First and second order semi-Markov chains for wind speed modeling

Outline
Introduction
Wind Speed Modeling with semi-Markov chains
Application to real data
Conclusions
First and second order semi-Markov chains
for wind speed modeling
Guglelmo D'Amic, Filippo Petroni, Flavio Praticco
by
Shokirov Nozir
Sabanci University
Shokirov Nozir First and second order semi-Markov chains for wind speed m

Outline
Introduction
Conclusions
Outline
1 Introduction
2 Wind Speed Modeling with semi-Markov chains
3 Application to real data
Hypothesis Test
Auto-correlation function
Probability density function and trajectories
4 Conclusions

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Introduction
Conclusions
Introduction

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Introduction
Conclusions
Wind Energy
Main Sources of Energy:
1 Fossils (78.4 %)
2 Nuclear ( 2.6 %)
3 Renewable (19.0 %)

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Introduction
Conclusions
How to harvest wind energy ?

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Introduction
Conclusions
What we need ?
1. A good stochastic model for wind speed is
needed to help both the optimization of turbine
design and to assist the system control to
predict the value of the wind speed in order to
position the blades quickly and correctly.
2. The possibility to have synthetic data of wind
speed is a powerful instrument to assist
designers to verify the structure of the wind
turbines or to estimate the energy recovered
from a specic site.

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Introduction
Conclusions
To generate synthetic data, Markov chains of rst
or higher order are often used.(Shamshad et
al. 2005, Nfaoui et al. 2004 , Kantz et
al. 2004, etc.).

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Introduction
Conclusions
Semi-Markov Chains
Semi-Markov chains are a generalization of Markov chains
allowing the times between transitions to occur at random
times according to any kind of distribution functions which
may depend on the current and the next visited state.
Pros
1 They have been widely used in the literature to model
natural phenomena (see D'Amico et al 2012,
G.Oprisan et al. 2003, Stenberg et al. 2006, etc).
2 Any kind of distribution is allowed
Cons
1 Availability of data to estimate the parameters of the
model which are more numerous.

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Introduction
Conclusions
Proposed Method
Three dierent semi-Markov chain models
First Order SMP:
Transition probabilities from two speed states (at time Tn
and Tn−1) depend on the initial state (the state at Tn−1),
nal state (the state at Tn ) and on the waiting time
(given by t = Tn − Tn−1).
Second Order SMP: Transition probabilities are
considered to depend also on the state the wind speed
was in before the initial state (which is the state at Tn−2).
Second Order SMP: Transition probabilities depends
on the three states at Tn−2, Tn−1 and Tn and on the
waiting times t1 = Tn−1 − Tn−2, and t2 = Tn − Tn−1.

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Introduction
Conclusions
Model
Consider a nite set of states
E = 1, 2, ..., S
and dene the following random variables:
Jn : Ω −→ E, Tn : Ω −→ N (1)
Jn denotes the wind speed at the n-th transition and by Tn the
time of the n-th transition of the wind speed process
We do the following conditional independence assumption:
P[Jn+1 = j, Tn+1 − Tn = t|σ(Js , Ts , Jn = k, Jn−1 = i, Tn − Tn−1 = x, 0 ≤ s ≤ n] (2)
P[Jn+1 = j, Tn+1 − Tn = t|Jn = k, Jn−1 = i, Tn − Tn−1 = x] :=x qi,k,j (t) (3)

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Conclusions
x qi,k,j (t) are stored in a matrix function q = (x qi,k,j (t)) called
the second order kernel in state and duration. The element
x qi,k,j (t) represents the probability that the next wind speed
will be in speed j at time t given that the current wind speed
is k and the previous wind speed state was i and the duration
in wind speed i before reaching the wind speed k was equal to
x units of time.
We can dene the cumulative second order kernel:
x Qi,k,j (t) =
t
s=1
(x qi,k,j (s)) (4)
Same is done for the cumulative distribution functions of the
waiting time in each state, given the state subsequently is
occupied.

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Conclusions
The second order(in state and duration) semi-Markov chain
can be dened as
Z(t) = (Z1
(t), Z2
(t)) = (JN(t)−1, JN(t)) (5)
For all
i, k, j ∈ E, t ∈ N
we dene the semi-Markov transition probabilities:
x φi,k,h,j (t) := P[JN(t) = j, JN(t)−1 = h|JN(0) = k, JN(0)−1 = i, TN(0) = 0, TN(0) − TN(0)−1=x ] (6)
Probabilities for duration eect are dened in similar manner.

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Conclusions
Hypothesis Test
1 The data used in this analysis are freely available from
http://www.lsi-lastem.it/meteo/page/dwnldata.aspx .
2 The database is then composed of about 230,000 wind
speed measures ranging from 0 to 16 m/s.
3 To be able to model the wind speed as a semi-Markov
process the state space of wind speed has been
discretized. In the example shown in this work we
discretized wind speed into 7 states chosen to cover
completely the distribution of wind speed.
4 From the discretized wind speeds we estimated the
probabilities P and G to generate synthetic trajectories by
means of Monte Carlo simulations for three semi-Markov
models.

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Hypothesis Test

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Hypothesis Test
Test
The semi-Markov hypothesis is tested by applying a
hypothesis test proposed by Stanberg et al.
At a signicance level of 95% the null hypothesis is
rejected for 28 out of 42 distributions.
The large values of the test statistic suggest the rejection
of the Markovian hypothesis in favor of the more general
semi-Markov one.

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Introduction
Conclusions
Hypothesis Test
Autocorrelation
If Z indicates wind speed, the time lagged (τ)
autocorrelation of wind speed is dened as
Σ(τ) =
Cov(Z(t + τ), Z(t))
Var(Z(t))

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Hypothesis Test
trajectories

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Hypothesis Test
pdf

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Conclusions
1.Wind speed is a stochastic process for which
completely satisfactory model is still lacking.
2.All three semi-Markov models reproduce the
statistical properties of wind speed data better
than the simple Markov chain.
3.To further decrease the dierence between the
auto correlation of real and synthetic data
probably a third/fourth order semi-Markov
chain would be needed, but this approach would
be computationally and data consuming,
4.semi-Markov models should be used when
dealing with wind speed data.

Outline
Introduction
Conclusions

First and second order semi-Markov chains for wind speed modeling

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