Introduction to Probabilistic Modeling and Statistical Characteristics of Aggregate Wind Power

Abstract The stochasticity of the electrical power output by wind turbines poses special challenges to power system operation and planning. Increasing penetration levels of wind and other weather-driven renewable resources exacerbate the uncertainty and variability that must be managed. This chapter focuses on the probabilistic modeling and statistical characteristics of aggregated wind power in large electrical systems. The mathematical framework for probabilistic models— accounting for geographic diversity and the smoothing effect—is developed, and the selection and application of parametric models is discussed. Statistical char- acteristics from several real systems with high levels of wind power penetration are provided and analyzed.

Keywords Copulas · Correlation · Geographic diversity · Smoothing effect ·

Wind generators · Wind power modeling


Wind turbines are classified as weather-driven renewable resources due to the dependency of their power output on local meteorological conditions [1]. These conditions are inherently transient and often erratic. Consequently, the power output by wind turbines—hereafter also simply referred to as ‘‘wind power’’—is appropriately characterized as being variable and uncertain. Variability refers to the unintentional tendency for wind power to change—perhaps rapidly—from one moment to the next, whereas uncertainty refers to the wide range of unknown future values of wind power.

The stochasticity of wind power is a concern for system operators, as the legacy electric grid was designed to be operated with primarily deterministic sources [2, 3]. Although stochastic, wind power often exhibits identifiable patterns and quantifiable statistical distributions, which can be modeled and exploited to better manage the system. These models, whether mathematically formalized or tacitly understood, have applications in several areas, including wind power forecast systems, stochastic unit-commitment programs, risk analysis, and Monte Carlo- based simulations for resource planning and research [46].

This chapter focuses on the aggregate system-wide wind power, rather than the wind power from individual wind plants or turbines. We are motivated to take this macro-level view because for many system operators it is the aggregate—not individual—wind power that is of utmost concern. Our goal is to identify and develop probabilistic models of aggregate wind power and analyze its statistical characteristics. More specifically, we use parametric distributions—probability density functions (pdf) and cumulative distribution functions (cdf)—to model the instantaneous and moment-to-moment variations of aggregate wind power.

The remainder of this chapter is organized as follows. Section 2 describes the general characteristics of aggregate wind power. Section 3 formulates an idealized probabilistic model of wind power output from an individual wind plant. Aspects of geographic diversity including correlation, dependency structures, and practical considerations are discussed in Sect. 4, leading to probabilistic models for instantaneous and moment-to-moment wind power variation in Sect. 5. Aggregate wind power data from four large systems are analyzed and discussed in Sect. 6. The concluding remarks are given in Sect. 7.

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