9. Interactions between actors: Mixed Complementarity Problems

In this section, we consider an approach to depict how relations between actors affect the outcome of the electricity markets. We will study electricity market situations as Nash and Nash-Cournot games. These games can be represented as a set of optimisation problems, describing the decision problems of each of the involved actors, that are linked together via a common coupling constraint. We can formulate this as a Mixed Complementarity Problem (MCP): the set of KKT conditions of the decision problems of the involved agents and the coupling constraint.

9.1. The market clearing problem as a Nash Game

Before we dive into the MCP formulation, we’ll revisit the market clearing problem, introduced in Section 5, formulated an optimization problem.

9.1.1. Revisiting the market clearing problem

In Section 5.5 we discussed the clearing of a day-ahead power market, from the perspective of a market operator, when the price-quantity bids and offers of the supply and demand agents are given. Using those price-quantity pairs, the market operator clears the market in a way that maximizes social welfare.

Let \((P_i^s, Q_i^s), (P_j^d, Q_j^d)\) be the price-quantity pairs of supply and demand agents. This market clearing problem can be formulated as :

(9.1)\[\begin{split}\begin{align} & \max \sum_{j \in J} P^d_j \cdot q^d_j - \sum_{i \in I} P^s_i \cdot q^s_i \tag{1} \\ & \text{s.t:}\\ & \sum_{j \in J} q^d_j - \sum_{i \in I} q^s_i = 0 \quad (\lambda) \tag{2}\\ & 0 \leq q^s_i \leq Q^s_i \quad (\underline{μ_i}, \overline{μ_i}) \quad \forall i \in I \tag{3}\\ & 0 \leq q^d_j \leq Q^d_j \quad (\underline{ν_j}, \overline{ν_j}) \quad \forall j \in J \tag{4}\\ \end{align}\end{split}\]

Note this is exactly the same problem that we formulated with “bids” and “offers” in Section 5.5, just with different notation.

This market clearing problem allows for determining the demand served \( \sum_{j \in J} q^d_j\) and supply cleared \(\sum_{i \in I} q^s_i\) in the market. The market clearing price \(\lambda\) is the dual variable associated with the supply-demand balance.

We can write the Karush Kuhn Tacker conditions for the market clearing problem of Eq. (9.1), as:

(9.2)\[\begin{split}\begin{align} & \sum_{j \in J} q^d_j - \sum_{i \in I} q^s_i = 0 \quad (\lambda) \tag{2} \\ & 0 \leq q^s_i \leq Q^s_i \quad (\underline{\mu_i}, \overline{\mu_i}) \quad \forall i \in I \tag{3} \\ & 0 \leq q^d_j \leq Q^d_j \quad (\underline{\nu_j}, \overline{\nu_j}) \quad \forall j \in J \tag{4} \\ & -P_j^d + \lambda - \underline{\nu_j} + \overline{\nu_j} = 0 \quad \forall j \in J \tag{5a} \\ & P_i^s - \lambda - \underline{\mu_i} + \overline{\mu_i} = 0 \quad \forall i \in I \tag{5b} \\ & -\underline{\nu_j} q_j^d = 0 \quad \forall j \in J \tag{6a} \\ & \overline{\nu_j} \cdot (q_j^d - \overline{Q_j^d}) = 0 \quad \forall j \in J \tag{6b} \\ & -\underline{\mu_i} q_i^s = 0 \quad \forall i \in I \tag{6c} \\ & \overline{\mu_i} \cdot (q_i^s - \overline{Q_i^s}) = 0 \quad \forall i \in I \tag{6d} \\ & \underline{\mu_i}, \overline{\mu_i} \geq 0 \quad \forall i \in I \tag{7a} \\ & \underline{\nu_j}, \overline{\nu_j} \geq 0 \quad \forall j \in J \tag{7b} \\ \end{align}\end{split}\]

9.1.2. The market clearing problem from the perspective of the participants

Above, we presented the clearing of the market as faced by the market operator, who aims to maximise social welfare, assuming known supply offers and demand bids. However, in real-life markets, supply and demand agents aim to maximize their own profit or utility.

The supply agents want to maximise their profit. We assume the electricity suppliers are price-taking, meaning they don’t influence the market price, and hence, consider this as a parameter in their own decision problem. The optimisation problem of each agents \(i\) in Eq. (9.3) reads:

(9.3)\[\begin{split}\begin{align} & \max (λ - VC_i) \cdot q^s_i \tag{8} \\ & \text{s.t:}\\ & 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{\mu_i}, \overline{\mu_i}) \tag{9} \\ \end{align}\end{split}\]

where :

\[\begin{align*} & VC_i = \text{Variable Cost of generator } i \\ & \overline{Q^s_i} = \text{the maximum output of generator } i \\ \end{align*}\]

The KKT conditions of the problem in Eq. (9.3) are the following :

(9.4)\[\begin{split}\begin{align} & VC_i - λ - \underline{μ_i} + \overline{μ_i} =0 \tag{10a} \\ & 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{μ_i}, \overline{μ_i}) \tag{9} \\ & -\underline{μ_i} \cdot q^s_i = 0 \tag{10b} \\ & \overline{μ_i} \cdot (q^s_i - \overline{Q^s_i}) = 0 \tag{10c} \\ & \underline{μ_i}, \overline{μ_i} \geq 0 \tag{10d} \\ \end{align}\end{split}\]

Similarly, the demand agents aim to maximize their utility, again considering the clearing price as given. In other words, the demand agents are also price-takers. The optimisation problem of each agent \(i\) in Eq. (9.5) reads:

(9.5)\[\begin{split}\begin{align} & \max (WTP_j - λ) \cdot q^d_j \tag{11} \\ & \text{s.t:}\\ & 0 \leq q^d_j \leq \overline{Q^d_j} \quad (\underline{ν_j}, \overline{ν_j}) \tag{12} \\ \end{align}\end{split}\]

where :

\[\begin{align*} & WTP_j = \text{Willingness to pay of demand agent } j \\ & \overline{Q^s_i} = \text{the maximum consumption capacity of generator } j \\ \end{align*}\]

The KKT conditions of the problem in Eq. (9.5) are :

(9.6)\[\begin{split}\begin{align} & -WTP_j + λ - \underline{ν_j} + \overline{ν_j} = \tag{13a} \\ & 0 \leq q^d_j \leq \overline{Q^d_j} \quad (\underline{ν_j}, \overline{ν_j}) \tag{12}\\ & - \underline{ν_j} \cdot q^d_j = 0 \tag{} \tag{13b} \\ & \overline{ν_j} \cdot (q^d_j - \overline{Q^d_j}) = 0 \tag{13c} \\ & \underline{ν_j}, \overline{ν_j} \geq 0 \tag{13d} \\ \end{align}\end{split}\]

Equations (9.3) and (9.5) presented the optimisation independently faced by each supply and demand agent. But as a basic principle, the total supply has to be equal to the total demand (Eq. (2) of the market clearing problem in Section 5.5). This is the linking constraint bringing together the different optimisation problems.

\[\begin{split}\begin{align} & \sum_{j \in J} q^d_j - \sum_{i \in I} q^s_i = 0 \quad (\lambda) \tag{2}\\ \end{align}\end{split}\]

Since the firms do not influence the price and compete on both prices and quantities, the electricity market can be described as a Nash game.

To determine the equilibrium, we have to simultaneously solve the supply-side problems (9.3), the demand-side problems (9.5) and the linking constraint (2). Thus, we have to find an optimal solution for all optimisation problems at once. This requires finding a solution to all Karush Kuhn Tucker conditions at once. We can concatenate the KKT conditions of the three problems as:

(9.7)\[\begin{split}\begin{align} & VC_i - λ - \underline{μ_i} + \overline{μ_i} =0 \tag{10a} \\ & 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{μ_i}, \overline{μ_i}) \tag{9} \\ & -\underline{μ_i} \cdot q^s_i = 0 \tag{10b} \\ & \overline{μ_i} \cdot (q^s_i - \overline{Q^s_i}) = 0 \tag{10c} \\ & \underline{μ_i}, \overline{μ_i} \geq 0 \tag{10d} \\ & -WTP_j + λ - \underline{ν_j} + \overline{ν_j} = \tag{13a} \\ & 0 \leq q^d_j \leq \overline{Q^d_j} \quad (\underline{ν_j}, \overline{ν_j}) \tag{12}\\ & - \underline{ν_j} \cdot q^d_j = 0 \tag{} \tag{13b} \\ & \overline{ν_j} \cdot (q^d_j - \overline{Q^d_j}) = 0 \tag{13c} \\ & \underline{ν_j}, \overline{ν_j} \geq 0 \tag{13d} \\ & \sum_{j \in J} q^d_j - \sum_{i \in I} q^s_i = 0 \quad (\lambda) \tag{2}\\ \end{align}\end{split}\]

By concatenating the optimality conditions of the supply and demand agent dispatch problems and the linking constraint, we have formed a Mixed Complementarity Problem.

9.1.3. Solving MCPs

There are three different approaches to solving the MCP and calculating the clearing price of the market:

• This MCP can be solved directly either with dedicated solvers such as PATH or as a feasibility problem (NLP or MILP) using optimisation. In simple cases, you can determine the equilibrium solution analytically by deriving best response functions (see Section 9.2). Larger problems, however, are very challenging to solve directly as we are - by definition, due to the complementarity conditions - dealing with non-linear problems.

• Apply price-search algorithms and iterative methods. The general principle of such methods is as follows: we start by making an educated guess on what the equilibrium price \(\lambda\) in the game is. We present this price to each of the agents in the game and solve their decision problems. Those solutions are optimal from their perspective, but may not ensure that demand and supply are matched (Eq. (2)). Based on the mismatch in demand and supply, we adjust the price \(\lambda\) accordingly and repeat this process until we have found the equilibrium price \(\lambda\). We will not cover these techniques here.

• For some MCPs, we can derive an equivalent optimisation problem (EOP) and solve it like we do other problems - using mathematical programming languages such as YALMIP or Pyomo and solvers such as Gurobi. This builds on the notion that if we can find an optimisation problem that is described by the same set of KKT conditions as the MCP, solutions to that optimisation problem will – by definition – also be solutions to the MCP.

We’ll illustrate the last approach on the market clearing problem. Comparing the KKTs of the concatenated (supply + demand + linking constraint) problem in Eq. (9.7) with those of the market clearing problem (market operator perspective) in Eq. (9.1), we can deduce that there is only a difference in the following constraints:

(9.8)\[\begin{align} & VC_i - λ - \underline{μ_i} + \overline{μ_i} =0 \tag{10a} \\ & P^s_i - \lambda - \underline{\mu_i} + \overline{\mu_i} = 0 \quad \forall i \in I \tag{5b} \\ & -WTP_j + λ - \underline{ν_j} + \overline{ν_j} = \tag{13a} \\ & -P^d_j + \lambda - \underline{\nu_j} + \overline{\nu_j} = 0 \quad \forall j \in J \tag{5a} \end{align}\]

However, we know that in competitive electricity markets generators bid their capacity at variable cost and demand agents at their willingness to pay. Hence, we can say \(VC_i = P^s_i\) and \(WTP_j = P^d_j\). Therefore, constraint (10a) is equal to constraint (5b) and (13a) to (5a). Through this observation, we can conclude that the KKTs of the concatenated dispatch problems (supply + demand+ linking constraint) are the same as those of the market clearing problem.

Since the two problems described by (9.7) and (9.1) have the same set of KKT conditions, the market clearing problem is identical to the dispatch optimisation problem faced by the supply and demand agents. Hence, a solution that satisfies the KKTs of the market clearing problem will also be a solution of the dispatch optimisation problem.

By concatenating the optimality conditions of the supply and demand agent dispatch problems and the linking constraint, we have formed a Mixed Complementarity Problem. Through the equivalence of the KKTs we have also proved that the market clearing problem is the Equivalent Optimisation Problem (EOP) of the Mixed Complementarity Problem ((9.7)).

Warning

Every optimisation has an equivalent MCP, but not every MCP has an equivalent optimisation problem.

9.1.4. Alternative notation using an inverse demand function

We can formulate the MCP problem using an inverse demand function. Instead of the price-quantity bids of the demand agents, we use a linear relationship between the price of electricity and the demanded quantity to represent the demand side of the market. The linear equation takes the form of :

(9.9)\[\begin{align} λ = \overline{λ} - β \cdot \sum_{j \in J} q^d_j \end{align}\]

From the market clearing constraint we know \(\sum_{j \in J} q^d_j = \sum_{i \in I} q^s_i\). Hence, the inverse demand function can be rewritten as \(λ = \overline{λ} - β \cdot \sum_{i \in I} q^s_i\). In this way, we have eliminated the second vector of decision variables \(q^d_j\). The \(\overline{λ}\) is the intercept of the demand curve and signifies the maximum willingness to pay and you can see the graphic interpretation of the inverse curve in Fig. 9.1. Taking all the above into account, the game can be finally reformulated as :

Fig. 9.1 Inverse demand function and supply curve, based on supply-side bid pairs.

(9.10)\[\begin{split}\begin{align} & \max \{(\lambda - VC_i) \cdot q^s_i \quad \forall i \in I \tag{8} \\ & \text{s.t.} \quad 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{\mu_i}, \overline{\mu_i}) \} \tag{9} \\ &\lambda = \overline{\lambda} - \beta \cdot \sum_{i \in I} q^s_i \tag{14} \\ \end{align}\end{split}\]

The KKT conditions of the decision problems of the suppliers and the inverse demand function jointly make up the MCP of this Nash game. Note that the inverse demand function participates as-is in the MCP, but is not part of the optimisation problem (it is outside of the curly brackets). That is the case since we are studying a Nash game, and therefore, the suppliers do not consider the impact of their actions on the price.

(9.11)\[\begin{split}\begin{align} & \{ VC_i - λ - \underline{μ_i} + \overline{μ_i} = 0 \tag{10a} \\ & 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{μ_i}, \overline{μ_i}) \tag{9} \\ & -\underline{μ_i} \cdot q^s_i = 0 \tag{10b} \\ & \overline{μ_i} \cdot (q^s_i - \overline{Q^s_i}) = 0 \tag{10c} \\ & \underline{μ_i}, \overline{μ_i} \geq 0 \tag{10d} \} \\ & λ = \overline{λ} - β \cdot \sum_{i \in I} q^s_i \tag{14} \\ \end{align}\end{split}\]

With the MCP defined, we can now derive the Equivalent optimisation problem to determine the equilibrium solution to the game in Eq. (9.10). Remember that the KKT conditions of the EOP have to match the MCP in order for the EOP to provide a solution to the initial problem. To derive the EOP, we make use of the graphical interpretation of the problem show in Fig. 9.1.

From the discussion above, we know that the Nash game between suppliers and consumers in perfectly competitive electricity markets will maximize social welfare in that market. Hence, we can formulate a candidate EOP that aims to maximize social welfare. Based on Fig. 9.2 and the fact that social welfare is the area between the demand and supply curve, social welfare can be computed as :

\[ SW = (ABDE) - (ABC) - \text{Area of sketched polygon} = \overline{\lambda} \cdot \sum_{i} q^s_i - \frac{1}{2}\beta(\sum_{i} q^s_i)^2 - \sum_{i} VC_i \cdot q^s_i \]

Fig. 9.2 Areas to calculate social welfare from inverse demand and supply curves.

The EOP should furthermore consider the constraints of the suppliers, resulting in the following optimisation problem:

(9.12)\[\begin{split}\begin{align} & \max \overline{\lambda} \cdot \sum_{i} q^s_i - \frac{1}{2}\beta (\sum_{i} q^s_i)^2 - \sum_{i} VC_i \cdot q^s_i \tag{15} \\ & \text{s.t}\\ & 0 \leq q^s_i \leq Q^s_i \quad (\underline{μ_i}, \overline{μ_i}) \quad \forall i \in I \tag{3} \\ \end{align}\end{split}\]

The equivalence of the MCP with the EOP can be demonstrated through the derivation of the KKTs for the EOP in Eq. (9.12). Note that we now have an optimisation problem with quadratic terms in the objective. However, these problems are still convex, hence, KKT conditions are – in all the cases that we consider – necessary and sufficient conditions for optimality. This also means that we can use solver software that supports solving quadratic problems, like Gurobi and Cplex, to efficiently obtain solutions.

9.2. Nash-Cournot games

Up to now, we have assumed a perfectly competitive market, meaning firms have zero market power and act as pure price-takers. This is demonstrated by the example in content:appendix:game-theory. This was a Nash game. On the contrary, in Nash-Cournot games, agents strategically anticipate how their own quantity decisions influence the price (and hence their profit or utility). Each agent takes the decisions of the other players as given, as in a Nash game. In this Section, we will describe how the optimisation problem changes, when agents anticipate the impact of their actions on price.

9.2.1. A simple Nash-Cournot game

To simplify notation, we will make use of an inverse demand curve to simulate the demand side. To allow agents to strategically anticipate how their actions influence the price, the inverse demand curve (14, below) now is part of the optimisation problem (note the position of the curly bracket). The Nash-Cournot game is:

(9.13)\[\begin{split}\begin{align} &\max \{(\lambda - VC_i) \cdot q^s_i \quad &\forall i \in I \tag{8} \\ & \text{s.t.} \quad 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{\mu_i}, \overline{\mu_i}) \tag{9} \\ &\lambda = \overline{\lambda} - \beta \cdot \sum_{i \in I} q^s_i \} \tag{14} \\ \end{align}\end{split}\]

The MCP associated with the Nash-Cournot game in Eq. (9.13) can be obtained by deriving the KKT conditions of the optimisation problems that make up the game:

(9.14)\[\begin{split}\begin{align} & \{ -\overline{λ} + β \cdot \sum_{i \in I} q^s_i + β \cdot q^s_i + VC_i - \underline{μ_i} + \overline{μ_i} = 0 \tag{10a'} \\ & 0 \leq q^s_i \leq \overline{Q^s_i} \quad (\underline{μ_i}, \overline{μ_i}) \tag{9} \\ & -\underline{μ_i} \cdot q^s_i = 0 \tag{10b} \\ & \overline{μ_i} \cdot (q^s_i - \overline{Q^s_i}) = 0 \tag{10c} \\ & \underline{μ_i}, \overline{μ_i} \geq 0 \tag{10d} \} \end{align}\end{split}\]

To understand the derivation of (10a’) in Eq. (9.14), we can say that the Lagrangian of (9.13) is:

\[ \mathcal{L} = -(λ - VC_i) \cdot q^s_i - \underline{\mu_i}q^s_i + \overline{\mu_i}(q^s_i - \overline{Q^s_i}) \]

Substituting equation (14) from Eq. (9.13), which defines the inverse demand function, into the Lagrangian, yields:

\[ \mathcal{L} = -\overline{λ}q^s_i + β \cdot q^s_i \cdot \sum_{i} q_i^s + VC_i \cdot q_i^s - \underline{\mu_i}q^s_i + \overline{\mu_i}(q^s_i - \overline{Q^s_i}) \]

Finally, differentiating the Lagrangian with respect to \(q^s_i\) yields (10a’) in (9.14):

\[ \frac{\partial \mathcal{L}}{\partial q_i^s} = -\overline{\lambda} + \beta \cdot \sum_{i} q_i^s + \beta \cdot q^s_i + VC_i - \underline{\mu_i} + \overline{\mu_i} \]

9.2.2. Obtaining Nash-Cournot Equilibria via equivalent optimisation problems

Nash-Cournot games can be solved with similar methods to Nash games, by either:

• directly solving the MCP: analytically/graphically using best response functions or via dedicated solvers;

• through iterative price search algorithms;

• through the derivation of an equivalent optimisation problem, which can be solved through optimisation algorithms.

How can the EOP be obtained in the case of a Nash-Cournot game? As always, this entails finding an optimisation problem that has the same KKT conditions as the MCP (9.14). In contrast to the Nash game, we cannot leverage any knowledge related to the equilibrium solution: Nash-Cournot equilibria are not welfare-maximizing. Comparing the MCP of the Nash game and the Nash-Cournot game, however, shows that they are identical with the exception of Eq. (10a) and Eq. (10a’).

\[\begin{split}\begin{align} & -\overline{λ} + β \cdot \sum_{i \in I} q^s_i + VC_i - \underline{μ_i} + \overline{μ_i} = 0 \tag{10a} \\ & -\overline{λ} + β \cdot \sum_{i \in I} q^s_i + β \cdot q^s_i + VC_i - \underline{μ_i} + \overline{μ_i} = 0 \tag{10a'} \\ \end{align}\end{split}\]

The optimality condition in the Nash-Cournot game has an extra term of the form \(β \cdot q^s_i\). With this information, we can now adjust the objective of EOP of the Nash game to ensure this additional term shows up in the KKT conditions. We do this by integrating the term that differentiates Eq. (10a) and Eq. (10a’) and adding it to the objective of the EOP of the Nash game.

The resulting EOP for the Nash-Cournot game with \(i \in I\) participants reads:

(9.15)\[\begin{split}\begin{align} & \max \overline{\lambda} \cdot \sum_{i} q^s_i - \frac{1}{2}\beta \sum_{i} \left(q^s_i\right)^2 - \frac{1}{2}\beta (\sum_{i} q^s_i)^2 - \sum_{i} VC_i \cdot q^s_i \tag{15'} \\ & \text{s.t}\\ & 0 \leq q^s_i \leq Q^s_i \quad (\underline{μ_i}, \overline{μ_i}) \quad \forall i \in I \tag{3} \\ \end{align}\end{split}\]

Note that the objective function of this EOP is no longer a measure of social welfare, in contrast to the objective of the EOP of the Nash game. The problem contains quadratic terms in the objective but is still convex.

9.3. Final remarks

With the introduction of the Mixed Complementarity Problems as a way to solve the Nash and Nash-Cournot games, we have introduced a paradigm shift in the way we study interactions between self-interested market participants. Up to now, we were considering the market only from the side of a central market operator that aims to clear the market. Now, with the techniques introduced in this section, we can study simple electricity markets as the interaction between different market actors. Specifically, through the Nash games, we can understand the participation of agents in a perfectly competitive market. To study how strategic behaviour and market power change market outcomes, we introduced the Nash-Cournot equilibrium.

For both Nash and Nash-Cournot games, we developed formulations of the game (i.e., the set of interrelated optimisation problems and the coupling constraint) and the corresponding mixed complementarity problem (i.e., the KKT conditions of the decision problems of the involved actors and the coupling constraint). We discussed how we can determine the equilibria in such games, focusing on equivalent optimisation problems. For the Nash game, we showed that the equivalent optimisation problem is a maximization of social welfare. We also obtained an equivalent optimisation problem for the Nash-Cournot game.

These games allow making relationships and interactions between market agents explicit. When the formulation of an optimisation problem is non-trivial, MCPs offer a more intuitive way of formalizing problems.

9.4. References