Ahmad M. Alshamrani

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Market power, defined as the ability to raise prices above competitive levels profitably, continues to be a prime concern in the restructured electricity markets. Market power must be mitigated to improve market performance and avoid inefficient generation investment, price volatility, and overpayment in power systems. For this reason, involving market power in the transmission expansion planning (TEP) problem is essential for ensuring the efficient operation of the electricity markets. In this regard, a methodological bilevel stochastic framework for the TEP problem that explicitly includes the market power indices in the upper level is proposed, aiming to restrict the potential market power execution. A mixed-integer linear/quadratic programming (MILP/MIQP) reformulation of the stochastic bilevel model is constructed utilizing Karush−Kuhn−Tucker (KKT) conditions. Wind power and electricity demand uncertainty are incorporated using scenario-based two-stage stochastic programming. The model enables the planner to make a trade-off between the market power indices and the investment cost. Using comparable results of the IEEE 118-bus system, we show that the proposed TEP outperforms the existing models in terms of market power indices and facilitates open access to the transmission network for all market participants.