# Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input

## Abstract

Wind and solar power generators are commonly described by a system of stochastic ordinary differential equations (SODEs) where random input parameters represent uncertainty in wind and solar energy. The existing methods for SODEs are mostly limited to delta-correlated random parameters (white noise). Here we use the Probability Density Function (PDF) method for deriving a closed-form deterministic partial differential equation (PDE) for the joint probability density function of the SODEs describing a power generator with time-correlated power input. The resulting PDE is solved numerically. A good agreement with Monte Carlo Simulations shows accuracy of the PDF method.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1233777

- Report Number(s):
- PNNL-SA-100191

Journal ID: ISSN 2166-2525; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Journal Name:
- SIAM/ASA Journal on Uncertainty Quantification

- Additional Journal Information:
- Journal Volume: 3; Journal Issue: 1; Journal ID: ISSN 2166-2525

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- PDF method, stochastic differential equations, power grid, uncertainty quantification

### Citation Formats

```
Wang, Peng, Barajas-Solano, David A., Constantinescu, Emil, Abhyankar, S., Ghosh, Donetta L., Smith, Barry, Huang, Zhenyu, and Tartakovsky, Alexandre M.
```*Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input*. United States: N. p., 2015.
Web. doi:10.1137/130940050.

```
Wang, Peng, Barajas-Solano, David A., Constantinescu, Emil, Abhyankar, S., Ghosh, Donetta L., Smith, Barry, Huang, Zhenyu, & Tartakovsky, Alexandre M.
```*Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input*. United States. doi:10.1137/130940050.

```
Wang, Peng, Barajas-Solano, David A., Constantinescu, Emil, Abhyankar, S., Ghosh, Donetta L., Smith, Barry, Huang, Zhenyu, and Tartakovsky, Alexandre M. Tue .
"Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input". United States. doi:10.1137/130940050.
```

```
@article{osti_1233777,
```

title = {Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input},

author = {Wang, Peng and Barajas-Solano, David A. and Constantinescu, Emil and Abhyankar, S. and Ghosh, Donetta L. and Smith, Barry and Huang, Zhenyu and Tartakovsky, Alexandre M.},

abstractNote = {Wind and solar power generators are commonly described by a system of stochastic ordinary differential equations (SODEs) where random input parameters represent uncertainty in wind and solar energy. The existing methods for SODEs are mostly limited to delta-correlated random parameters (white noise). Here we use the Probability Density Function (PDF) method for deriving a closed-form deterministic partial differential equation (PDE) for the joint probability density function of the SODEs describing a power generator with time-correlated power input. The resulting PDE is solved numerically. A good agreement with Monte Carlo Simulations shows accuracy of the PDF method.},

doi = {10.1137/130940050},

journal = {SIAM/ASA Journal on Uncertainty Quantification},

issn = {2166-2525},

number = 1,

volume = 3,

place = {United States},

year = {2015},

month = {9}

}

DOI: 10.1137/130940050

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.