# A non-autonomous optimal control model of renewable energy production under the aspect of fluctuating supply and learning by doing

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## Abstract

Given the constantly raising world-wide energy demand and the accompanying increase in greenhouse gas emissions that pushes the progression of climate change, the possibly most important task in future is to find a carbon-low energy supply that finds the right balance between sustainability and energy security. For renewable energy generation, however, especially the second aspect turns out to be difficult as the supply of renewable sources underlies strong volatility. Further on, investment costs for new technologies are so high that competitiveness with conventional energy forms is hard to achieve. To address this issue, we analyze in this paper a non-autonomous optimal control model considering the optimal composition of a portfolio that consists of fossil and renewable energy and which is used to cover the energy demand of a small country. While fossil energy is assumed to be constantly available, the supply of the renewable resource fluctuates seasonally. We further on include learning effects for the renewable energy technology, which will underline the importance of considering the whole life span of such a technology for long-term energy planning decisions.

## Keywords

Optimal control Nonlinear dynamical systems Renewable energy Learning by doing## 1 Introduction

Facing the impacts of climate change, the rapid economic growth coming along with a higher energy demand as well as the fact that one of the main contributors to the constantly increasing green house gas emissions is given by the energy sector, the possibly biggest problem of this century will be to find a carbon-low, sustainable, and simultaneously secure energy supply. Therefore, the incentives for developing and improving renewable energy technologies have changed during the past decades. Originally, the driving force has been given by the rapidly narrowing horizon of depletion of fossil fuels. However, due to the development of new extraction techniques and the discovery of new sources nowadays, the threats of global warming play a major role. Mitigation policies supporting investments into renewable energy technologies should reduce the emissions and slow down the global warming process. The available alternatives of energy generation in the future, however, strongly depend on structural and technological changes together with the accompanying investment decisions right now, because the development and the diffusion of a new technology is a time-intensive dynamic process (cf. Harmon 2000). This underlines the importance of timely planing for energy technology choices. In contrast to conventional energy generation, renewable energy technologies have high investment costs and, therefore, investment decisions for a new technology are often postponed until they get cheaper. This, however, strongly restricts the scale of alternative energy generation (cf. Berglund and Söderholm 2006; Rong-Gang 2013). Therefore, it is important to consider the whole life span of a new technology for energy planning decisions to include the diffusion process and the cost reduction that come along with implementing the new technology. Another challenge of renewable energy generation is the fact that the supply of renewable sources is not constant at all but fluctuating.

To investigate this issue we consider a small country in which a representative decision maker of the energy sector optimizes a portfolio consisting of fossil and renewable energy. We postulate for simplicity that full information about the energy demand that has to be covered is available and that it is stationary, as done in Coulomb (2011). Instead of assuming that the energy demand depends on the GDP of the country, as done in Chakravorty et al. (2012), or on the electricity price, we follow Messner (1997) and consider the energy demand to be exogenous and, further on, constant. Given this demand and considering the fact that the supply of the used renewable sources is fluctuating seasonally, the representative energy sector decision maker optimizes this portfolio to find the most cost-effective solution. We focus especially on solar energy and follow Chakravorty et al. (2006) in omitting completely the possibility of storage so that the generated energy has to be used immediately or otherwise is lost.

In the literature of recent years, some important developments in macroeconomics and energy economics can be observed, dealing with the issue of technological change. While in some modeling approaches technological change, if considered at all, has been included as an exogenous increase in energy conversion efficiency, more recently the aim has been to model it endogenously, especially in form of learning-by-doing effects which sometimes is also considered as technological learning (see for example Chakravorty et al. 2008, 2011; Köhler et al. 2006; Messner 1997; Reichenbach and Requate 2012). To include the aspects of learning by doing in our model, we use a log-linear learning curve to model decreasing investment costs as a function of accumulated experience.

As we consider in our approach the seasonal fluctuations in the supply of renewable sources, this optimal control problem with one state and two controls exhibits a particular mathematical property by being non-autonomous. Solving this problem by applying Pontryagin’s maximum principle, we are looking for a periodic solution that solves the non-autonomous canonical system, which makes the problem numerically sophisticated and which differs from the usual steady-state analysis of autonomous approaches.

The paper is organized as follows: We briefly present first the concept of learning by doing in energy planing models in Sect. 2. In Sect. 3 we then give a detailed description of the model formulation, while Sect. 4 deals with the solution of the problem. The numerical results are presented and interpreted in Sect. 5. As it turns out that the optimal long-run solution is sensitive with respect to the fossil energy price, the learning coefficient, as well as to geographical site specific parameters, we conduct a sensitivity analysis with respect to these parameters in Sect. 6. Finally, we summarize our findings, give conclusions and a brief outlook on future work in Sect. 7.

## 2 The learning curve concept

The development of the learning curve originates from Wright (1936) who observed that in airplane-manufacturing the number of working-hours spent for the production of an airframe is a decreasing function of the total number of the previously produced airframes of the same type. In other words, this means that the unit costs of labor declined with experience measured in cumulative output. Later, Arrow (1962) used cumulative gross investments in form of cumulative production of capital goods as an index of experience so that each new machine produced and used in the production process changes the production environment and leads to a learning process with continual incentive. There exist some other references in the literature, however, stating that interruptions of the production process could also cause negative learning effects, referred to as forgetting by not doing (e.g. see Argote et al. 1990; Argote and Epple 1990; Benkard 2000; Epple et al. 1991), and, hence, rather net investments are a better index for experience. In all different forms, the learning curve concept has been applied in many fields of research and has become an important tool to measure cost-effectiveness of technologies. Given the goal of achieving adequate technology policies to mitigate climate change, the implementation of endogenous technological change via the learning curve in models of future energy scenarios is essential (e.g. see Gerlagh et al. 2003; Grübler and Messner 1998). The learning curve provides an important tool to measure the cost effectiveness of policy decisions to support new technologies. It connects expected future costs with current investments so that the cost of the new technology depends on earlier developments reflected by the cumulative capacity. This comes along with the path dependence of technological competition.

*t*, \(K_{t}\) is the cumulative capacity at time

*t*, \(K_{0}\) is the initial cumulative capacity at time \(t=0\), and \(C_{0}\) are the initial investment costs. This scaling expresses that for an initially low cumulative capacity, it takes more efforts and investments to produce a given level of energy than for an initially high cumulative capacity (cf. Van der Zwaan et al. 2002). Taking the logarithm of Eq. (1) yields an expression which can be estimated econometrically to get a reasonable value for \(\alpha \), and, therefore, for the learning-by-doing rate LDR. This, of course, strongly depends on the type of technology and is crucial for the speed of learning (a survey on estimates of learning rates for a set of energy technologies can be found in McDonald and Schrattenholzer 2001). Equation (1) is also referred to as the single- or one-factor-learning curve. The so-called break-even point is reached, when so much experience is accumulated that the new technology gets competitive with the conventional one.

## 3 The model

To investigate the challenges of including renewable energy into a power system under the aspect of learning by doing, we consider an economy of a small country in which both fossil and renewable energy can be used as perfect substitutes to cover an exogenously given energy demand. Due to the size of the country, we assume that there are no or at least not enough available fossil resources and, therefore, fossil energy has to be imported from other countries for the current market price. As far as renewable energy is concerned, harvesting is for free and the generation is possible within the own country. In contrast to fossil energy, which is assumed to be constantly available, the supply of renewable energy seasonally fluctuates. In order to use this renewable energy resource, capital is necessary for the energy generation for which investments have to be undertaken. We consider for our model a representative energy sector decision maker who chooses the optimal energy portfolio composition for the whole country. It is postulated that this representative energy sector decision maker has full information about the energy demand that has to be covered at each point of time. Therefore, he/she decides on the most cost-effective portfolio consisting of these two energy types, taking into account the seasonal fluctuations of renewable energy supply, the investment costs for renewable energy generation capital which decline with experience, and the import costs of fossil energy. One important implication of the size of the country is that the representative energy sector decision maker is assumed to be a price taker and, therefore, his/her decision has no influence on the market price.

*t*which therefore makes the problem non-autonomous.

*E*

^{1}and no further uncertainties are included, it is postulated that the demand has to be completely satisfied with the portfolio of fossil \(E_{F}(t)\) and renewable \(E_{S}(t,K_{S}(t))\) energy. Shortfalls are not allowed while surpluses are possible. However, as we do not include the possibility of storage, this implies that surpluses are lost and cannot be further used.

^{2}This balance is included in the model by the mixed path constraint

## 4 Solution

### 4.1 Canonical system and necessary first order conditions

*current-value Hamiltonian*

^{3}which reads as

*Lagrangian*(augmented current-value Hamiltonian) which reads as

^{4}the mixed case where both types of energy are used for the coverage with \(E_{F}(t)\), \(I_{S}(t)>0\) and \(E_{F}(t)+E_{S}(t,K_{S}(t))-E=0\), and the renewable case, where no more fossil energy is used in addition to renewable energy to cover the demand, meaning that \(E_{F}(t)=0\), \(I_{S}(t)>0\), and \(E_{S}(t,K_{S}(t))-E\ge 0\) holds. Inserting the corresponding values for the Lagrange multipliers yields the three different canonical systems, with the fossil case as

### 4.2 Periodic solution

*x*(

*t*) is a solution of the canonical system, also \(x(t+k)\) for every integer

*k*is a solution. Due to this periodicity in the dynamics, a candidate for the optimal long-run solution of the problem in (3), which is the solution to which each optimal solution is converging to over time, is given by a periodic solution with the period length of 1 year. In order to find such candidates, we first determine the instantaneous equilibrium points (cf. Ju et al. 2003), which are calculated for the general canonical system in (4) and (5) as the solution of the differential equation system

*n*switching times \(\tau _{1},\dots \tau _{n}\), which satisfy

*i*with \(i=1,\dots ,n+1\). For the numerical solution of the system, for each arc \(i+1\) we use the time transformation

### 4.3 Stability

*J*(

*t*) is the Jacobian matrix evaluated at the periodic solution \(\Gamma (t)\),

^{5}Calculating the Jacobian matrix for the mixed and the renewable case yields

### 4.4 Numerical continuation of optimal paths

*F*is spanning the orthogonal complement to the stable eigenspace (see Grass 2012) and

*T*is the truncation time of the path. The condition in (24) guarantees that the solution ends on the linearized stable manifold to which the vector

*F*is orthogonal (for a more detailed analysis of the so-called asymptotic boundary condition see Lentini and Keller 1980).

Parameter values used for the numerical analysis

Interpretation | Parameter | Value |
---|---|---|

Investment costs | | 0.6 |

Adjustment costs | | 0.3 |

Energy demand | | 2000 |

Fossil energy price | \({{ p}_{ F}}\) | 0.051 |

Discount rate | | 0.04 |

Learning coefficient | \(\alpha \) | 0.25 |

Depreciation rate | \(\delta _{S}\) | 0.03 |

Initial investment costs | \(\epsilon \) | 1 |

Degree of efficiency | \(\eta \) | 0.2 |

Maximal radiation increment | \(\nu \) | 4.56 |

Minimal radiation in winter | \(\tau \) | 0.79 |

## 5 Results

Multiple periodic solutions for \({{ p}_{ F}} = 0.051\)

Solution | \(K_{S}^{*}(0)\) | \(I_{S}^{*}(0)\) | \(E_{F}^{*}(0)\) | Eigenvalues | Objective function (in \(10^3\)) |
---|---|---|---|---|---|

Fossil | 0.0000 | 0.0000 | 2000.00 | {0.9704, 1.0725} | \(-\)2.4500 |

Mixed low | 2.0797 | 0.0623 | 1999.67 | {1.0182+0.0645i, | \(-\)2.4491 |

1.0182-0.0645i} | |||||

Mixed high | 30.6739 | 0.9201 | 1995.15 | {0.9827, 1.0591} | \(-\)2.4351 |

The time-control paths and the time-state paths of the two saddle-type solutions are shown in more detail in Fig. 3. To correctly understand the fluctuating investment path shown in Fig. 3a, it is important to distinguish between acquisition investments and maintenance investments. Remember that we have included depreciation in the state equation in (3a) and, consequently, maintenance investments are necessary to keep the capital in a good condition. Therefore, along a path leading into a long-run periodic solution, both, acquisition investments for new capital as well as maintenance investments for the already accumulated capital, are necessary to increase the capital stock. In the periodic solution itself, however, the desired capital stock level is already reached and only maintenance investments are required. While in the fossil solution no maintenance investments \(I_{S}(t)\) are made, one can see their seasonality for the high mixed solution. Their fluctuations result from the time lag between the depreciation process and the optimal timing of maintenance investments, as the incentive for maintaining the capital is higher shortly before the global radiation peak in summer. Therefore, the investment decision of the high mixed solution in Fig. 3a can be interpreted as follows: In winter, the global radiation is weak. Consequently, the benefit of the capital stock with respect to renewable energy generation and, hence, the incentive for high maintenance investments is low. For this reason, investments are kept on a low level. However, as soon as global radiation goes up in spring, the benefit of the capital stock gets higher. Due to the quadratic investment costs, however, high maintenance investments at once are expensive. Therefore, maintenance investments slowly increase already in winter and reach a peak in spring to have the capital in best condition during summer where global radiation reaches its maximum. This can be seen in Fig. 3b. Further on, renewable energy generation increases in this period and less fossil energy is needed to cover the demand, shown in Fig. 3c. Over summer, investments decline again as the need for maintenance gets lower and they reach their minimum in autumn just before they start to increase again in preparation for the next year. One can see that the optimal investment decision exhibits the same seasonality as the global radiation but it is shifted along the time axis by the expenditure of time for maintenance activities, so that the fluctuation of the capital stock coincides with the one of global radiation.

Such seasonal maintenance planning can be observed in various fields of energy generation. For hydro storage power stations, for example, the seasonality is given by the natural inflows, which usually are higher in early spring due to snow melting and lower over summer during droughts. Also here, maintenance activities are mainly done during the less productive period in summer, in order to keep opportunity costs of a reduced machine availability low. Also thermic power plants usually have their revisions during periods of reduced operating hours due to a lower energy demand (e.g. summer).

Summing up the obtained solutions, we have two periodic solutions being of saddle-type whose areas of attraction probably are separated by an indifference threshold point induced by the unstable focus in between. Indifference threshold points are points in the state space at which the paths leading into different optimal long-run solutions have the same objective value. Therefore, at these points one is indifferent between the two solutions. For more details on indifference threshold points (see Grass et al. 2008; Kiseleva 2011; Kiseleva and Wagener 2010).

### 5.1 Calculation of the indifference threshold point

### 5.2 Economic interpretation of the indifference threshold point

The occurrence of an indifference threshold point is an important result of this analysis as the optimal long-run periodic solution depends on the initial capital stock with which optimization is started.

Figure 5 shows how the indifference threshold point separates the areas of attraction of the mixed and the fossil periodic solution. If the initial capital stock exactly lies on the indifference threshold point \(K_{S}^\mathrm{ITP}\), the paths to both periodic solutions are equally expensive. Therefore, the decision maker is indifferent between increasing investments \(I_{S}(t)\) and moving towards the mixed periodic solution with a higher capital stock and a lower fossil energy amount during the summer period on the one hand, and stopping investments and moving towards the fossil periodic solution where all the energy demand is covered with fossil energy on the other hand. If the initial capital stock is higher than the indifference threshold point \(K_{S}^{\mathrm{ITP}}\), it is optimal to move up towards the mixed periodic solution, if it is lower, the fossil long-run periodic solution is optimal. The reason for this change lies within the learning-by-doing effect. If the initial capital stock is high enough, the reduction of the investment costs due to the learning-by-doing effect can compensate the cost of additional capital accumulation and, therefore, it is optimal to increase the capital stock, which even enforces this effect but at a decreasing rate. If, however, the initial capital stock is low, the learning-by-doing effect on the investment costs is too weak to compensate the costs for additional capital accumulation. Therefore, it is profitable to reduce investments and, hence, the capital stock, and increase the share of fossil energy used to cover the energy demand until finally, the fossil optimal long-run periodic solution is reached. This initial state-dependent separation of the areas of attraction is also known as history dependence, as the optimal long-run periodic solution is determined by the accumulation effort for renewable energy capital in the past.

This result points out the difficulty of introducing a new energy technology into the market. While conventional energy types already are competitive and have low prices due to the high experience accumulated over years, the investment costs for new technologies are very high. As no experience exists at the beginning, these high investment cost would have to be paid over some period of time during which the new technology definitely is not profitable, until finally at least some reduction due to accumulated experience is archived which would be the very first step on the long way towards the break even point. This aspect underlines the importance of subsidies and other kind of financial support that is necessary during the starting-up period to help new technologies over this barrier. Such temporary incentives, also known as directed technical change, can be fundamental for the inclusion of clean technologies. See for this Acemoglu et al. (2012), where it is shown that for an economy consisting of a clean and a dirty technology being sufficiently substitutable, environmental regulation is essential to avoid an environmental disaster. As soon as the clean technology is adequately advanced, however, no further regulation is needed anymore as profit-maximizing production will automatically shift to the clean technology. As in contrast to this, our model approach does not consider such regulations, it, therefore, would never be optimal to start with the renewable energy technology from the very beginning. If no experience exists to reduce the initially high investment costs, fossil energy always is less cost intensive and, as no further restrictions are included like CO\(_{2}\) performance standards for example, no switch to a cleaner energy technology would happen. Only, if there is already a sufficiently high level of experience when optimization is started, further investments are profitable.

### 5.3 Breakeven analysis

As accumulated experience improves the technical processes and hence reduces the necessary financial effort, the technology gets more profitable. However, it can take a long time until full competitiveness with the conventional technology is achieved, which happens at the so-called break-even point.

## 6 Sensitivity analysis

### 6.1 Fossil energy price \({{ p}_{ F}}\)

*t*and for the current parameter set and the fossil solution is given as \({{ p}_{ F}}=0.0678\). For higher values of \({{ p}_{ F}}\), however, a fossil-mixed solution still can be feasible if the part along which the Lagrange multiplier would be negative is replaced by a mixed arc. As soon as the Lagrange multiplier is negative already at the point of time where \(\lambda (t)\) reaches its minimum, however, also no feasible fossil-mixed solution exists, which is for the current parameter set at \({{ p}_{ F}}=0.069\). For fossil energy prices \({{ p}_{ F}}{>}0.069\), the optimal long-run periodic solution is given by the high mixed periodic solution. Figure 8 shows what happens if the fossil energy price \({{ p}_{ F}}\) increases even beyond 0.07. As renewable energy generation progressively gets profitable due to the reduced investment costs by the accumulated experience as well as the comparatively more expensive fossil energy, a strong increase in renewable energy generation capital can be observed. However, still both energy types are needed over the whole period to cover the given energy demand. At \({{ p}_{ F}}=0.5613\), renewable energy generation capital is so high that during summer, when global radiation reaches its maximum, the demand even can be covered without fossil energy. At this point, the feasible boundary of the mixed case is reached and, from this fossil energy price on, a periodic solution exists that consists of two mixed arcs and a renewable arc in between. Figure 9 shows such a mixed/renewable solution in more detail for a fossil energy price of \({{ p}_{ F}}=0.8\). Along these mixed/renewable solutions, the demand over some time interval in summer is covered only by renewable energy, while in winter fossil energy still is needed in addition. If the fossil energy price increases even more, there is still an increase in the stock of renewable energy capital, however, obviously at a decreasing rate. The reason for this is that the marginal benefit of an additional unit of renewable energy capital declines. Remember that generated surpluses beyond the energy demand cannot be used because storage is not included in the model. Therefore, a further increase of the capital stock only is profitable along the mixed arcs, where the necessary amount of fossil energy can be reduced by slightly increasing renewable energy generation. But as the global radiation at the switching times between the arcs gets lower, the closer they are to 0 and 1, more and more renewable energy capital is necessary to compensate. Although the investment costs of renewable energy capital decline with the increasing capital stock and reduce at least the financial effort for this compensation, this saturation effect occurs.

Figure 7 further on shows that a turning point occurs at \({{ p}_{ F}}=0.044\) in the mixed solution. To investigate how the optimal vector field changes here, we consider the local behavior of the monodromy matrix. Figure 10 shows the norm of the eigenvalues of each long-run periodic solution along the \({{ p}_{ F}}\)-axis. The eigenvalues belonging to the fossil long-run periodic solution are shown in dark gray. As we already have shown in Sect. 4.3, the monodromy matrix and hence the eigenvalues of any fossil solution are independent on the periodic solution itself as no state nor co-state occurs in the Jacobian for this case. Hence, the eigenvalues of the fossil long-run periodic solution in Fig. 10 are constant over the fossil energy price \({{ p}_{ F}}\) and are given as \(e_{1}=e^{-\delta _{S}},~e_{2}=e^{r+\delta _{S}}\). As one eigenvalue lies within and the other one outside the unit circle, which in the figure is plotted as black horizontal line, the fossil solution is of saddle-type over its whole interval of existence. The probably most interesting result can be observed for the mixed solutions. The eigenvalues corresponding to the upper mixed long-run periodic solution are shown in Fig. 10 as black line, where again one is lying within and the other one outside the unit circle which specifies the solutions to be of saddle-type. The lower the fossil energy price \({{ p}_{ F}}\), the higher gets the stable eigenvalue until finally, at \({{ p}_{ F}}=0.044\), it crosses the unit circle. Hence, a fold-bifurcation occurs (see for more details Reithmeier 1991) and the stability of the mixed long-run periodic solution changes from saddle-point stability to unstable. The two eigenvalues outside the unit circle are plotted as light gray lines in Fig. 10. At the beginning, they are still real and, hence, the lower mixed periodic solution is an unstable node, but very soon they get complex and the mixed periodic solution turns into an unstable focus. At \({{ p}_{ F}}=0.0612\), the lower mixed periodic solution merges into the fossil-mixed solution whose eigenvalues are shown as light gray dotted line. Also here, the eigenvalues are complex and their real parts are outside of the unit circle, which specifies this solutions as unstable focus as well.

### 6.2 Learning coefficient \(\alpha \)

As already mentioned, not only the fossil energy price plays an important role how the optimal portfolio composition looks like, but also the reducing impact of the learning-by-doing effect on the investment costs of renewable energy, which is determined by the learning coefficient \(\alpha \). In literature, many research papers can be found that investigate the correct height of learning coefficients for different types of technologies. However, opinions strongly differ. To analyze how sensitive the optimal portfolio composition is to different assumptions on the learning coefficient, we conduct in this section the same analysis as in the previous one, but this time with respect to the learning coefficient \(\alpha \).

### 6.3 Global radiation intensity

So far we have investigated the impact of price and learning-by-doing effects on the optimal portfolio composition. However, we completely have fixed site-specific aspects concerning the supply of global radiation for the previous analysis. Therefore, an interesting aspect on which we focus on in the following is as to how the solutions change when geographical conditions vary.

Estimates for \(\tau \) and \(\nu \)

\(\tau \) | \(\nu \) | |
---|---|---|

Austria | 0.79 | 4.56 |

Scenario 1 | 0.21 | 4.08 |

Scenario 2 | 1.35 | 5.64 |

In order to investigate the changes in the optimal portfolio composition when site-specific parameters change, we conduct the same sensitivity analysis with respect to the fossil energy price \({{ p}_{ F}}\), as done in Sect. 6.1, and compare the different outcomes.

### 6.4 Sensitivity analysis for Scenarios 1 and 2

Figure 13 shows the results of the sensitivity analysis for Scenarios 1 and 2, respectively, compared to the results we have obtained for the parameters estimated for Austria.

Second, we investigate Scenario 2 with a higher intensity of global radiation. Also for this case, the qualitative outcome does not change, but again the price boundaries are of special interest. While the interval, in which all three long-run periodic solutions exist and the area of attraction is separated by an indifference threshold point, started at \({{ p}_{ F}}=0.0446\) in the original set and at \({{ p}_{ F}}=0.0609\) in Scenario 1, one can observe from Fig. 13 that this here happens already at \({{ p}_{ F}}=0.0328\). As the supply of global radiation is higher, the investment costs per unit of power for an equal capital stock here are even lower than for the other two cases. Hence, investments into renewable energy get profitable already at a lower fossil energy price. For this reason, also the indifference threshold curve has shifted to the left. The high mixed solution in Scenario 2 gets dominant at \({{ p}_{ F}}=0.0449\), a price at which in the original set a mixed portfolio just starts to be an alternative to the pure fossil one, not to mention Scenario 1 where this possibility does not exist at all at this price level. Starting at \({{ p}_{ F}}=0.0495\), the high mixed solution is the optimal long-run periodic solution. Here, the slope, with which the high mixed periodic solution increases with the fossil energy price, is higher compared to the basic scenario for Austria. Due to the higher global radiation, more renewable energy can be generated and, hence, a higher renewable energy capital stock is profitable already at a lower fossil energy price. Consequently, the interval, in which the indifference threshold curve separates the areas of attraction of the two periodic solutions being of saddle-type, gets smaller as the capital stock, at which the mixed periodic solution with research starts to dominate the fossil one, is reached at a lower fossil energy price.

Varying the intensity of the site-specific global radiation has shown some interesting aspects. While in all three cases, the original parameter set as well as the two scenarios, the intensity of the learning-by-doing effect is exactly the same, the outcomes and their possible consequences for political decisions are completely different. In reality, of course, the price boundaries between which the indifference threshold curve separates the areas of attraction are hard to observe and the only indicator for subsidy decisions might be given by the current fossil energy price. Assume, that in all three cases a price level of \({{ p}_{ F}}=0.05\) is given and subsidies are set to foster renewable energy generation. While for the original parameter set at this price level indeed the indifference threshold curve occurs and the subsidies could help to achieve a switch to renewable energy generation, for the southern country in Scenario 2 the mixed portfolio is already the only optimal solution at this price. Consequently, renewable energy generation here would be over subsidized. Otherwise, for the northern country in Scenario 1, renewable energy generation is not at all an option at this price level as the fossil solution is the only optimal solution and the subsidies here would be completely ineffective. This shows how sensitive the effectiveness of subsidies is to country-specific conditions. Same would also apply for taxes on fossil energy as the shift in the fossil energy price, that is necessary to enable a switch to renewable energy generation, depends on the country-specific situation as well.

## 7 Conclusions

We have investigated in this paper how a small country’s optimal composition of a portfolio consisting of fossil and renewable (solar) energy looks like, when the effect of learning by doing reduces the investment costs due to accumulated experience. Modeling the problem as a non-autonomous optimal control model, we have included a one-factor log-linear learning curve into the objective function so that the accumulated renewable energy capital, which is supposed to reflect the collected experience, has a diminishing impact on the investment costs. Further on, we postulated seasonally varying renewable energy supply and a well-known energy demand that has to be covered.

The obtained results have shown that the fluctuating supply of the renewable resource is one of the major challenges for renewable energy generation. Even if the break even point for the renewable energy technology could be reached where it gets competitive, the shortfalls in winter still have to be covered with fossil energy. This, however, implies another problem which can be seen at the current market situation. As no resource costs occur for renewable energy generation, the variable costs are almost zero, well in contrast to fossil energy. Therefore, a too high renewable energy generation could jeopardize the competitiveness of fossil energy and, hence, its capability to cover these shortfalls. However, as long as the renewable energy generation is not autarkic, for example due to the help of supportive storage systems, fossil energy is urgently needed as backup.

Sensitivity analysis with respect to the fossil energy price \({{ p}_{ F}}\) has shown that there exist price intervals in which multiple periodic solutions occur, and whose areas of attraction are separated by an indifference threshold point. Further on, it turns out that these results are not only sensitive to the fossil energy price but also to the intensity of the learning-by-doing effect as well as on geographical conditions concerning the global radiation.

The occurrence of an indifference threshold point yields important aspects for the economic interpretation of the obtained results. We have seen that whether investments into renewable energy generation capital are worthwhile or not depends on the initial capital stock. Due to this history dependence, investments into renewable energy generation from the very beginning never would be optimal in our approach as the initial investment costs are too high. The level of the capital stock, at which such investments get profitable, shifts even further up if global radiation is lower, as for the northern countries, or if the learning-by-doing effect is weaker, meaning that the learning coefficient is assumed to be lower. One important conclusion of these results is that financial support in form of subsidies during the starting up period of a new technology could play a major role for the successful introduction of this technology into the market. The profitability, however, strongly depends on the site-specific conditions.

The most important aspect for an increasing renewable energy generation in our model has been given by accumulated experience. This learning-by-doing effect, however, has been restricted to the considered country itself. In reality, of course, learning by doing is not only a national but also an international issue. Knowledge spillovers between different countries would even enforce the learning effect and, hence, the adaption of renewable energy technology would happen much faster. Synergies between different countries, therefore, could shift the indifference threshold point to the left and, consequently, a lower subsidy effort would be necessary to support the inclusion of renewable energy generation into the portfolio.

While international cooperation can be supportive for accumulating experience with renewable energy generation, the results from the sensitivity analysis with respect to the global radiation intensity have shown that the effectiveness of a subsidy system strongly depends on national conditions. This means that a cross-border subsidies policy might overlook such tiny but important differences between countries, which could make the subsidies ineffective, as we have seen in Sect. 6.3, and, therefore, could be counterproductive. Consequently, this implies that subsidy policies should remain a national issue while international cooperation and knowledge exchange on renewable energy technologies should be fostered.

Experience in this approach has been the driving force for the reduction of investment costs. But this is not the only source for technological learning. Of course also research and development efforts could foster the competitiveness of a new technology, which implies accumulation of knowledge and, hence, an additional reduction in investment costs. The extension of the model with this aspect will be of special interest in one of our future works.

## Footnotes

- 1.
Remember that, for simplicity, the energy demand is assumed to be constant and stationary. Therefore, here no time argument appears. We also have analyzed similar approaches with a seasonally fluctuating energy demand (peaks in winter and/or summer due to heating and air conditioning), but comparing the results has shown that the observed effects remain the same, only the quantitative portfolio composition might change a little bit. For more details about the different model approaches that we have considered see also Moser (2014).

- 2.
In practice, of course, small surpluses generally would be traded on the market. However, in times of great surpluses as it sometimes occurs around Christmas, prices often turn negative which also comes along with great losses. Therefore, we do not further include the trading aspect in our model but consider such losses in form of sunk investment costs.

- 3.
Note that from here on we often omit the time argument in the function arguments for the ease of exposition.

- 4.
Note that for the fossil case the generated renewable energy \(E_{S}(t,K_{S}(t))\) still is included in the energy balance equation. This is because renewable energy at the beginning of the path could still contribute to the portfolio if there is an initially positive capital stock. As no further investments are done, however, the capital stock will decline over time and the contribution of renewable energy gets negligibly small in the long-term. If, in contrast, the initial capital stock is zero, the contribution is zero along the whole path.

- 5.
For discounted optimal control problems (\(r>0\)), unstable solutions are the only possible limit sets of the canonical system. Consequently, solutions of saddle-type and their stable paths are considered to be the most important candidates for optimal solutions of such problems, in case it is assumed that an optimal solution converges to its limit set. For more details on this see Feichtinger and Hartl (1986) and Grass et al. (2008).

## Notes

### Acknowledgments

We thank Alexia Prskawetz for the fruitful discussions and remarks. Further on, we would like to thank the referees and editors for their valuable comments. This research was partly supported by the Austrian Science Fund (FWF) under Grant No. P25979-N25 and is an extract out of the Ph.D. thesis (Moser 2014).

## References

- Acemoglu D, Aghion P, Bursztyn L, Hemous D (2012) The environment and directed technical change. Am Econ Rev 102(1):131–166. doi: 10.1257/aer.102.1.131. URL http://www.aeaweb.org/articles.php?doi=10.1257/aer.102.1.131
- Argote L, Beckman SL, Epple D (1990) The persistence and transfer of learning in industrial settings. Manag Sci 36(2):140–154CrossRefGoogle Scholar
- Argote L, Epple D (1990) Learning curves in manufacturing. Science 247:920–924. doi: 10.1126/science.247.4945.920
- Benkard CL (2000) Learning and forgetting: the dynamics of aircraft production. Am Econ Rev 90(4):1034–1054CrossRefGoogle Scholar
- Arrow KJ (1962) The economic implications of learning by doing. Rev Econ Stud 29(3):155–173. URL http://ssrn.com/abstract=1506343
- Berglund C, Söderholm P (2006) Modeling technical change in energy system analysis: analyzing the introduction of learning-by-doing in bottom-up energy models. Energy Policy 34(12):1344–1356. doi: 10.1016/j.enpol.2004.09.002. URL http://www.sciencedirect.com/science/article/pii/S0301421504002927
- Chakravorty U, Leach A, Moreaux M (2008) “Twin peaks” in energy prices: A hotelling model with pollution and learning. IDEI Working Papers 52, Institut d’ Économie Industrielle (IDEI), Toulouse. URL http://ideas.repec.org/p/ide/wpaper/10017.html
- Chakravorty U, Leach A, Moreaux M (2011) Would hotelling kill the electric car? J Environ Econ Manag 61(3):281–296. URL http://ideas.repec.org/a/eee/jeeman/v61y2011i3p281-296.html
- Chakravorty U, Magné B, Moreaux M (2006) A hotelling model with a ceiling on the stock of pollution. J Econ Dyn Control 30(12):2875–2904. doi: 10.1016/j.jedc.2005.09.008. URL http://www.sciencedirect.com/science/article/pii/S0165188905002101
- Chakravorty U, Magné B, Moreaux M (2012) Resource use under climate stabilization: can nuclear power provide clean energy? J Public Econ Theory 14(2):349–389. URL http://ideas.repec.org/a/bla/jpbect/v14y2012i2p349-389.html
- Coulomb R, Henriet F (2011) Carbon price and optimal extraction of a polluting fossil fuel with restricted carbon capture. Working paper 322, Banque de France, Paris. URL http://ideas.repec.org/p/bfr/banfra/322.html
- Deshmukh MK, Deshmukh SS (2008) Modeling of hybrid renewable energy systems. Renew Sustain Energy Rev 12(1):235–249. doi: 10.1016/j.rser.2006.07.011. URL http://www.sciencedirect.com/science/article/pii/S1364032106001134
- Epple D, Argote L, Devadas R (1991) Organizational learning curves: a method for investigating intra-plant transfer of knowledge acquired through learning by doing. Organ Sci 2(1):58–70. doi: 10.1287/orsc.2.1.58
- Feichtinger G, Hartl RF (1986) Optimale Kontrolle ökonomischer Prozesse: Anwendungen des Maximumprinzips in den Wirtschaftswissenschaften. de Gruyter, BerlinCrossRefGoogle Scholar
- Feichtinger G, Hartl RF, Kort PM, Veliov VM (2006) Anticipation effects of technological progress on capital accumulation: a vintage capital approach. J Econ Theory 126(1):143–164. doi: 10.1016/j.jet.2004.10.001. URL http://www.sciencedirect.com/science/article/pii/S0022053104002121
- Gerlagh R, Van der Zwaan B (2003) Gross world product and consumption in a global warming model with endogenous technological change. Resour Energy Econ 25(1):35–57. URL http://ideas.repec.org/a/eee/resene/v25y2003i1p35-57.html
- Grass D (2012) Numerical computation of the optimal vector field: exemplified by a fishery model. J Econ Dyn Control 36(10):1626–1658. URL http://www.sciencedirect.com/science/article/pii/S0165188912000966
- Grass D, Caulkins JP, Feichtinger G, Tragler G, Behrens DA (2008) Optimal control of nonlinear processes: with applications in drugs, corruption, and terror. Springer, Berlin. URL http://books.google.com/books?id=M7qGPmzrVAkC
- Grübler A, Messner S (1998) Technological change and the timing of mitigation measures. Energy Econ 20(5–6):495–512 (1998). doi: 10.1016/S0140-9883(98)00010-3. URL http://www.sciencedirect.com/science/article/pii/S0140988398000103
- Hale J, Koçak H (1991) Dynamics and Bifurcations, Springer, New YorkCrossRefGoogle Scholar
- Harmon C (2000) Experience curves of photovoltaic technology. IIASA Interim Report IR-00-014. http://www.iiasa.ac.at/publication/more_IR-00-014.php
- Hartley PR, Medlock KB, Temzelides T, Zhang X (2010) Innovation, renewable energy, and macroeconomic growth. Working paper, James A. Baker III Institute for Public Policy, Rice University, HoustonGoogle Scholar
- Ju N, Small D, Wiggins S (2003) Existence and computation of hyperbolic trajectories of aperiodically time dependent vector fields and their approximations. Int J Bifurc Chaos 13(6):1449–1457. doi: 10.1142/S0218127403007321
- Kiseleva T, Wagener FOO (2010) Bifurcations of optimal vector fields in the shallow lake model. J Econ Dyn Control 34(5):825–843CrossRefGoogle Scholar
- Kiseleva T (2011) Structural analysis of complex ecological economic optimal control problems. Ph.D. thesis, University of Amsterdam, Center for Nonlinear Dynamics in Economics and Finance (CeNDEF)Google Scholar
- Köhler J, Grubb M, Popp D, Edenhofer O (2006) The transition to endogenous technical change in climate-economy models: a technical overview to the innovation modeling comparison project. Energy J:17–56. URL http://ideas.repec.org/a/aen/journl/2006se-a02.html
- Lentini M, Keller HB (1980) Boundary value problems on semi-infinite intervals and their numerical solution. SIAM J Numer Anal 17(4):577–604CrossRefGoogle Scholar
- McDonald A, Schrattenholzer L (2001) Learning rates for energy technologies. Energy Policy 29(4):255–261. doi: 10.1016/S0301-4215(00)00122-1. URL http://www.sciencedirect.com/science/article/pii/S0301421500001221
- Messner S (1997) Endogenized technological learning in an energy systems model. J Evol Econ 7(3):291–313. URL http://ideas.repec.org/a/spr/joevec/v7y1997i3p291-313.html
- Moser E (2014) Renewable energy generation when supply fluctuates seasonally and the effects of learning: an optimal control approach. Ph.D. thesis, Vienna University of TechnologyGoogle Scholar
- Nema P, Nema RK, Rangnekar S (2009) A current and future state of art development of hybrid energy system using wind and PV-solar: a review. Renew Sustain Energy Rev 13(8):2096–2103. doi: 10.1016/j.rser.2008.10.006. URL http://www.sciencedirect.com/science/article/pii/S1364032108001755
- Rasmussen TN (2001) CO2 abatement policy with learning-by-doing in renewable energy. Resour Energy Econ 23(4):297–325. URL http://ideas.repec.org/a/eee/resene/v23y2001i4p297-325.html
- Reichenbach J, Requate T (2012) Subsidies for renewable energies in the presence of learning effects and market power. Resour Energy Econ 34(2):236–254. doi: 10.1016/j.reseneeco.2011.11.001. URL http://www.sciencedirect.com/science/article/pii/S0928765511000649
- Reithmeier E (1991) Periodic solutions of nonlinear dynamical systems, vol 1483., Lecture Notes in Mathematics Springer, BerlinGoogle Scholar
- Rong-Gang C (2013) An optimization model for renewable energy generation and its application in china: a perspective of maximum utilization. Renew Sustain Energy Rev 17(C):94–103. URL http://ideas.repec.org/a/eee/rensus/v17y2013icp94-103.html
- SODA (2014) Solar radiation data. URL http://www.soda-is.org/eng/services/service_invoke/gui.php?xml_descript=hc1_month.xml&Submit=HC1month
- Van der Zwaan BCC, Gerlagh R, Klaassen G, Schrattenholzer L (2002) Endogenous technological change in climate change modelling. Energy Econ 24(1):1–19. URL http://ideas.repec.org/a/eee/eneeco/v24y2002i1p1-19.html
- Wright TP (1936) Factors affecting the cost of airplanes. J Aeronaut Sci 3(4):122–128CrossRefGoogle Scholar
- ZAMG (2012) Klimadaten. http://www.zamg.ac.at/fix/klima/oe71-00/klima2000/klimadaten_oesterreich_1971_frame1.html

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