It is well known that the large-scale access of wind power to the power system will affect the economic and environmental objectives of power generation scheduling, and also bring new challenges to the traditional deterministic power generation scheduling because of the intermittency and randomness of wind power. In order to deal with these problems, a multiobjective optimization dispatch method of wind-thermal power system is proposed. The method can be described as follows: A multiobjective interval power generation scheduling model of wind-thermal power system is firstly established by describing the wind speed on wind farm as an interval variable, and the minimization of fuel cost and pollution gas emission cost of thermal power unit is chosen as the objective functions. And then, the optimistic and pessimistic Pareto frontiers of the multi-objective interval power generation scheduling are obtained by utilizing an improved normal boundary intersection method with a normal boundary intersection (NBI) combining with a bilevel optimization method to solve the model. Finally, the optimistic and pessimistic compromise solutions is determined by a distance evaluation method. The calculation results of the 16-unit 174-bus system show that by the proposed method, a uniform optimistic and pessimistic Pareto frontier can be obtained, the analysis of the impact of wind speed interval uncertainty on the economic and environmental indicators can be quantified. In addition, it has been verified that the Pareto front in the actual scenario is distributed between the optimistic and pessimistic Pareto front, and the influence of different wind power access levels on the optimistic and pessimistic Pareto fronts is analyzed.

All points on the Pareto boundary can be obtained by solving the weighted sum rate maximization problem for some weighted coefficients. Unfortunately, the problem is non-convex and difficult to solve without performing an exhaustive search. In this paper, we propose an optimal distributed beamforming strategy for the two-user multiple-input single-output (MISO) interference channel (IC). Through minimizing the interference signal power leaked to the other receiver for fixed useful signal power received at the intended receiver, the original non-convex optimization problem can be converted into a family of convex optimization problems, each which can be solved in distributed manner with only local channel state information at each transmitter. After some conversion, we derive the closed-form solutions to all Pareto optimal points based on a game-theoretic viewpoint which indicates that linear combinations of the maximum-ratio transmit (MRT) and zero-forcing (ZF) beamforming strategies can achieve any point on the Pareto boundary of the rate region for the two-user MISO interference channel, and the only computation involved is to solve a basic quadratic equation. Finally, the result is validated via numerical simulations.

A conical area evolutionary algorithm (CAEA) is presented to further improve computational efficiencies of evolutionary algorithms for bi-objective optimization. CAEA partitions the objective space into a number of conical subregions and then solves a scalar subproblem in each subregion that uses a conical area indicator as its scalar objective. The local Pareto optimality of the solution with the minimal conical area in each subregion is proved. Experimental results on bi-objective problems have shown that CAEA offers a significantly higher computational efficiency than the multi-objective evolutionary algorithm based on decomposition (MOEA/D) while CAEA competes well with MOEA/D in terms of solution quality.

When we try to solve Multiobjective Optimization Problems (MOPs) using an evolutionary algorithm, the Pareto Genetic Algorithm (Pareto GA) introduced by Goldberg in 1989 has now become a sort of standard. After the first introduction, this approach was further developed and lead to many applications. All of these approaches are based on Pareto ranking and use the fitness sharing function to maintain diversity. On the other hand in the early 50's another scheme was presented by Nash. This approach introduced the notion of Nash Equilibrium and aimed at solving optimization problems having multiobjective functions that are originated from Game Theory and Economics. Since the concept of Nash Equilibrium as a solution of these problems was introduced, game theorists have attempted to formalize aspects of the equilibrium solution. The Nash Genetic Algorithm (Nash GA), which is introduced by Sefrioui, is the idea to bring together genetic algorithms and Nash strategy. The aim of this algorithm is to find the Nash Equilibrium of MOPs through the genetic process. Another central achievement of evolutionary game theory is the introduction of a method by which agents can play optimal strategies in the absence of rationality. Not the rationality but through the process of Darwinian selection, a population of agents can evolve to an Evolutionary Stable Strategy (ESS) introduced by Maynard Smith in 1982. In this paper, we propose Game theory based Co-Evolutionary Algorithm (GCEA) and try to find the ESS as a solution of MOPs. By applying newly designed co-evolutionary algorithm to several MOPs, the first we will confirm that evolutionary game can be embodied by co-evolutionary algorithm and this co-evolutionary algorithm can find ESSs as a solutions of MOPs. The second, we show optimization performance of GCEA by applying this model to several test MOPs and comparing with the solutions of previously introduced evolutionary optimization algorithms.