**1. Introduction**

Nowadays, governments are developing ambitious goals toward the future green and sus‐ tainable sources of energy. In the U.S., the penetration level of wind energy is expected to be 20% by the year 2030 [1]. Several European countries already exhibit the adoption level in the range of 5%–20% of the entire annual demand. Also with further developments in the solar cells technology and lower manufacturing costs, the outlook is that the photovoltaic (PV) power will possess a larger share of electric power generation in the near future. Gridconnected PV is ranked as the fastest-growing power generation technology [2]. PV gener‐ ates pollution-free and very cost-effective power which relies on a free and abundant source of energy.

Due to the increasing wind and solar penetrations in power systems, the impact of system variability has been receiving increasing research focus from market participants, regulators, system operators and planners with the aim to improve the controllability and predictability of the available power from the uncertain resources. The produced power from these re‐ sources is often treated as non-dispatchable and takes the highest priority of meeting de‐ mand, leaving conventional units to meet the remaining or net demand. This issue makes the optimum scheduling of power plants in power system cumbersome as embeds the sto‐ chastic parameters into the problem to be handled. The unpredictability along with poten‐ tial sudden changes in the net demand, may face operators with technical challenges such as ramp up and down adaptation and reserve requirement problems [3-4].

© 2013 Abedi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Abedi et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Several investigations aiming at handling the uncertain nature of wind and solar energy re‐ sources have been reported. Basically, the methods found in the literature can be classified into three groups: methods that deal with the prediction of uncertain variables as an input data pre-processing, methods that use stochastic scenario-based approach within the optimi‐ zation procedure to cover all the outcomes per the probable range of uncertain variables, and methods based on a combination of these two approaches. The studies presented in [5-7] can be mentioned as one of the most recent efforts lying in the first group. In [5-6] an Artificial Neural Network (ANN) forecast technique is employed and followed by risk anal‐ ysis based on the error in the forecast data. Then, the so called pre-processed data is directly taken as the input to the optimization process. Relying on the forecast tools, such methods suffer from high inaccuracy or ex-ante underestimation of the available power which in‐ creases the scheduled generation and reserve costs. Anyway, this approach is useful as it ac‐ counts for the temporal correlation between the random variables representative of each time step of the scheduling period, in terms of time-series models. On the other hand, in [8-9] which belong to the second group, the focus is on the stochastic scenario analysis rath‐ er than the forecasting methods. The usage of this approach also has its own advantages, as it tries to model the likely range of values for the random variables. However, the efficiency of this approach largely depends on the accuracy and reliability of their probabilistic analy‐ sis; based on which the potential scenarios are built.

tial correlation of the power generated by several wind farms or PV farms that are spread over different locations in the power system. A final framework is developed to perform the stochastic analysis of the random variables to be input into the stochastic optimization proc‐

Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain

Renewables

103

http://dx.doi.org/10.5772/45849

In order to generate sample data for random variables, the random behavior should be si‐ mulated somehow that the model follows the historical data pattern with the most homolo‐ gy to real data. In order to specify the pattern of a random variable, the PDF should be obtained. There are two classes of methods to determine the PDF of a random variable in‐ cluding parametric and non-parametric methods [15]. In parametric methods, the data sam‐ ples are fitted to one of the well-known standard PDFs (such as Normal, Beta, Weibull, etc.) so that the most possible adaptation between the PDF and the existing data is achieved. The values associated with the PDF parameters are evaluated using Goodness of Fit (GoF) meth‐ ods such as Kolmogorov-Smirnov test [16]. On the other hand, the nonparametric methods

The use of parametric methods in some studies in which simulation of probabilistic models for wind and solar data is included have been reported, as in [6, 12]. Similarly, authors in [9] employ a fixed experimental equation to represent the PDF of wind data. However, this ap‐

**1.** The parametric methods may show significant deviation to the actual distribution of data, mainly because the actual distribution does not characterize the underlying sym‐ metry in the standard PDFs. As an example, Figure1 shows the distribution function for yearly solar irradiation sample data at 11 AM in a region. As seen in the figure, the parametric distribution fittings are not capable of modeling the right side skewness in the actual distribution, which will reveal considerable error in the outcoming samples.

**2.** Some random variables in general and particularly solar irradiation and wind speed are very time-dependent in behavior. In other words, their patterns change with different time periods, months and seasons. Hence, the nonparametric approach is advantageous in terms of time period adaptation, because it does not consider a specific type of distri‐ bution. However, the parametric approach tries to nominate a certain type of PDF to each random phenomenon in all circumstances. For instance, it is common to associate a Weibull pattern to wind speed data, which may not be the most appropriate option to

Based on the aforementioned facts, in this study, it is desired to obtain the most accurate dis‐ tribution model taking the advantage of Kernel Density Estimation (KDE), categorized as a

ess, as discussed in the following sections.

**2. Methodology of data processing**

do not employ specific well-known PDF models.

be generalized to all time periods.

non-parametric method.

proach to PDF estimation can bring about some defects as follows:

**2.1. Probability distribution function and data sampling**

The most effective approach is associated with the third group, which applies the advantag‐ es of both forecast techniques and scenario-based optimization approach. Reference [10] presents a computational framework for integrating a numerical weather prediction (NWP) model in stochastic unit commitment/economic dispatch formulations that describes the wind power uncertainty. In [11], the importance of stochastic optimization tools from the viewpoint of the profit maximization of power generation companies is investigated. The exposed financial losses regarding the wind speed forecast errors are discussed. A stochastic model is also presented in [12]which uses a heuristic optimization method for the reduction of random wind power scenarios. The wind speed data is assumed to follow the normal PDF. A similar approach is introduced in [13] whereas the wind speed error distribution is considered as a constant percentage of the forecasted data. In [14], the Auto-Regressive Moving Average (ARMA) time series model was chosen to estimate the wind speed volatili‐ ty. Based on the model, the temporal correlation of wind speed at a time step with respect to the prior time steps is well analyzed.

In this chapter, the authors present a framework for stochastic modeling of random process‐ es including wind speed and solar irradiation which are involved in the power generation scheduling optimization problems. Based on a thorough statistical analysis of the accessible historical observations of the random variables, a set of scenarios representing the available level of wind and solar power for each time step of scheduling are estimated. To this aim, the Kernel Density Estimation (KDE) method is proposed to improve the accuracy in model‐ ing the Probability Distribution Function (PDF) of wind and solar random variables. In ad‐ dition, the concept of aggregation of multi-area wind/solar farms is analyzed using Copula method. Taking the advantage of this method, we can reflect the interdependency and spa‐ tial correlation of the power generated by several wind farms or PV farms that are spread over different locations in the power system. A final framework is developed to perform the stochastic analysis of the random variables to be input into the stochastic optimization proc‐ ess, as discussed in the following sections.
