A mathematical model for the diusion of photovoltaics

Outline

A mathematical model for the diffusion ofphotovoltaics

Viviana Fanelli, Lucia Maddalena

Dipartimento di Scienze Economiche Matematiche e StatisticheUniversita di Foggia

Rome, June, 30th 2008ISS on “Risk Measurement and Control 2008”

Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia

A mathematical model for the diffusion of photovoltaics

Outline

Outline

1. The Innovation Diffusion Theory

2. A model for the diffusion of photovoltaics

3. The stability of the equilibrium

4. Conclusions

5. Future Research

6. References



The Innovation Diffusion Theory

Part I





The diffusion of an innovation is the process by which an innovation iscommunicated through certain channels over time among the membersof a social system (Rogers, 2005)




The adoption process

The adoption process

♦Knowledge;♦Persuasion;♦Decision or evaluation;♦Implementation;♦Confirmation or adoption.

Communication channels

Historical distinctionbetween

technological break thoughts (innovations)and

engineering refinements (improvements)(De Solla Price, 1985)




Marketing problems

Marketing problems

1960:To develop marketing strategy in order to promote the adoption of a newproduct and penetrate the market.




Marketing problems

The Bass model, 1969

dA(t)

dt= p(m − A(t)) +

q

mA(t)(m − A(t)) (1)

p is a parameter that takes into account as the new adopters joint themarket as a result of external influences: activities of firms in the market,advertising, attractiveness of the innovation,q refers to the magnitude of influence of another single adopter




Time delay model with stage structure

Stage structure model

Time delay models with stage structure for describing the newtechnology adoption process

W. Wang, and P. Fergola, and S. Lombardo, and G. Mulone, 2006

Beretta (2001)




Time delay model with stage structure

Photovoltaics diffusion

Photovoltaics technology: “critical nature” → its diffusion must besupported by the government policy of incentives

⇓The use of public subsides accelerate market penetration, but actually

the costs of photovoltaics are still so high and the incentives aresubstantially very low

⇓The process of diffusion is stopped at the first stage and it cannot

develop.



The model

Part II

A model for the diffusion of photovoltaics



The model

THE MODEL

Aim

We will propose a time delay model to simulate the adoption processwhen individuals in the social system are influenced by external factors(government incentives and production costs) and internal factors(interpersonal communication). We will find a final level of adopters andwe will carry out a qualitative analysis to study the stability of modelequilibrium solution.



The model

THE MODEL

The model

X The total population C consists of the adopters of thenew technology, A, and of the potential adopters, P.

X i is a government incentive and c represents theproduction costs, eη(i−c) = a is an external factor ofinfluence.

X τ is the average time for an individual to evaluate if toadopt or not the technology. Then, the knowledge and theawareness of technology occur at time t − τ and in theinterval [t − τ, t] the individual decides whether to adoptor not the technology.



The model

THE MODEL

The model

X ρ is the percentage of individuals that don’t adopt thetechnology after the test period, e−ρτ = k is the fractionof individuals that are still interested in the adoption afterτ .

X α is the valid contact rate of adopters with potentialadopters, δ the discontinuance rate of adopters and γ avalid contact rate between the adopters at time t andthose at time t − τ .



The model

THE MODEL

The model

A(t) + P(t) = C⇓

dA(t)

dt= [a + αA(t − τ)] (C−A(t−τ))k−δA(t)+γA(t)A(t−τ), ∀τ > 0

(2)



The model

THE MODEL

The equilibrium

A∗ =(αCk − ak − δ)−

√(αCk − ak − δ)2 − 4(γ − αk)akC

2(γ − αk)



The model

THE MODEL

Sufficient condition for a positive equilibrium

TheoremIf (γ − αk) < 0, that is if the growth factor due to the interpersonalcommunication inside the adopters class contributes in increasing thenumber of adopters less than the factor of imitation, then the time delaydifferential equation admits an unique positive equilibrium.



The stability of the equilibrium

Stability

Stability

Letx(t) = A(t)− A∗.

Then

dx(t)

dt= a0x(t) + b0x(t − τ) + f (x(t − τ), x(t))

where

a0 = (γA∗ − δ),

b0 = (αkC − ak − 2αkA∗ + γA∗),

f (x(t − τ), x(t)) = γx(t)x(t − τ)− αkx2(t − τ).




Stability

Stability

Poincare-Liapunov Theorem⇓

The characteristic equation of the delayed differential equation linear partis

− λτeλτ + a0τeλτ + b0τ = 0

⇓Hayes Theorem




Stability

Stability

TheoremThe equilibrium is asymptotically stable if

τ 2 <θ2

[φ(α, δ, γ, a, k, C)]2 + 2(γA∗ − δ)φ(α, δ, γ, a, k, C)

where φ(α, δ, γ, a, k, C) =p

(αkC − δ − ak)2 − 4akC(γ − αk) and θ is theroot of θ = (γA∗ − δ)τ tan θ, such that(

0 < θ < π2

when A∗ > δγ,

π2

< θ < π when δγ− φ(α,δ,γ,a,k,C)

2γ< A∗ < δ

γ.




Stability switch

Stability switch

We apply the methods proposed by Beretta (2006) and used by W.Wang,P. Fergola, S. Lombardo, and G. Mulone (2006) to demonstrate the timedelay differential equation admits stability switches.




Stability switch

Procedure

Consider the following first order characteristic equation

D(λ, τ) = a(τ)λ + b(τ) + c(τ)e−(λτ) = 0, ∀τ > 0 (3)

wherea(τ) = −τ , b(τ) = τa0(τ) and c(τ) = τb0(τ)and the condition b(τ) + c(τ) 6= 0 is fulfill because b0(τ) 6= a0(τ).




Stability switch

Procedure

If λ = iω with ω > 0 is a root of our characteristic equation, we musthave

F (ω, τ) := τω2 + (τa0(τ))2 − (τb0(τ))2 = 0, ∀τ > 0 (4)

ω(τ) =√

b0(τ)2 − a0(τ)2

which is true only if

αk√

(αCk − ak − δ)2 − 4(γ − ak)akC + γ(αCk − ak + δ)

αkδ> 2




Stability switch

Procedure

Consider

sinϑ(τ) =ω(τ)a(τ)

c(τ)=

√b0(τ)2 − a0(τ)2

b0(τ)

and

cosϑ(τ) = −b(τ)

c(τ)= −a0(τ)

b0(τ)

where ϑ(τ) ∈ (0, 2π)




Stability switch

Switches

Set

τn(τ) =ϑ(τ) + 2nπ

ω(τ)for n ∈ {0, 1, ...}

Then STABILITY SWITCHES occur at the zeros of the followingfunction

Sn(τ) = τ − τn(τ), n ∈ {0, 1, ...}. (5)

Since it is difficult to obtain the zeros of Sn(τ) analytically, we fix theparameters and use numerical calculations.If n = 0, C = 20, δ = 0.2, α = 0.1, a = 0.1, k = e−0.2τ and γ = 0.05,then τ = 1.32.



Conclusions

Part IV

Conclusions



Conclusions

Conclusions

Conclusions I

Conclusions

I We have proposed a mathematical model with stage structure tosimulate adoption processes. The model includes the awarenessstage, the evaluation stage and the decision-making stage.

I We have found the final level of adopters for certain parameters and,from studying the local stability of the equilibrium, we have foundthat the final level of adopters is unchanged under smallperturbations (if the evaluation delay is small).

I The model can be used to predict the number of adopters for certainparameters and economic and financial estimations about marketprofit may be made if the equilibrium is stable.



Conclusions

Conclusions

Conclusions I

Conclusions

I We have shown that the model admits stability switches. Thismeans the number of adopters could fluctuate in time. (This is aneffective characteristic of the new technology markets and, inparticular, of the photovoltaics market).



Future Research

Part V

Future Research



Future Research

Future Research

Future Research I

First The existence of stability switches suggests to investigatethe existence of periodic solutions and their stability.

Second Since there exist stochastic factors in the environment aswell as in the interior system and the action of thesefactors can cause a random adoption pattern of the newtechnology, we wish to investigate the effect of stochasticperturbations for the model.



References

References I

F.M. Bass.A new product growth model for consumer durables.Management Sci., 15:215–227, January 1969.

R. Bellman and K.L. Cooke.Differential-difference equations.Academic Press, New York, 1963.

E. Beretta and Y. Kuang.Geometric stability switch criteria in delay differential system withdelay-dependent parameters.SIAM J. Math.Anal., 33:1144–1165, 2001.



References

References II

D. de Solla Price.The science/technology relationship, the craft of experimentalscience, and policy for the improvement of high technologyinnovation.Research Policy, 13:3–20, 1984.

J.K. Hale and S.M.V. Lunel.Introduction to Functional Differential Equations.Springer-Verlag, New York, 1993.

V. Mahajan, E. Muller, and F.M. Bass.New-Product Diffusion Models.in Eliashberg, J, Lilien, G.L (Eds),Handbooks in Operations Researchand Management Science, Elsevier Science Publishers, New York,NY, 1993.



References

References III

E.M. Rogers.Diffusion of Innovations.The Free Press, New York, 4th edition, 1995.

W. Wang, P. Fergola, S. Lombardo, and G. Mulone.Mathematical models of innovation diffusion with stage structure.Applied Mathematical Modelling, 30:129–146, 2006.



A mathematical model for the diusion of photovoltaics

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