# A Markov-Switching Vector Autoregressive Stochastic Wind Generator for Multiple Spatial and Temporal Scales

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## Abstract

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## 1. Introduction

## 2. Stochastic Wind Generator Model

#### 2.1. Markov-Switching Vector Autoregressive Model

- The original ${u}_{t,j}$ and ${v}_{t,j}$ components for $t=1,\dots ,n$ and $j=1,\dots ,p$ do not follow a Gaussian distribution, so they are first each transformed to normality with a Gaussian copula as follows:
- (a)
- The empirical cumulative distribution function (ecdf) of each component of ${\mathbf{y}}_{t}$ is obtained with:$${\widehat{F}}_{{u}_{j}}\left(a\right)=\frac{1}{n}\sum _{t=1}^{n}{\mathrm{\U0001d7d9}}_{[{u}_{t,j}<a]}$$
- (b)
- The transformed values of ${u}_{t,j}$, denoted ${u}_{t,j}^{\prime}$, are ${u}_{t,j}^{\prime}={\Phi}^{-1}({\widehat{F}}_{{u}_{j}}({u}_{t,j}))$, and similarly for ${v}_{t,j}$, where ${\Phi}^{-1}(\xb7)$ is the inverse of the standard normal cumulative distribution function.

- Then, we remove the seasonality and diurnal variability from the transformed ${u}_{t,j}$ and ${v}_{t,j}$ components of ${\mathbf{y}}_{t}$ individually using a generalized additive model (GAM) [43] with:$${u}_{t,j}^{\prime}={\beta}_{0}+s\left({m}_{t}\right)+s\left({d}_{t}\right)+s\left({h}_{t}\right)+{\u03f5}_{t,j}$$
`R mgcv`package [44]. Define detrended residuals as ${u}_{t,j}^{r}={u}_{t,j}^{\prime}-{\widehat{u}}_{t,j}^{\prime}$ and ${v}_{t,j}^{r}={v}_{t,j}^{\prime}-{\widehat{v}}_{t,j}^{\prime}$, and the corresponding detrended vector is ${\mathbf{y}}_{t}^{r}=\phantom{\rule{3.33333pt}{0ex}}{({u}_{t,1}^{r},{v}_{t,1}^{r},{u}_{t,2}^{r},{v}_{t,2}^{r},\dots ,{u}_{t,p}^{r},{v}_{t,p}^{r})}^{T}$. The diurnal wind cycle can have a substantial impact on sizing and modeling integrated renewable systems, so it is important to model it properly [32,45,46], but note that the term $s\left({h}_{t}\right)$ is removed from Equation (2) when fitting a trend for the daily averages. - Depending on the number of locations wherein it is desired to simulate the wind vector, we take one of two approaches to choosing the number of “regimes” in the Markov-switching model.
- $p=1$
- : Plot the wind rose of the observed wind speed and direction. Let the number of modes in the joint distribution of speed and direction be the number of regimes.
- $p>1$
- : Average the observed wind speed and wind directions across all p sites at each time t. Plot the wind rose of the averaged speed and directions, and let the number of regimes equal the number of modes in the joint distribution. Note that the circular mean of directions is taken whenever an average of directions is required [47].

- Given the number of regimes, K, we must classify the observations belonging to each one. We do this with an unconstrained Gaussian mixture model (GMM) clustering approach [48] applied to the observed transformed u and v components, ${u}_{t,j}^{\prime}$ and ${v}_{t,j}^{\prime}$. Here, the components of the mixture model are assumed to be multivariate normal distributions with means ${\mu}_{k}$, covariance matrices ${\Sigma}_{k}$ and mixing proportions ${\tau}_{k}$ for $k=1,\dots ,K$. The GMM is able to model ellipsoidal clusters of any size and orientation. We use the mclust package in R to perform the clustering [49], but we note here that clustering the 10-min data, which has 52,560 observations, fails, due to the size of the dataset. Thus, we cluster the hourly data and apply each hour’s cluster assignment to all 10-min observations within the corresponding hour. Secondly, when $p>1$, we construct two sets of regimes based on the following sets of values:
- (a)
- the mean of the transformed u and v components across all p locations, defined as ${\overline{u}}_{t}^{\prime}=\frac{1}{p}{\sum}_{j=1}^{p}{u}_{t,j}^{\prime}$ and ${\overline{v}}_{t}^{\prime}=\frac{1}{p}{\sum}_{j=1}^{p}{v}_{t,j}^{\prime}$ for $t=1,\dots ,n$; and
- (b)
- the transformed u and v components of all p locations, ${\mathbf{y}}_{t}^{\prime}={({u}_{t,1}^{\prime},{v}_{t,1}^{\prime},{u}_{t,2}^{\prime},{v}_{t,2}^{\prime},\dots ,{u}_{t,p}^{\prime},{v}_{t,p}^{\prime})}^{T}.$

- Use the subsets of observations identified in Step 4 to obtain least-squares estimates of the parameters in Equation (3), the Markov-switching autoregressive model (MSVAR) of order one:$${\mathbf{y}}_{t}^{r}={\mathbf{A}}_{{r}_{t}}{\mathbf{y}}_{t-1}^{r}+\u03f5\left({r}_{t}\right);\phantom{\rule{42.67912pt}{0ex}}\u03f5\left({r}_{t}\right)\sim N(0,{\Sigma}_{{r}_{t}})$$
- The transition probability matrix, $\mathbf{P}$, is estimated using the identified clusters and the observed proportion of instances in which the cluster assignments switch,$${\widehat{p}}_{jk}=\frac{{\sum}_{t=1}^{n}{\mathrm{\U0001d7d9}}_{[{r}_{t+1}=k|{r}_{t}=j]}}{{\sum}_{t=1}^{n}{\mathrm{\U0001d7d9}}_{[{r}_{t}=j]}}.$$
- Given the parameter estimates of ${\mathbf{A}}_{{r}_{t}}$, ${\Sigma}_{{r}_{t}}$ and $\mathbf{P}$ from Step 6, simulate a new set of values, denoted ${\tilde{\mathbf{y}}}_{t}^{r}$, from Equation (3).
- Add back the estimated trend from Equation (2) to obtain:$${\tilde{\mathbf{y}}}_{t}^{\prime}={\tilde{\mathbf{y}}}_{t}^{r}+{({\widehat{u}}_{t,1}^{\prime},{\widehat{v}}_{t,1}^{\prime},{\widehat{u}}_{t,2}^{\prime},{\widehat{v}}_{t,2}^{\prime},\dots ,{\widehat{u}}_{t,p}^{\prime},{\widehat{v}}_{t,p}^{\prime})}^{T}.$$
- Transform the ${\tilde{\mathbf{y}}}_{t}^{\prime}$ back into the original units by:$${\tilde{u}}_{t,j}={\widehat{F}}_{{u}_{j}}^{-1}(\Phi \left({\tilde{u}}_{t,j}^{\prime}\right))$$
- As a final step, we convert the u and v components of ${\tilde{\mathbf{y}}}_{t}$ into speed and direction, as these are usually more interpretable quantities upon which to perform validation.

## 3. Simulation Scenarios

#### 3.1. Data Description

**Table 1.**Summary of location names and the number of missing observations for each temporal aggregation level. Note that the 10-min, hourly and daily data have 52,560, 8,760 and 365 observations in a year, respectively.

Acronym | Name | 10-min | Hourly | Daily |
---|---|---|---|---|

AUG | Augspurger | 1,422 | 236 | 9 |

BID | Biddle Butte | 0 | 0 | 0 |

BUT | Butler Grade | 2,690 | 444 | 14 |

CNK | Chinook | 2,687 | 444 | 14 |

FOR | Forest Grove | 0 | 0 | 0 |

GDH | Goodnoe Hills | 2,687 | 444 | 14 |

HOO | Hood River | 0 | 0 | 0 |

HOR | Horse Heaven | 0 | 0 | 0 |

KEN | Kennewick | 0 | 0 | 0 |

MAR | Mary’s Peak | 4,081 | 673 | 23 |

MEG | Megler | 0 | 0 | 0 |

HEB | Mt. Hebo | 2,148 | 351 | 12 |

NAS | Naselle Ridge | 201 | 30 | 0 |

ROO | Roosevelt | 281 | 46 | 1 |

SML | Seven Mile Hill | 2,687 | 444 | 14 |

SHA | Shaniko | 199 | 32 | 0 |

SUN | Sunnyside | 0 | 0 | 0 |

TIL | Tillamook | 0 | 0 | 0 |

TRO | Troutdale | 0 | 0 | 0 |

WAS | Wasco | 0 | 0 | 0 |

#### 3.2. Spatial and Temporal Scales

- Coastal: Tillamook and Mt. Hebo;
- East of Cascades: Kennewick, Butler and Horse Heaven;
- West of Cascades: Biddle Butte and Troutdale.

**Table 2.**Number of bivariate time series for each spatial and temporal scale for which the model is used to simulate data.

Temporal | Spatial Scales | ||
---|---|---|---|

Scales | Individual | Local | Regional |

10-min | 3 | 3 | 1 |

Hourly | 3 | 3 | 1 |

Daily | 3 | 3 | 1 |

#### 3.3. Spatial Locations

## 4. Validation

- the distribution of speed, direction, u and v;
- the temporal autocorrelation of the u and v components;
- the diurnal variability of the u and v components;
- the joint distribution of speed and direction; and
- the correlation between the u and v components.

- histograms of speed, direction, u and v of the observed data with the average count per bin taken across all 100 simulations overlaid;
- autocorrelation (ACF) and partial autocorrelation (PACF) plots of the observed u and v components with the average ACF and PACF for each lag taken over the 100 simulations overlaid (e.g., the average of 100 lag-1 autocorrelations is taken to obtain the plotted value), and diurnal variability can also be assessed with the ACF plots;
- wind roses of the observed speed and direction and wind roses of the average number of simulated observations across all 100 simulations occurring in each speed and direction bin;
- the observed correlation between the u and v components and the average correlation between the u and v components across all 100 simulations; and
- a heat map of the spatial correlations among the observed u and v components and a heat map of the average of the spatial correlations across all 100 simulations.

#### 4.1. Spatial and Temporal Scales

**Figure 2.**Histograms of observed direction, speed, u and v (first, second, third and fourth rows) for 10-min, hourly and daily time resolutions (left, center and right columns) for Wasco. The average simulated count for each variable is given by each overlaid red curve.

**Figure 3.**Observed autocorrelation functions (ACFs) (top two rows) and and partial autocorrelation functions (PACFs) (bottom two rows) of u and v with 10-min, hourly and daily in the left, center and right columns, respectively, for Wasco. The average simulated correlation for each variable is given by each overlaid red curve.

**Figure 4.**Wind roses of speed and direction for the observed (left) and the average across the 100 simulations (right) at Wasco.

Time Scale | ${\widehat{A}}_{1}$ | ${\widehat{A}}_{2}$ | ${\widehat{\Sigma}}_{1}$ | ${\widehat{\Sigma}}_{2}$ | ||||
---|---|---|---|---|---|---|---|---|

10-min | 0.99 | 0.01 | 0.96 | −0.10 | 0.03 | 0.00 | 0.04 | −0.02 |

0.01 | 0.95 | 0.00 | 0.81 | 0.00 | 0.10 | −0.02 | 0.15 | |

Hourly | 0.97 | 0.07 | 0.89 | −0.34 | 0.08 | 0.02 | 0.08 | −0.05 |

0.04 | 0.85 | −0.02 | 0.49 | 0.02 | 0.32 | −0.05 | 0.24 | |

Daily | 0.36 | −0.07 | 0.61 | 0.47 | 0.62 | −0.08 | 0.80 | 0.49 |

0.03 | 0.19 | 0.28 | 0.46 | −0.08 | 0.49 | 0.49 | 0.70 |

**Figure 5.**ACF of the detrended, transformed residuals of u and v for the hourly Wasco scenario for the overall series (top row) and each of the two regimes (second and third rows).

**Figure 6.**Observed ACFs (top row) and PACFs (bottom row) of hourly u and v for Tillamook. The average simulated correlation for each variable is given by each overlaid red curve.

**Figure 7.**Wind roses of hourly direction and speed for the observed (left) and average across the 100 simulations (right) at Tillamook (top) and Sunnyside (bottom).

**Table 4.**Observed overall correlations between the u and v components for the simulation scenarios summarized in Table 2 and the corresponding average simulated correlations.

Temporal | Spatial | Observed Overall Corr. | Simulated Overall Corr. |
---|---|---|---|

Tillamook | −0.37 | −0.42 | |

Sunnyside | −0.03 | −0.17 | |

Wasco | 0.02 | −0.23 | |

10-min | Coast | −0.11 | −0.10 |

East | 0.55 | 0.42 | |

West | 0.19 | 0.21 | |

Region | 0.24 | 0.21 | |

Tillamook | −0.39 | −0.47 | |

Sunnyside | −0.01 | −0.28 | |

Wasco | 0.02 | −0.30 | |

Hourly | Coast | −0.10 | −0.09 |

East | 0.56 | 0.42 | |

West | 0.19 | 0.18 | |

Region | 0.23 | 0.10 | |

Tillamook | −0.25 | −0.14 | |

Sunnyside | 0.19 | 0.15 | |

Wasco | 0.14 | −0.13 | |

Daily | Coast | −0.06 | −0.06 |

East | 0.58 | 0.39 | |

West | 0.21 | 0.05 | |

Region | 0.28 | 0.40 |

#### 4.2. Spatial Locations

**Figure 8.**Observed correlation of u and v across space for the set of four sites (top two rows) and the set of 10 sites (bottom two rows) in the left column. In the middle column are the average simulated correlations for both u and v for the second regime definition in Step 4(b). The last column shows the difference between the observed and average simulated correlations. The sites are organized from west to east along the horizontal axis.

**Figure 9.**Observed correlation of speed across space for the set of four sites (top row) and the set of 10 sites ( bottom row) in the left column. In the middle column are the average simulated correlations for the speed for the second regime definition in Step 4(b). The last column shows the difference between the observed and average simulated correlations. The sites are organized from west to east along the horizontal axis.

**Figure 10.**The left column is the observed distribution of wind speed and direction; the center column is the average across all 100 simulations of four sites simulated simultaneously based on the second regime definition in Step 4(b); and the right column is the average across all 100 simulations of these four sites for the 10 sites simulated simultaneously based on the second regime definition.

## 5. Discussion

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Hering, A.S.; Kazor, K.; Kleiber, W.
A Markov-Switching Vector Autoregressive Stochastic Wind Generator for Multiple Spatial and Temporal Scales. *Resources* **2015**, *4*, 70-92.
https://doi.org/10.3390/resources4010070

**AMA Style**

Hering AS, Kazor K, Kleiber W.
A Markov-Switching Vector Autoregressive Stochastic Wind Generator for Multiple Spatial and Temporal Scales. *Resources*. 2015; 4(1):70-92.
https://doi.org/10.3390/resources4010070

**Chicago/Turabian Style**

Hering, Amanda S., Karen Kazor, and William Kleiber.
2015. "A Markov-Switching Vector Autoregressive Stochastic Wind Generator for Multiple Spatial and Temporal Scales" *Resources* 4, no. 1: 70-92.
https://doi.org/10.3390/resources4010070