The uncertainty of wind power brings many challenges to the operation and control of power systems, especially for the joint operation of multiple wind farms. Therefore, the study of the joint probability density function (JPDF) of multiple wind farms plays a significant role in the operation and control of power systems with multiple wind farms. This research was innovative in two ways. One, an adaptive bandwidth improvement strategy was proposed. It replaced the traditional fixed bandwidth of multivariate nonparametric kernel density estimation (MNKDE) with an adaptive bandwidth. Two, based on the above strategy, an adaptive multi-variable non-parametric kernel density estimation (AMNKDE) approach was proposed and applied to the JPDF modeling for multiple wind farms. The specific steps of AMNKDE were as follows: First, the model of AMNKDE was constructed using the optimal bandwidth. Second, an optimal model of bandwidth based on Euclidean distance and maximum distance was constructed, and the comprehensive minimum of these distances was used as a measure of optimal bandwidth. Finally, the ordinal optimization (OO) algorithm was used to solve this model. The scenario results indicated that the overall fitness error of the AMNKDE method was 8.81% and 11.6% lower than that of the traditional MNKDE method and the Copula-based parameter estimation method, respectively. After replacing the modeling object the overall fitness error of the comprehensive Copula method increased by as much as 1.94 times that of AMNKDE. In summary, the proposed approach not only possesses higher accuracy and better applicability but also solved the local adaptability problem of the traditional MNKDE.
Competition on distributed generation (DG) investments among multiple stakeholders in a distribution system results in incompleteness of market information, in which each stakeholder does not have full knowledge on investment and operation decisions of other participants. It leads to an incomplete information game among multiple stakeholders. This paper discusses a multilateral incomplete information game based approach to study distribution system planning while considering both supply and demand sides competitions. Profit models of three types of stakeholders, including DG investors (i.e., DG units are investor-owned), electricity consumers, and the distribution company, are constructed. The interaction among the stakeholders and their gaming behavior are further studied under the context of multilateral incomplete information. Bayesian Nash equilibrium form of the multilateral incomplete information game is obtained via Harsanyi transformation. An improved co-evolutionary algorithm is adopted to find the Bayesian Nash equilibrium. Simulation results on a modified IEEE 33-bus test system show that, compared with the complete information game method, the proposed approach presents higher expected profits and more accurate planning schemes. Indeed, the proposed approach reflects the realistic planning process of distribution systems under a deregulated competitive environment, and it ensures fairness of competition among self-interested independent stakeholders while guaranteeing their individual performance. INDEX TERMS Distribution system planning, multilateral incomplete information game, Bayesian Nash equilibrium, Harsanyi transformation, co-evolutionary algorithm. NOMENCLATURE A. INDICES
How to obtain the optimal decision-making scheme based on the investment behavior of various stakeholders is an important issue that needs to be solved urgently in incremental distribution network planning. To this end, this article introduces the virtual player “Nature” to realize the combination of the game theory and robust optimization and proposes an incremental distribution network source–load–storage collaborate planning method with a multi-agent game. First, the planning and decision-making models of a DG investment operator, a distribution network (DN) company, power consumers, and a distributed energy storage (DES) investment operator are constructed, respectively. Then the static game behaviors between the DG investment operator and distribution network company, as well as the DG investment operator and the DES investment operator, are analyzed based on the transfer relations between these four participants. At the same time, robust optimization is used to deal with the uncertainty of the DG output, and the virtual player “Nature” is introduced to study the dynamic game behavior between the DG investment operator and the distribution company. Finally, a dynamic–static joint game planning model is proposed. The simulation results verify the correctness and effectiveness of the proposed method.
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