Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

The Value of Better Wind Information in Investment Decisions Michail Chronopoulos1

Gunnar S. Eskeland1

1 Norwegian

School of Economics, Department of Business and Management Science, Bergen, Norway Research supported by the CenSES and Nord-Star research project at SNF

9 September 2014

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Background  Wind farm investment is faced with considerable uncertainties  Price uncertainty  Regulatory risk  Wind speed  Technological uncertainty  Uncertainties for commodities such as electricity, natural gas, and oil are reasonably well known

    

Geometric Brownian motion (Pindyck, 1999) Mean–reverting processes (Deng et al., 2001) Regime switching models (Karakatsani and Bunn, 2008) Stochastic volatility (Heydari and Siddiqui, 2010) Two–factor models (Schwartz and Smith, 2000)

 Uncertainties pertaining to R&D in new technologies are less well understood  The future development path of RE technologies is likely to be different from their progress in the past (Jamasb and K¨ohler, 2008)  Modelling wind uncertainty  Adkins and Paxson (2013) adopt geometric Brownian motion to model production uncertainty for a renewable energy facility  Howell et al., (2011) adopt Brownian motion to model wind resource uncertainty

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Motivation  Wind speed uncertainty is typically represented by a Weibull distribution (Covelle et al., 2011)

 Better wind information may result in lower volatility and a different mean ex ante  Wind speed profile remains uncertain and forecasting error increases with the horizon 





N(n , m ) 

t



N(n , m ) N(n, m) 



•n

•

n

•n



n n n   m ≤m ≤m

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Motivation  Regional changes in climate are likely to result in increased wind speeds (Snyder et al., 2008)  Increased concentrations of atmospheric CO2 have resulted in stronger winds (Abraham et al., 2014)  We describe wind resource uncertainty via a geometric Brownian motion

 An exploration of the wind resource may reduce uncertainty over the power

80

80

70

70

60

60

50

50

Output, Pt

Output, Pt

production, thereby affecting the decision to invest

40 30 20

30 20

10 0 0

40

10

10

20

Time, t

30

40

0 0

10

20

30

40

Time, t

 Technological and productivity advances may make wind increasingly valuable with time

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Objective  We address the problem of optimal wind farm investment and analyse the impact of

 Wind uncertainty

 computer modeling limitations  limited availability of empirical data  incomplete details about terrain wind flow patterns and wind speeds

 Survey lags of fixed or random length

 Investment lags typically reflect the time to build (Bar–Ilan and Strange, 1996 ; Gollier et al., 2005)

 Allowing for a time interval prior to investment may reduce wind uncertainty by allowing for more accurate information regarding wind speed  Results indicate that

 Survey facilitates investment  Reduction in the discounted expected value of the project due to delay  Lower wind uncertainty mitigates this effect

 An increase in the efficiency of the survey lowers the required abandonment threshold

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Assumptions  Assessments of the average wind speed are modelled via GBM dPt = μPt dt + σPt dWt , P0 ≡ P > 0

 μ : annual growth rate (small or zero)  σ : annual volatility  dWt : increment of the standard Brownian motion

 We assume that the output price Et is fixed (fixed feed–in tariffs).  ρ : exogenous discount rate  τ, τ : optimal time of investment and abandonment respectively  Pτ , Pτ : optimal investment and abandonment thresholds respectively  F(·) : option value  V(·) : project value  Investment cost I = Iw + Is  Iw is the cost of the wind farm

 Is is the cost of the survey, Is = ηh

 The reduction in wind uncertainty may be linear or hyperbolic (c1 and c2 are constants)  σ = c1 − ψ × c2

 σ=

c1 ψa +1 ,

ψ ∈ [0, 1]

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Investment under Wind Uncertainty without Survey Investment



-

F (P; E)

V (P; E)

···

• t 0 τ   waiting region active project  At time τ, the firm has the option to incur a fixed cost, I, and, thus, receive an uncertain payoff. The expected NPV of the project is   ∞ EP V (P; E) = EP −I e−ρt EPt dt − I = ρ−μ 0 where EP is the expectation operator which is conditional on P  The value of the option to invest is ⎧ β 1 ⎪ ⎪ if P < Pτ ⎨AP , F (P; E) = ⎪ ⎪ ⎩ EP − I, if P ≥ P ρ−μ τ 2

where β1 is the positive root of the quadratic 12 σ β(β − 1) + μβ − ρ = 0 and 1−β1

A=

Pτ

E

β1 (ρ − μ)

and

Pτ =

β1 I(r − μ) β1 − 1 E

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Option and Project Value  High wind uncertainty increases the opportunity cost of investment, thereby raising the required investment threshold Parameter values μ = 0.01, ρ = 0.1, I = 2000, E = 10, c1 = 0.2, c2 = 0.1 2500

Option Value, Project Value

2000 1500

Pτ = 30, σ = 0.2 Pτ = 26.83, σ = 0.15

1000

Pτ = 24, σ = 0.1

500 0 −500 −1000

Option Value Project Value Optimal Investment Threshold

−1500 −2000 0

5

10

15

20

Output, Pt

25

30

35

40

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Investment under Wind Uncertainty and Survey Lags  More accurate wind information may reduce uncertainty in power production.  Once the survey is complete, i.e., t = h, we can have ( )  Ph < Pτ ⇒ Investment is further deferred

 Ph ≥ Pτ ⇒ Investment must be exercised immediately ( )

Investment

@   @ R β1

( )  Ph  - V P ;E ( ) P



τ

τ

• • 0 h τ  survey lag waiting region  The expected value of the project at t = h is ⎧ β1 ⎪  ⎪ Ph ( ) ( ) ⎪

 ⎪ ⎪ V Pτ ; E , if Ph < Pτ ⎨ ( ) P F Ph ; E = ⎪ ⎪

τ  ⎪ ( ) ⎪ ⎪ ⎩V Ph ; E , if Ph ≥ Pτ

 V Ph ; E

 V Ph ; E

··· ···

t

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Investment under Wind Uncertainty and Survey Lags (cont’d)  The expected project value at t = 0 is ⎡⎛ ⎞ ⎢⎢⎢⎜⎜ P ⎟⎟β1   ⎜⎜⎜ h ⎟⎟⎟ ⎢ ⎢ EP F Ph ; E = EP ⎢⎢⎣⎜⎝ ( ) ⎟⎠ P 

+ EP

τ

EPh ρ−μ

⎡ ( ) ⎤⎤  ⎢⎢⎢ EP ⎥⎥⎥⎥ ⎢⎢⎢⎢ τ − I ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ × PP P < P( ) h τ ⎣ρ − μ ⎦⎦   ( ) − I × PP P h ≥ P τ

where I = Iw + Is (h)  Iw is the investment cost for the wind farm

 Is (h) is the cost of the survey

 By the definition of conditional probability and the characteristics of a GBM process (Etheridge, 2002), we have ⎛ ( ) ⎞ ⎞ ⎛  ⎜⎜⎜ P ⎟⎟⎟ ⎟⎟ ⎜⎜⎜ 1 2 ⎜⎜⎝ τ ⎟⎟⎠ ⎟⎟⎟ μ − h − ln σ ⎜ ⎜⎜⎜ 2 P ⎟  ⎟⎟⎟ ( ) ⎜⎜ ⎜ PP Ph ≥ Pτ = Φ ⎜⎜⎜ ⎟⎟⎟⎟ √ σ h ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎝ ⎠ where Φ is the cumulative distribution function of a standard normal random variable ( )

 The optimal investment threshold, Pτ , is obtained by solving the optimisation problem   max EP F Ph ; E ( ) ≥P τ

P

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Impact of Survey Lags on Wind Uncertainty and Investment

 Given a maximum survey horizon of 10 years and that initially σ = 0.2, we have  σ = 0.2 − ψ × 0.1

 σ= where ψ =

0.2 ψa +1 h 10 , i.e.,

0.2

ψ ∈ [0, 1] ⇒ σ ∈ [0.1, 0.2] 31

Optimal Investment Threshold

σ = 0.2 − ψ × 0.1 σ = ψ0.2 a +1

0.19 0.18

Volatility, σ

0.17 0.16 0.15

a=1

0.14 a = 0.5

0.13 a = 0.3

0.12 0.11 0.1 0

a = 0.1

2

4

6

Survey Lag, h

8

10

30 ()

Pτ 29

Pτ (σ = 0.2)

28 27

α=1 26 25 24 0

α = 0.5 α = 0.3 α = 0.1 2

4

6

Survey Lag, h

8

10

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Optimal Survey Period and Probability of Investment

 A longer survey period lowers the required investment threshold, thereby raising the likelihood of investment  The optimal survey period is decreasing in the survey’s efficiency

10

0.3

Probability of Investment

9

Survey Length

8 7 6 5 4 3 2 0

0.25

0.2

0.15

α α α α

0.1

Is = 0 Is = 10 × h 0.2

0.4

0.6

Efficiency

0.8

1

0.05 0

2

4

6

Survey Lag, h

8

= = = =

1 0.5 0.3 0.1 10

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Investment under a Random Survey Period  The lifetime of the survey is exponentially distributed, i.e., h ∼ exp(λ)

 λ is the intensity of the Poisson process ! "  h is independent of the process Pt , t ≥ 0 Completion of exploration



G(P; E)

- 

Investment F (P; E)

- 

V (P; E)

• • h 0 τ  survey lag waiting region  Within an infinitesimal time interval dt

 the exploration process will be complete with probability λdt  with probability 1 − λdt the exploration process will continue G(P; E)

=

(1 − ρdt)λdtEP [F(P + dP; E)] + (1 − ρdt)(1 − λdt)EP [G(P + dP; E)]

t

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Investment under a Random Survey Period  Via Itˆo’s lemma, we obtain the differential equation for G(P; E) 1 2 2  σ P G (P; E) + μPG (P; E) − (λ + ρ)G(P; E) + λF(P; E) 2

=

0

 The differential equation for G(P; E) must be solved for each expression of F(P; E), i.e., β 1 2 2  σ P G (P; E) + μPG (P; E) − (λ + ρ)G(P; E) + λAP 1 2

1 2 2  σ P G (P; E) + μPG (P; E) − (λ + ρ)G(P; E) + λV(P; E) 2

=

0

=

0

 Solving for G(P; E) we have

G (P; E)

=

⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨ β1 AP − ⎪ ⎪ ⎪ ⎪ ⎪ λPE ⎩

η ρ+λ

(ρ−μ)(ρ+λ−μ)

δ

δ

if P < Pτ

+ BP 1 + CP 2 , if Pτ ≤ P < Pτ −

η+λI ρ+λ

δ

+ HP 2 ,

if P ≥ Pτ

where Pτ , B, C, H are obtained numerically via value–matching and smooth–pasting conditions.

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Option and Project Value

 Uncertainty over the outcome of the survey lowers the expected value of the project  As λ increases, G(P; E) converges to F(P; E) 150

Value Function, G(P ; E)

Option Value, Project Value

100

Option Value Project Value Optimal Investment Threshold Pτ = 15

λ = 0.3

50

λ = 0.2 λ = 0.1 0

Pτ = 12.7 −50

Pτ = 9.81 Pτ = 8.63

−100 0

2

4

6

8

10

Output, Pt

12

14

16

18

20

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Impact of λ on Optimal Investment and Abandonment  The expected duration of the survey is E(h) = 1/λ  σ=

c1

a

(1/λ) +1

 The impact of λ on the required investment threshold is non–monotonic  Higher λ lowers the likelihood of abandonment 9

α α α α

14.5

= = = =

1 0.5 0.3 0.1

Optimal Abandonment Threshold

Optimal Investment Threshold

15

14

13.5

13

12.5

12

0.2

0.4

0.6

Survey Lag, h

0.8

1

8 7

α α α α

= = = =

1 0.5 0.3 0.1

6 5 4 3 2

0.2

0.4

0.6

Survey Lag, h

0.8

1

Introduction

Assumptions

Analytical Formulation

Conclusions/Future Work

Summary  We investigate the potential of reducing wind resource uncertainty and analyse the impact of better wind information on wind farm investment  Survey lags in combination with reduced uncertainty have an ambiguous impact on the required investment threshold

 Lower wind uncertainty increases the incentive to invest  A survey lowers the discounted expected value of the project, thereby raising the required investment threshold

 Higher likelihood of success decreases the required abandonment threshold by lowering the cost of the survey  Further work  Stochastic reduction in volatility via regime switching (Boomsma et al., 2012)

 Combine wind resource with electricity price uncertainty  Sequential investment in storage Research supported by the CenSES and Nord-Star research project at SNF

Introduction

Assumptions

Analytical Formulation

Thank you !

Conclusions/Future Work