Adopting a zonal structure of electricity market requires specification of zones' borders. One of the approaches to identify zones is based on clustering of Locational Marginal Prices (LMP). The purpose of the paper is twofold: (i) we extend the LMP methodology by taking into account variable weather conditions and (ii) we point out some weaknesses of the method and suggest their potential solutions. The offered extension comprises simulations based on the Optimal Power Flow (OPF) algorithm and twofold clustering method. First, LMP are calculated by OPF for each of scenario representing different weather conditions. Second, hierarchical clustering based on Ward's criterion is used on each realization of the prices separately. Then, another clustering method, i.e. consensus clustering, is used to aggregate the results from all simulations and to find the global division into zones. The offered method of aggregation is not limited only to LMP methodology and is universal.

We present a comprehensive study of utility function of the minority game in its efficient regime. We develop an effective description of state of the game. For the payoff function g(x) = sgn(x) we explicitly represent the game as the Markov process and prove the finitness of number of states. We also demonstrate boundedness of the utility function. Using these facts we can explain all interesting observable features of the aggregated demand: appearance of strong fluctuations, their periodicity and existence of preferred levels. For another payoff, g(x) = x, the number of states is still finite and utility remains bounded but the number of states cannot be reduced and probabilities of states are not calculated. However, using properties of the utility and analysing the game in terms of de Bruijn graphs, we can also explain distinct peaks of demand and their frequencies.

Abstract-One of the methodologies that carry out the division of the electrical grid into zones is based on the aggregation of nodes characterized by similar Power Transfer Distribution Factors (PTDFs). Here, we point out that satisfactory clustering algorithm should take into account two aspects. First, nodes of similar impact on cross-border lines should be grouped together. Second, cross-border power flows should be relatively insensitive to differences between real and assumed Generation Shift Key matrices. We introduce a theoretical basis of a novel clustering algorithm (BubbleClust) that fulfills these requirements and we perform a case study to illustrate social welfare consequences of the division.

Generalization of the minority game to more than one market is considered. At each time step, every agent chooses one of its strategies and acts on the market related to this strategy. If the payoff function allows for strong fluctuation of utility then market occupancies become inhomogeneous with preference given to this market where the fluctuation occurred first. There exists a critical size of agent population above which agents on bigger market behave collectively. In this regime, there always exists a history of decisions for which all agents on a bigger market react identically.

Abstract-Adopting a zonal structure of electricity market requires specification of zones' borders. In this paper we use social welfare as the measure to assess quality of various zonal divisions. The social welfare is calculated by Market Coupling algorithm. The analyzed divisions are found by the usage of extended Locational Marginal Prices (LMP) methodology presented in paper [1], which takes into account variable weather conditions. The offered method of assessment of a proposed division of market into zones is however not limited to LMP approach but can evaluate the social welfare of divisions obtained by any methodology.