## Description

Karlsruhe Institute of Technology

Institute for Automation and Applied Informatics

TUTORIAL I: TIME SERIES ANALYSIS

Problems which are discussed in the tutorial are marked with .

Exercises to study at home are marked with .

Questions on both kind of tasks are answered in the tutorial and all exercises are relevant for the exam.

PROBLEM I.1 (PROGRAMMING) – DATA ANALYSIS

The following data are made available to you on the course home page and the github repository :

de_data.csv, gb_data.csv, eu_data.csv, (wind.csv, solar.csv, load.csv).

They describe (quasi-real) time series for wind power generation W(t), solar power generation S(t) and load L(t) in Great Britain (GB), Germany (DE) and Europe (EU). The time step is 1h and the time series are several years long.

(a) Check that the wind and solar time series are normalized to ’per-unit of installed capacity’, and that the load time series is normalized to MW.

(b) For all three regions, calculate the maximum, mean, and variance of the time series.

(d) Resample the time series to daily, weekly and monthly data points and visualise them in plots. Can you identify some recurring patterns?

(e) For all three regions, plot the duration curve for W(t), S(t), L(t).

(f) For all three regions, plot the probability density function of W(t), S(t), L(t).

(g) Recurring patterns of time series can also be visualised more rigorously by applying a Fourier Transform. Apply a (Fast) Fourier Transform to the the three time series X ∈ W(t), S(t), L(t):

Z T

X˜ (ω) = X(t)eiωt dt .

0

For all three regions, plot the energy spectrum as a function of ω. Discuss the relationship of these results with the findings obtained in (b)-(f).

(h) Normalize the time series to one, so that hW i = hSi = hLi = 1. Now, for all three regions, plot the mismatch time series

∆(t) = γαW(t) + γ(1 −α)S(t) − L(t)

for the same winter and summer months as in (c). Choose α ∈ {0.0,0.5,0.75,1.0} with γ = 1, and γ ∈ {0.5,0.75,1.0,1.25,1.5} with α = 0.75.

Which configuration entails the lowest mismatch on average and in extremes?

(i) For all three regions, repeat (b)-(g) for the mismatch time series.

PROBLEM I.2 (ANALYTICAL) – EFFECT OF SEASONALITY

Figure 1 shows approximations to the seasonal varia-

1, and the constants have the values 0 0.2 0.4 0.6 0.8 1

2π Figure 1: Seasonal variations of wind

ω = T = 1year

T and solar power generation W(t) AW = 0.4 AS = 0.75 AL = 0.1 and S(t) , and load L(t)

around the mean 1 .

(a) What is the seasonal optimal mix α, which minimizes

dt,

(b) How does the optimal mix change if we replace AL → −AL? (c) Now assume that there is a seasonal shift in the wind signal

W(t) = 1 + AW cos (ωt−φ) .

Express the optimal mix α as a function of φ.

(d) A constant conventional power source C(t) = 1 −γ is now introduced. The mismatch then becomes

∆(t) = γ[αW(t) + (1 −α)S(t)] + C(t) − L(t).

Analogously to (a), find the optimal mix α as a function of 0 ≤ γ ≤ 1, which minimizes h∆2 i.

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