating costs less discounted generation investment costs; see the objective function in (2a). Constraint (2b) models the balance of electricity generation and sales, capacity constraints are modeled in (2c), and upper bounds on the capacity investments are given in (2d). As it can be seen in (2c), generation investments affect the capacity constraints. Here and in what follows, dual variables are denoted by Greek letters and are given in parentheses next to the constraints. Finally, (2e) and (2f) ensure nonnegativity of generation, sales, and capacity investments. Since dual variables of simple nonnegativity constraints will be directly eliminated later, we do not state them here explicitly.
Transmission system operator
The model of the transmission system operator (TSO) is given by
$$\begin{aligned} \max _{\theta ,\Delta T} \quad&\sum _{(i,j)\in A} \left( \pi _j - \pi _i\right) B_{ij} \left( \theta _i-\theta _j\right) - \sum _{(i,j)\in A} c_{ij}^\text {exp}\Delta T_{ij} \end{aligned}$$
(3a)
$$\begin{aligned} \text {s.t.}\quad&B_{ij} \left( \theta _i-\theta _j \right) \le T_{ij} + \Delta T_{ij}, \quad (i,j) \in A,&[\lambda _{ij}^+] \end{aligned}$$
(3b)
$$\begin{aligned}&-B_{ij} \left( \theta _i-\theta _j\right) \le T_{ij} + \Delta T_{ij}, \quad (i,j) \in A,&[\lambda _{ij}^-] \end{aligned}$$
(3c)
$$\begin{aligned}&\Delta T_{ij} \le \Delta T_{ij}^+, \quad (i,j) \in A,&[\gamma _{ij}] \end{aligned}$$
(3d)
$$\begin{aligned}&-\pi \le \theta _i \le \pi , \quad i \in I\setminus \{i_0\},&[\varepsilon _i^-, \varepsilon _i^+] \end{aligned}$$
(3e)
$$\begin{aligned}&\theta _{i_0} = 0,&[\xi ] \end{aligned}$$
(3f)
$$\begin{aligned}&\Delta T_{ij} \ge 0, \quad (i,j) \in A, \end{aligned}$$
(3g)
where \(\theta = (\theta _i)_{i\in I}\) is the vector of all phase angles in the network and \(\Delta T= (\Delta T_{ij})_{(i,j)\in A}\) comprises all transmission line capacity investments. The objective of the TSO is to effectively distribute the transmission system services considering lossless DC network constraints and to optimize its revenues obtained due to these operations. The TSO’s revenue optimization in this manner, in fact, ensures that firms cannot use market power to obtain more transmission rights in the competitive market; see [30]. In other words, the system operator works as an arbitrageur who benefits from price differences between nodes. Furthermore, in this model, the TSO also decides on transmission line capacity investments \(\Delta T_{ij}\) for all transmission lines \((i,j) \in A\). The objective function (3a) denotes the revenue of the TSO, calculated as the price differences multiplied by power flows less discounted transmission line expansion costs. Constraints (3b) and (3c) model lossless DC power flow. Upper bounds on the transmission line expansion are given in (3d) and (3e) represents lower and upper bounds on the phase angles \(\theta _i\), \(i \in I\).Footnote 1 The phase angle of the reference bus \(i_0\) is fixed in (3f) to ensure a unique physical solution and, finally, (3g) ensures nonnegativity of capacity investments.
Market clearing
As market clearing conditions we use the nodal flow balance equations
$$\begin{aligned} d_i - \sum _{f \in F} x_{fi} + \sum _{(i,j)\in A} B_{ij} \left( \theta _i - \theta _j\right) -\sum _{(j,i)\in A}B_{ji}\left( \theta _j-\theta _i\right) = 0, \quad i \in I. \end{aligned}$$
(4)
Note that demand \(d_i\) at node i is the sum of all firms’ sales to that node, i.e., \(d_i = \sum _{f \in F} s_{fi}\).
A mixed complementarity market equilibrium model
The market equilibrium model including generation and transmission investments is mainly taken from [9] and it is presented as the following MCP, which is obtained by concatenating the optimality conditions (that are both necessary and sufficient in our case) of all players and the market clearing conditions.Footnote 2
$$\begin{aligned} 0\le d_i \perp&\ \pi _i-p_i(d_i)\ge 0,&i\in I, \end{aligned}$$
(5a)
$$\begin{aligned} 0\le s_{fi} \perp&\ \nu _f-\pi _i \ge 0,&f \in F, \ i \in I, \end{aligned}$$
(5b)
$$\begin{aligned} 0\le x_{fi} \perp&\ c_{fi}^{\text {op}} - \nu _f + \mu _{fi} \ge 0,&f \in F, \ i \in I_f, \end{aligned}$$
(5c)
$$\begin{aligned} 0\le \Delta K_{fi} \perp&\ c_{fi}^{\text {inv}} - \mu _{fi} + \delta _{fi} \ge 0,&f \in F, \ i \in I_f, \end{aligned}$$
(5d)
$$\begin{aligned} \nu _f \text { free} \perp&\ \sum _{i \in I} s_{fi} - \sum _{i \in I_f} x_{fi} = 0,&f \in F, \end{aligned}$$
(5e)
$$\begin{aligned} 0\le \mu _{fi} \perp&\ K_{fi} + \Delta K_{fi}-x_{fi}\ge 0,&f \in F, \ i \in I_f, \end{aligned}$$
(5f)
$$\begin{aligned} 0\le \delta _{fi} \perp&\ \Delta K_{fi}^+ - \Delta K_{fi}\ge 0,&f \in F, \ i \in I_f, \end{aligned}$$
(5g)
$$\begin{aligned} 0\le \Delta T_{ij} \perp&\ c_{ij}^{\text {exp}} - \lambda _{ij}^- - \lambda _{ij}^+ + \gamma _{ij} \ge 0,&(i,j)\in A, \end{aligned}$$
(5h)
$$\begin{aligned} 0\le \lambda _{ij}^+ \perp&\ T_{ij} + \Delta T_{ij}-B_{ij} \left( \theta _i-\theta _j \right) \ge 0,&(i,j)\in A, \end{aligned}$$
(5i)
$$\begin{aligned} 0\le \lambda _{ij}^- \perp&\ T_{ij} + \Delta T_{ij}+B_{ij} \left( \theta _i-\theta _j\right) \ge 0,&(i,j)\in A, \end{aligned}$$
(5j)
$$\begin{aligned} 0\le \gamma _{ij} \perp&\ \Delta T_{ij}^+-\Delta T_{ij}\ge 0,&(i,j)\in A, \end{aligned}$$
(5k)
$$\begin{aligned} 0\le \varepsilon _i^{+} \perp&\ \pi -\theta _i \ge 0,&i \in I\setminus \{i_0\}, \end{aligned}$$
(5l)
$$\begin{aligned} 0\le \varepsilon _i^{-} \perp&\ \theta _i + \pi \ge 0,&i \in I\setminus \{i_0\}, \end{aligned}$$
(5m)
$$\begin{aligned} \theta _i \text { free} \perp&\ \sum _{(i,j)\in A} B_{ij} \left( \pi _j-\pi _i\right) -\sum _{(j,i)\in A} B_{ji} \left( \pi _i-\pi _j\right) \nonumber \\&\ +\sum _{(i,j)\in A} B_{ij} \left( \lambda _{ij}^- - \lambda _{ij}^+ \right) - \sum _{(j,i)\in A} B_{ji} \left( \lambda _{ji}^--\lambda _{ji}^+\right) \nonumber \\&\ - \varepsilon _i^{+} + \varepsilon _i^{-} = 0,&i \in I\setminus \{i_0\}, \end{aligned}$$
(5n)
$$\begin{aligned} \theta _{i_0} \text { free} \perp&\ \sum _{(i,j)\in A} B_{ij} \left( \pi _j-\pi _i\right) -\sum _{(j,i)\in A} B_{ji} \left( \pi _i-\pi _j\right) \nonumber \\&\ +\sum _{(i,j)\in A} B_{ij} \left( \lambda _{ij}^- - \lambda _{ij}^+ \right) - \sum _{(j,i)\in A} B_{ji} \left( \lambda _{ji}^--\lambda _{ji}^+\right) =0,&\end{aligned}$$
(5o)
$$\begin{aligned} \xi \text { free} \perp&\ \theta _{i_0} = 0,&\end{aligned}$$
(5p)
$$\begin{aligned} \pi _i \text { free} \perp&\ d_i - \sum _{f \in F} x_{fi} + \sum _{(i,j)\in A} B_{ij} \left( \theta _i - \theta _j\right) \nonumber \\&\ -\sum _{(j,i)\in A}B_{ji}\left( \theta _j-\theta _i\right) = 0,&i \in I. \end{aligned}$$
(5q)
Note that the market clearing conditions are equipped with the beforehand exogenously given market prices as dual variables. Thus, we obtain a system in the primal variables \(d, s, x, \Delta K, \theta , \Delta T\) and in the dual variables \(\nu , \mu , \delta , \lambda ^+, \lambda ^-, \gamma , \varepsilon ^{-}, \varepsilon ^{+}, \xi , \pi \). A solution of this system, by construction, corresponds to solutions of the separate optimization problems presented in Sects. 2.2–2.4 that also satisfy the market clearing conditions (4). Thus, a solution of (5) is a market equilibrium and \(\pi = (\pi _i)_{i \in I}\) is the vector of market clearing nodal prices.
An equivalent welfare maximization problem
It is well-known that the MCP (5), which models the wholesale electricity market under perfect competition, is equivalent to the welfare maximization problem (WMP)
$$\begin{aligned} \max _{z} \quad&\sum _{i\in I} \int _0^{d_i} p_i(\omega ) \,\mathrm {d}\omega - \sum _{f\in F} \left( \sum _{i\in I_f} c_{fi}^{\text {op}}x_{fi} + \sum _{i\in I_f} c_{fi}^{\text {inv}}\Delta K_{fi}\right) - \sum _{(i,j)\in A}c_{ij}^{\text {exp}}\Delta T_{ij} \end{aligned}$$
(6a)
$$\begin{aligned} \text {s.t.}\quad&\text {Consumers: } (1\mathrm{b}) \quad \text {for all } i \in I, \end{aligned}$$
(6b)
$$\begin{aligned}&\text {Generating firms: } (2\mathrm{b})\text {--}(2\mathrm{f}) \quad \text {for all } f \in F, \end{aligned}$$
(6c)
$$\begin{aligned}&\text {TSO: } (3\mathrm{b})\text {--}(3\mathrm{g}), \end{aligned}$$
(6d)
$$\begin{aligned}&\text {Market clearing: } (\mathrm{4}) \end{aligned}$$
(6e)
with variables \(z = (d^\top , s^\top , x^\top , \Delta K^\top , \theta ^\top , \Delta T^\top )^\top \) as before. The equivalence can be shown by comparing the first-order optimality conditions of Problem (6)—which are, again, necessary and sufficient—with the MCP (5) and by identifying the dual variables of the market clearing conditions in (6e) with the equilibrium prices \(\pi _i\), \(i\in I\), of the MCP.
An equivalent variational inequality
In this section, we also present an equivalent formulation of the MCP model given in Sect. 2.6 as a variational inequality (VI). In general, the latter is given as the following problem. Given a feasible set \(K \subseteq \mathbb {R}^n\) and a vector-valued mapping \(G : \mathbb {R}^n \rightarrow \mathbb {R}^n\), the variational inequality problem \(\text {VI}(G,K)\) is to find a vector \(z^* \in K\) that satisfies
$$\begin{aligned} G(z^*)^\top (z - z^*) \ge 0 \quad \text {for all } z \in K. \end{aligned}$$
(7)
One advantage of VI formulations (compared to MCPs) is that only primal variables appear in the formulation. In the context of the market equilibrium problem studied so far, the feasible set is given by the feasible sets of all players in the market equilibrium problem and the market clearing conditions, i.e.,
$$\begin{aligned} K = \{z:(6\mathrm{b})\text {--}(6\mathrm{e}) \text { are satisfied}\}. \end{aligned}$$
(8)
Note that this set is a convex polyhedron. The variable vector of the VI thus is given by z and the VI’s mapping G is defined as
$$\begin{aligned} G(v) = \begin{pmatrix} -(p_i(d_i))_{i \in I}\\ 0\\ (c_{fi}^\text {op})_{f \in F, i \in I_f}\\ (c_{fi}^\text {inv})_{f\in F, i\in I_f}\\ 0\\ (c_{ij}^\text {exp})_{(i,j)\in A} \end{pmatrix}, \end{aligned}$$
where 0 here stands for the zero vector in appropriate dimension.
We close this section with some brief comments on existence and uniqueness of market equilibria. Existence can be easily shown using the VI approach of this section. Since the function G is continuous and because a nonempty, convex, and compact set \({\tilde{K}} \subseteq K\) exists that contains all solutions of the \(\text {VI}(G,K)\), standard VI theory can be applied that ensures the existence of a solution. Since the VI is equivalent to the MCP (5) and to the welfare maximization problem (6), this also implies the existence of solutions for these two formulations. The situation is much more complicated when it comes to uniqueness of solutions. To the best of our knowledge, there is no result in the literature that can be applied directly to the setting studied in this paper. For a related long-run model without DC power flow constraints, uniqueness of market equilibria is shown in [28]. Moreover, uniqueness of the solution of a short-run model, again without DC power flow constraints is proven in [42] for the case of transport costs. However, the most related study is given in [41]. There, a short-run market equilibrium model is analyzed that also incorporates DC power flow constraints. It is shown that equilibria are, in general, not unique. Thus, we do not expect uniqueness of solutions for the setting considered in this paper.