**2. The effect of renewable energy on wholesale electricity market**

The electricity generated from renewable energies (RES-E) is physically integrated into the wholesale electricity market.

In wholesale electricity daily market (which sets the 89% of final electricity price in Spain), electricity generation selling companies determine, for every generation unit, the offered amount and price according to their short-term marginal cost, which is the variable cost of producing one extra unit of electricity (including the fuel, the emissions, and the operation and maintenance costs).

The supply curve is constructed according to the "merit order" of plants of different technologies in generation markets, ranking capacity from the cheapest to the most expensive (in terms of marginal costs). Plants with low marginal cost are used to cover base demand; plants with intermediate marginal cost operate in the middle of the merit generation and finally plants with high marginal cost are used to cover demand peaks. In parallel, electricity consumers establish the demanded amount.

Finally, supply and demand settle at the same marginal kWh cost of electricity. Thus, in the wholesale market all of the electricity producers get the same price according to the marginal kWh cost of electricity.

Suppose a wholesale market without RES-electricity plants (this is illustrated in Figure 2 below). The figure shows how the different types of production units typically offer electricity at different costs corresponding to their short-term marginal cost.

The production units with the lowest short-term marginal cost are nuclear plants followed by technologies based on coal and gas as combined heat and power or condensing power plants. The resulting price is the price level at which the supply and demand curves intersect.

RES-E is physically integrated into the market and will influence wholesale electricity price. These production technologies are characterized by having lower short-term marginal cost than other conventional technologies.

Therefore, their entrance in the electricity markets can allow the reduction of the wholesale electricity prices because they displace the marginal technology based on nuclear and fossil fuel. The introduction of RES-E thus changes the structure of power supply shifting the whole curve to the right and decreases marginal prices, due to the increased supply at low variable costs as we can see in Figure 3.

In addition, environmental costs related to CO2 emissions in electricity generation usually have a significant negative effect on electricity price as a CO2 emission trading scheme (ETS) exists. The substitution of conventional electricity generation by renewable energies could reduce the costs derived from environmental emissions and the electricity price.

Additional RES-E substitute electricity from fossil fuels, and thus CO2-emissions are reduced. The demand for emission reductions is lowered; as a result the CO2 price is also reduced and consequently the wholesale price for electricity decreases (Rathmann, 2007).

Fig. 2. Wholesale Electricity market without RES-E

The electricity generated from renewable energies (RES-E) is physically integrated into the

In wholesale electricity daily market (which sets the 89% of final electricity price in Spain), electricity generation selling companies determine, for every generation unit, the offered amount and price according to their short-term marginal cost, which is the variable cost of producing one extra unit of electricity (including the fuel, the emissions, and the operation

The supply curve is constructed according to the "merit order" of plants of different technologies in generation markets, ranking capacity from the cheapest to the most expensive (in terms of marginal costs). Plants with low marginal cost are used to cover base demand; plants with intermediate marginal cost operate in the middle of the merit generation and finally plants with high marginal cost are used to cover demand peaks. In

Finally, supply and demand settle at the same marginal kWh cost of electricity. Thus, in the wholesale market all of the electricity producers get the same price according to the

Suppose a wholesale market without RES-electricity plants (this is illustrated in Figure 2 below). The figure shows how the different types of production units typically offer

The production units with the lowest short-term marginal cost are nuclear plants followed by technologies based on coal and gas as combined heat and power or condensing power plants. The resulting price is the price level at which the supply and demand curves

RES-E is physically integrated into the market and will influence wholesale electricity price. These production technologies are characterized by having lower short-term marginal cost

Therefore, their entrance in the electricity markets can allow the reduction of the wholesale electricity prices because they displace the marginal technology based on nuclear and fossil fuel. The introduction of RES-E thus changes the structure of power supply shifting the whole curve to the right and decreases marginal prices, due to the increased supply at low

In addition, environmental costs related to CO2 emissions in electricity generation usually have a significant negative effect on electricity price as a CO2 emission trading scheme (ETS) exists. The substitution of conventional electricity generation by renewable energies could

Additional RES-E substitute electricity from fossil fuels, and thus CO2-emissions are reduced. The demand for emission reductions is lowered; as a result the CO2 price is also reduced and consequently the wholesale price for electricity decreases (Rathmann, 2007).

reduce the costs derived from environmental emissions and the electricity price.

electricity at different costs corresponding to their short-term marginal cost.

**2. The effect of renewable energy on wholesale electricity market** 

parallel, electricity consumers establish the demanded amount.

wholesale electricity market.

and maintenance costs).

marginal kWh cost of electricity.

than other conventional technologies.

variable costs as we can see in Figure 3.

intersect.

Fig. 3. Wholesale Electricity market with RES-E

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 245

The objective of the *quotas of negotiable green certificates* is that the produced energy from renewable sources can be converted in an integral part of the electricity market. For it, the government establishes the obligation that distributors have to acquire a certain percentage of their supply, generally that increases in the time, from renewable energy (fixed quota of

Feed in tariffs promotion support is placed in most of the European Union countries (see Table 1) with the exception of UK, Sweden, Italy, Belgium, and Poland where Quota-based

Feed-in tariffs Quotas of negotiable

Denmark X X (wind)

Latvia X X X (wind)

Luxembourg X

Spain X Sweden X

Table 1. Mechanisms of renewable energy promotion used in European Union

Therefore, the countries of European Union have introduced various support mechanisms to the renewable production technologies as there is not a consensus about which instrument is the most suitable. However, the experience shows that the development of a feed-in tariff system, that allows to guarantee an attractive profitability of the renewable installations, is

Germany X

Hungary X (Possible in the future)

Italy X (photovoltaic solar) X

PoGeothermal X

Romania X

UK X

effective in the promotion of the renewable energies.

France X X (>12, except wind)

green certificates Auctions

systems are now in place (European Commission, 2008).

Belgium X

Austria X

Bulgaria X Chipre X Czech Republic X

Estonia X

Greece X

IreGeothermal X

Lithuania X

Malta X HolGeothermal X

Portugal X

Slovakia X Slovenia X

Source: European Comission, 2008.

electricity).

With increasing RES-E, the effect on wholesale market prices is starting to become significant in some countries, notably Denmark, Spain, and Germany (Klessmann et al., 2008, Sáenz de Miera et al., 2008). However, there is not clear evidence about the effect of the development of renewable energies entail on the final electricity prices.

A large share of RES-E power generally gives lower electricity prices, as it has been shown, reducing the profitability of investing in new electricity capacity (RES are characterised by high capital investment). If RES-E generators are exposed to market prices, it directly affects their market revenues.

In this context, the participation of the governments is necessary in the initial phase of the introduction of the new production technologies. It will allow to secure their development and to protect them from the direct competition of the conventional technologies.

Reiche & Bechberger (2004) identify a number of success conditions for an increased use of RES: long-term planning security for investors, technology-specific remuneration for green power, strong efforts in the field of the power supply systems (grid extension, fair access to the grid, etc.) and measures to reduce local resistance against RES projects.

Therefore a higher use of renewable energies could reduce the wholesale electricity prices but as the development of RES-E is mainly driven by public renewable support schemes, which are financed via the electricity market, it could increased the final price paid by consumers.

## **3. The promotion of the renewable energies in spain in the framework of the European Union**

Therefore, as the majority of renewable energy technologies are not profitable at current energy prices, their development is mainly driven by different public renewable support schemes: feed in tariffs, quota obligations, green-certificate trading, fiscal measures as tax benefits, investment grants, etc (a classification of the existing promotion strategies for renewals is provided in Haas et al., 2011).

Directive 2009/77/CE allows to each member state for the choice of mechanism of renewable energy promotion that can be better adapted to its characteristics. A revision of these measurements, in EU, shows the establishment of three types of mechanisms:


By means of the *feed-in tariffs*, renewable energy generators have right to sell all their production in the electricity wholesale market and to obtain, for it, a retribution based on a fixed price or in the daily price of electricity market plus an incentive that compensates the environmental value of the renewable production.

With the *competitive auction systems*, the regulator reserves a proportion of market for the production of renewable energy and develops the competition between generators that use these resources. Distributors have the obligation of acquiring the total of produced quantity in that reserved market. Therefore, by means of this mechanism, competition is centered on the price due to the production offers are sorted in an increasing order of prices until the proposal quantity was reached.

The objective of the *quotas of negotiable green certificates* is that the produced energy from renewable sources can be converted in an integral part of the electricity market. For it, the government establishes the obligation that distributors have to acquire a certain percentage of their supply, generally that increases in the time, from renewable energy (fixed quota of electricity).

Feed in tariffs promotion support is placed in most of the European Union countries (see Table 1) with the exception of UK, Sweden, Italy, Belgium, and Poland where Quota-based systems are now in place (European Commission, 2008).


Source: European Comission, 2008.

244 Modeling and Optimization of Renewable Energy Systems

With increasing RES-E, the effect on wholesale market prices is starting to become significant in some countries, notably Denmark, Spain, and Germany (Klessmann et al., 2008, Sáenz de Miera et al., 2008). However, there is not clear evidence about the effect of the

A large share of RES-E power generally gives lower electricity prices, as it has been shown, reducing the profitability of investing in new electricity capacity (RES are characterised by high capital investment). If RES-E generators are exposed to market prices, it directly affects

In this context, the participation of the governments is necessary in the initial phase of the introduction of the new production technologies. It will allow to secure their development

Reiche & Bechberger (2004) identify a number of success conditions for an increased use of RES: long-term planning security for investors, technology-specific remuneration for green power, strong efforts in the field of the power supply systems (grid extension, fair access to

Therefore a higher use of renewable energies could reduce the wholesale electricity prices but as the development of RES-E is mainly driven by public renewable support schemes, which are financed via the electricity market, it could increased the final price paid by consumers.

**3. The promotion of the renewable energies in spain in the framework of the** 

Therefore, as the majority of renewable energy technologies are not profitable at current energy prices, their development is mainly driven by different public renewable support schemes: feed in tariffs, quota obligations, green-certificate trading, fiscal measures as tax benefits, investment grants, etc (a classification of the existing promotion strategies for

Directive 2009/77/CE allows to each member state for the choice of mechanism of renewable energy promotion that can be better adapted to its characteristics. A revision of

By means of the *feed-in tariffs*, renewable energy generators have right to sell all their production in the electricity wholesale market and to obtain, for it, a retribution based on a fixed price or in the daily price of electricity market plus an incentive that compensates the

With the *competitive auction systems*, the regulator reserves a proportion of market for the production of renewable energy and develops the competition between generators that use these resources. Distributors have the obligation of acquiring the total of produced quantity in that reserved market. Therefore, by means of this mechanism, competition is centered on the price due to the production offers are sorted in an increasing order of prices until the

these measurements, in EU, shows the establishment of three types of mechanisms:

and to protect them from the direct competition of the conventional technologies.

the grid, etc.) and measures to reduce local resistance against RES projects.

development of renewable energies entail on the final electricity prices.

their market revenues.

**European Union** 

feed-in tariffs,

competitive auctions and

proposal quantity was reached.

renewals is provided in Haas et al., 2011).

the quotas of negotiable green certificates.

environmental value of the renewable production.

Table 1. Mechanisms of renewable energy promotion used in European Union

Therefore, the countries of European Union have introduced various support mechanisms to the renewable production technologies as there is not a consensus about which instrument is the most suitable. However, the experience shows that the development of a feed-in tariff system, that allows to guarantee an attractive profitability of the renewable installations, is effective in the promotion of the renewable energies.

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 247

By means of this new regulation, the reward of each renewable production technology is not homogenous but is given by the produced amount and the temporal horizon of each plant. Likewise, a reference premium and the upper and lower limits are established for every renewable production technology that participates in the wholesale market (see Table 3).

**Thermal solar** 0,03 0,03 0,03 12 12 18,74 19,059 19,912 19,912 25,4 34,397 25,403

<= 5 MW 3,16 2,87 2,87 2,89 2,66 3,603 3,665 3,829 3,829 2,929 8,494 7,127

Energy crops 3,04 2,76 2,76 2,78 3,32 3,603 3,655 3,829 3,829 11,529 16,63 15,41

forestry wastes 8,211 13,31 12,09 (a) Photovoltaic solar plants under R.D. 661 and below 100 kW under R.D. 436 do not have the premium

Table 3. Fixed premiums established in the Spanish electricity industry (sales to market) (in

In Spain the government support of renewable energies has suppose that Spain becomes a

The importance of renewable energies in Spain is observed in Figure 4. It shows the

pioneering and leader country in the integration of renewable energies

electricity generated by sources as percentage of the total in the year 2009.

8,211 13,31 12,09

3,603 3,665 3,829 3,829

Sea <= 5 MW 3,603 3,665 3,829 3,829 8,43 16,4

**2007 R.D. 436** 

**2007 R.D. 661 Reference**

**2007 R.D. 661 Upper limit** 

**2007 R.D. 661 Lower limit** 

**Technology 1999 2000 2001 2002 2003 2004 2005 2006**

<= 100 kW a a a a

<= 10 MW 18,74 19,059 19,912 19,912

<= 50 MW 18,74 19,059 19,912 19,912

< 5 kW 36 36 36 36 36 > 5 kW 18 18 18 18 18

**Production** 

**Photovoltaic solar** 

> 100 kW and

> 10 MW and

Geothermal > 5 MW and <= 50 MW

Sea >= 5 MW **Biomass** 

Biomass from agricultural wastes

Biomass from

Source: Del Río (2008).

hundredth part of Euro/kWh).

option.

**Wind**  Geothermal

It is the case of Spain that has been characterized by introducing this mechanism from the first phases of the promotion in the renewable production technologies.The feed-in tariff1 , in this country, entails two possibilities in the sale of electricity generated by renewable energies:



Source: Del Río (2008).

Table 2. Feed-in tariffs established in the Spanish electricity industry (sale option to distributor) (in hundredth part of Euro/kWh).

<sup>1</sup>The current legal framework of renewable energy in Spain is the Royal-Decree 661/2007. Later, Royal Decree 1578/2008 establishes a new tariff system for the photovoltaic solar energy. The new feed-in tariff system is based on the location of this type of plants: plants located on covers (type I) and plants located on the ground (type II). Order ITC/1723/2009 establishes an actualization of the tariffs and the premiums fixed for the renewable production technologies based on cogeneration and wastes. Likewise, the Royal Decree 1614/2010 establishes a restriction of the equivalent hours of functioning in the installations of wind production and thermal solar with right to premium and it supposes an updating of their premiums.

By means of this new regulation, the reward of each renewable production technology is not homogenous but is given by the produced amount and the temporal horizon of each plant. Likewise, a reference premium and the upper and lower limits are established for every renewable production technology that participates in the wholesale market (see Table 3).


(a) Photovoltaic solar plants under R.D. 661 and below 100 kW under R.D. 436 do not have the premium option.

Source: Del Río (2008).

246 Modeling and Optimization of Renewable Energy Systems

It is the case of Spain that has been characterized by introducing this mechanism from the first phases of the promotion in the renewable production technologies.The feed-in tariff1 , in this country, entails two possibilities in the sale of electricity generated by renewable energies:

 to sell their surplus of electricity energy to a distributor where the reward will be given in the way of a feed-in tariff and it is calculated as a percentage of the medium or

 to sell their production surplus in the electricity production market or by means of a bilateral contract where the reward will be given by the negotiated market price, an

<= 100 kW 41,441 42,149 44,038 44,038 44,038

<= 10 MW 21,621 21,991 22,976 22,976 41,75

<= 50 Mw 21,621 21,991 22,976 22,976 22,976 **Thermal solar** 21,621 21,991 22,976 22,976 26,937

<= 5 MW 6,62 6,26 6,26 6,28 6,21 6,486 6,597 6,892 6,892 7,322

6,486 6,597 6,892

Sea >= 5 MW 6,486 6,597 6,892 6,892 7,8

Energy crops 6,5 6,15 6,15 6,17 6,85 6,486 6,597 6,892 6,892 15,889

Table 2. Feed-in tariffs established in the Spanish electricity industry (sale option to

1The current legal framework of renewable energy in Spain is the Royal-Decree 661/2007. Later, Royal Decree 1578/2008 establishes a new tariff system for the photovoltaic solar energy. The new feed-in tariff system is based on the location of this type of plants: plants located on covers (type I) and plants located on the ground (type II). Order ITC/1723/2009 establishes an actualization of the tariffs and the premiums fixed for the renewable production technologies based on cogeneration and wastes. Likewise, the Royal Decree 1614/2010 establishes a restriction of the equivalent hours of functioning in the installations of wind production and thermal solar with right to premium and it supposes an updating of their premiums.

6,73 6,36 6,36 6,38 6,49 6,486 6,597 6,892 6,892 6,89

Sea <= 5 MW 6,486 6,597 6,892

**R.D. 436** 

**2007 R.D. 661** 

**Technology 1999 2000 2001 2002 2003 2004 2005 2006 <sup>2007</sup>**

reference electricity tariff every year (see Table 2) or

incentive for their participation and a fixed premium.

< 5 kW 39,6 39,6 39,6 39,6 39,6 > 5 kW 21,6 21,6 21,6 21,6 21,6

**Production** 

**Photovoltaic solar** 

> 100 kW and

> 10 MW and

Geothermal > 5 MW and <= 50 MW

**Biomass** 

Biomass from agricultural wastes Biomass from forestry wastes

Source: Del Río (2008).

distributor) (in hundredth part of Euro/kWh).

**Wind**  Geothermal

> Table 3. Fixed premiums established in the Spanish electricity industry (sales to market) (in hundredth part of Euro/kWh).

In Spain the government support of renewable energies has suppose that Spain becomes a pioneering and leader country in the integration of renewable energies

The importance of renewable energies in Spain is observed in Figure 4. It shows the electricity generated by sources as percentage of the total in the year 2009.

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 249

Spanish electricity market liberalization has been progressively adopted since 1998 and

The liberalization has the purpose of increasing efficiency of the energy sector following in a reduction of electricity prices. However, as we have seen previously RES-E affects the electricity prices in the liberalized market. We investigate this effect over the period 2003-

Additionally, it is important to account others electricity generation technologies in empirical analysis as different technologies will result in different marginal costs and then different electricity prices. As it is showed in figure 5, generation is greatest proportion of electricity tariff costs, so the majority of variation in end-user prices should be accounted for

The goal of this section is to estimate the effect of several variables (m) related to electricity generation by RES and other sources on electricity prices (*y*) by using data of several years

coefficients to be estimated (vector (m+1)x1, included a constant term) and u the vector of

As dependent variable we use the *Electricity prices for Household consumers* (*y*). This indicator measures electricity prices charged to final consumers, which are defined as follows: Average national price in Euro per MWh without taxes applicable for the first semester of each year for medium size household consumers (Consumption Band Dc with annual

The following explanatory variables (m) related to the participation of different energies in

the electricity generation and thus in the wholesale electricity market are proposed:

, being X a matrix TXm, *y* a matrix TX1,

the vector of

2008 as legal opening electricity market in Spain finish in 2003.

Fig. 5. Breakdown of electricity tariff costs. 2008

consumption between 2500 and 5000 kWh).

(2003-2008, T=6): *y X u*

disturbances.

finished 1st of July 2007.

by generation prices.

Fig. 4. Electricity generation by source (2009).

In fact, Spain is the second European country in terms of wind installed capacity and production, only behind Germany as we can see in Table 4. Spain is the fourth country in the world in terms of installed wind power after the US, Germany and China

The Spanish Renewable Energy Plan 2011-2020 processed by the Industry and Energy Council of the Spanish government and the Spanish Institute for Energy Diversification and Saving-IDEA shows that wind energy will continue to play a dominant role, accounting for 52% of renewable electricity production in 2020 (on- and offshore considered jointly).


Table 4. Installed wind energy (MW). Source: Eurobserver

#### **4. The impact of RES-E on electricity prices. A maximum entropy econometric estimated model**

Therefore, the increased participation of renewable energies is an important factor to explain the final electricity price in the liberalized electricity market. In this section, the effect of RES-E on electricity prices in Spain is explored by using a maximum entropy econometric model. The used data set are provided by Eurostat during the period 2003-2008 (available at the web site http://epp.eurostat.ec.europa.eu).

Pumping; 0,8%

Natural gas; 24,7%

In fact, Spain is the second European country in terms of wind installed capacity and production, only behind Germany as we can see in Table 4. Spain is the fourth country in

The Spanish Renewable Energy Plan 2011-2020 processed by the Industry and Energy Council of the Spanish government and the Spanish Institute for Energy Diversification and Saving-IDEA shows that wind energy will continue to play a dominant role, accounting for 52% of renewable electricity production in 2020 (on- and offshore considered jointly).

 **2001 2002 2003 2004 2005 2006 2007 2008 2009**  Germany 8754 11994 14609 16629 18415 20622 22475 23903 25777 Spain 3397 4891 5945 8317 10028 11630 15151 16740 19149 Italy 697 788 904 1132 1718 2123 2726 3737 4185 France 94 153 249 382 756 1737 2482 3542 4521 Sweden 293 328 399 452 493 519 831 1021 1560 Unitd Kingdom 474 552 649 933 1565 1961 2477 3406 4051 Portugal 125 194 297 537 1047 1681 2150 2862 3535 Denmark 2417 2889 3115 3125 3129 3135 3142 3166 3481

the world in terms of installed wind power after the US, Germany and China

37,2%

Table 4. Installed wind energy (MW). Source: Eurobserver

**econometric estimated model** 

the web site http://epp.eurostat.ec.europa.eu).

**4. The impact of RES-E on electricity prices. A maximum entropy** 

Therefore, the increased participation of renewable energies is an important factor to explain the final electricity price in the liberalized electricity market. In this section, the effect of RES-E on electricity prices in Spain is explored by using a maximum entropy econometric model. The used data set are provided by Eurostat during the period 2003-2008 (available at

Fig. 4. Electricity generation by source (2009).

Petroleum p.; 6,9%

Coal; 12,6%

Nuclear ; 17,8%

Renewables;

Spanish electricity market liberalization has been progressively adopted since 1998 and finished 1st of July 2007.

The liberalization has the purpose of increasing efficiency of the energy sector following in a reduction of electricity prices. However, as we have seen previously RES-E affects the electricity prices in the liberalized market. We investigate this effect over the period 2003- 2008 as legal opening electricity market in Spain finish in 2003.

Additionally, it is important to account others electricity generation technologies in empirical analysis as different technologies will result in different marginal costs and then different electricity prices. As it is showed in figure 5, generation is greatest proportion of electricity tariff costs, so the majority of variation in end-user prices should be accounted for by generation prices.

Fig. 5. Breakdown of electricity tariff costs. 2008

The goal of this section is to estimate the effect of several variables (m) related to electricity generation by RES and other sources on electricity prices (*y*) by using data of several years (2003-2008, T=6): *y X u* , being X a matrix TXm, *y* a matrix TX1, the vector of coefficients to be estimated (vector (m+1)x1, included a constant term) and u the vector of disturbances.

As dependent variable we use the *Electricity prices for Household consumers* (*y*). This indicator measures electricity prices charged to final consumers, which are defined as follows: Average national price in Euro per MWh without taxes applicable for the first semester of each year for medium size household consumers (Consumption Band Dc with annual consumption between 2500 and 5000 kWh).

The following explanatory variables (m) related to the participation of different energies in the electricity generation and thus in the wholesale electricity market are proposed:

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 251

inversion, there is more than one vector P making the solution feasible. Therefore the problem is ill-posed and there is no basis for picking a particular solution vector for P from the feasible set. Thus, by asking for a particular set of probabilities considered as most likely, it seems reasonable to favour the one that could have been generated in the greatest number

The definition of the entropy measure H(P) and the formulation of the Entropy Maximization problem can help to estimate a unique P distribution since the principle of Maximum Entropy provides a basis for transforming the sample information into a

The measures of entropy H(P) quantify the uncertainty associated with a random experiment. In particular, given a random variable X with values xi and probability

The value of the entropy is maximum when all the values xi have the same probability (and then P is a uniform distribution). This situation would be justified by the Laplace Indifference Principle, according to which the uniform distribution is the most suitable representation of the knowledge when the random variable is completely unknown. Nevertheless, sometimes the ignorance of the probability distribution of X is not absolute and there is some partial information on the distribution such as the mean, variance, moments or some characteristics which can be formulated as equality constraints. In such a case, it is possible to estimate the probability distribution through the application of the Maximum Entropy principle (Jaynes 1957) choosing the distribution for which the available

Thus, if certain values *<sup>r</sup> a* (r=1..., s) associated with functions ( ) *<sup>r</sup> g X* of the values of X are known but its distribution is unknown, the problem consists of estimating a nonnegative

By solving the maximization problem it is possible to obtain the estimated probabilities <sup>1</sup> <sup>ˆ</sup> ˆ ˆ , , *Pp p <sup>n</sup>* . The maximum entropy distribution does not have a closed-form solution and therefore numerical optimization techniques must be used to compute the probabilities. Working towards a criterion for recovering the parameters of the regression model related to

a specific independent variable is more significant than others, the related probability distribution (P) would be the uniform (according to Laplace Indifference Principle). However, the principle of maximum entropy provides a basis for using the sample information in a probability distribution P that reflects the uncertainty about the individual independent variable. Therefore, the problem consists of estimating a nonnegative distribution P by

1

*S S n ii*

*HP Hp p p p*

*n i i p* 

1

1 ,..., - log *n*

*i*

1

*n i i p* 

1

, if there is no evidence that

, maximizing the value

 .

, Shannon's measure of

probability distribution that reflects the uncertainty about the individual outcomes.

distribution *P p p* <sup>1</sup> ,..., *<sup>n</sup>* with 0 *<sup>i</sup> p* (i=1..., n) and

entropy (Shannon 1948) is defined as: <sup>1</sup>

information is just sufficient to obtain the probability assignment.

distribution that fulfils the conditions 0 *<sup>i</sup> p* for i=1..., n and

electricity price in the general inverse problem *y X u XP u*

of the entropy.

of ways given the available data.


Moreover, the following variables are also used:


The estimation of by regression techniques requires that the number of observations was superior to the number of independent variables (T>m). Nevertheless, information is limited (T=6 and m=9-including a constant term) and when trying to estimate the electricity price models through regression procedures a dimensionality problem arises. Therefore, in a situation of limited sample data the estimation of the model by regression procedures (OLS) is not possible as the problem is undetermined or ill-posed. However, when these circumstances of small amount of information available make it unfeasible to estimate the model *yX u* through OLS procedures the Maximum Entropy Econometric approach allows to recover the estimates of 1 2 , ,..., *<sup>m</sup>* in the corresponding parameterized model without making distributional assumptions. The approach consists of developing a nonlinear inversion procedure (Golan et al. 1996) which requires the application of the tools provided by the Information Theory (Shannon, 1948; Jaynes 1957a, 1957b).

#### **4.1 Maximum entropy econometric approach**

Consider a regression-based method: *y X u* in a situation of limited sample data where m>T. A probability distribution should be used in order to represent partial and limited information regarding the individual observations so they are consistent with the observed sample data. Therefore, following Golan et al. 1996 it is possible to define an inverse general problem for recovering defined as: *y X u XP u* , where 1 *Pp p* : ( , , )' *<sup>m</sup>* is a mdimensional vector of unknown terms related to the probability distribution. The main objective is to estimate a probability distribution P given the limited information and minimal distributional assumptions and therefore recover as ˆ ˆ *P* .

However, as the number of observations (T) is smaller than the number of independent variables (m), in order to recover P by using traditional procedures of mathematical inversion, there is more than one vector P making the solution feasible. Therefore the problem is ill-posed and there is no basis for picking a particular solution vector for P from the feasible set. Thus, by asking for a particular set of probabilities considered as most likely, it seems reasonable to favour the one that could have been generated in the greatest number of ways given the available data.

The definition of the entropy measure H(P) and the formulation of the Entropy Maximization problem can help to estimate a unique P distribution since the principle of Maximum Entropy provides a basis for transforming the sample information into a probability distribution that reflects the uncertainty about the individual outcomes.

The measures of entropy H(P) quantify the uncertainty associated with a random experiment. In particular, given a random variable X with values xi and probability

distribution *P p p* <sup>1</sup> ,..., *<sup>n</sup>* with 0 *<sup>i</sup> p* (i=1..., n) and 1 1 *n i i p* , Shannon's measure of

entropy (Shannon 1948) is defined as: <sup>1</sup> 1 ,..., - log *n S S n ii i HP Hp p p p* .

The value of the entropy is maximum when all the values xi have the same probability (and then P is a uniform distribution). This situation would be justified by the Laplace Indifference Principle, according to which the uniform distribution is the most suitable representation of the knowledge when the random variable is completely unknown. Nevertheless, sometimes the ignorance of the probability distribution of X is not absolute and there is some partial information on the distribution such as the mean, variance, moments or some characteristics which can be formulated as equality constraints. In such a case, it is possible to estimate the probability distribution through the application of the Maximum Entropy principle (Jaynes 1957) choosing the distribution for which the available information is just sufficient to obtain the probability assignment.

Thus, if certain values *<sup>r</sup> a* (r=1..., s) associated with functions ( ) *<sup>r</sup> g X* of the values of X are known but its distribution is unknown, the problem consists of estimating a nonnegative distribution that fulfils the conditions 0 *<sup>i</sup> p* for i=1..., n and 1 1 *n i i p* , maximizing the value

of the entropy.

250 Modeling and Optimization of Renewable Energy Systems

 *Greenhouse gas emissions by Energy industries* as a total of Greenhouse gas emissions. The emission trading schemes affect the short-term marginal cost of energy industries, increasing the wholesale electricity prices and thus, the household electricity price.

 *Energy dependency*: Spain has a high dependence (around 80% levels) on imported resources such as crude oil and natural gas, therefore electricity prices are linked to

superior to the number of independent variables (T>m). Nevertheless, information is limited (T=6 and m=9-including a constant term) and when trying to estimate the electricity price models through regression procedures a dimensionality problem arises. Therefore, in a situation of limited sample data the estimation of the model by regression procedures (OLS) is not possible as the problem is undetermined or ill-posed. However, when these circumstances of small amount of information available make it unfeasible to estimate the

> 

 defined as: *y X u XP u* 

m>T. A probability distribution should be used in order to represent partial and limited information regarding the individual observations so they are consistent with the observed sample data. Therefore, following Golan et al. 1996 it is possible to define an inverse general

dimensional vector of unknown terms related to the probability distribution. The main objective is to estimate a probability distribution P given the limited information and

However, as the number of observations (T) is smaller than the number of independent variables (m), in order to recover P by using traditional procedures of mathematical

without making distributional assumptions. The approach consists of developing a nonlinear inversion procedure (Golan et al. 1996) which requires the application of the tools

 , ,..., 

provided by the Information Theory (Shannon, 1948; Jaynes 1957a, 1957b).

by regression techniques requires that the number of observations was

through OLS procedures the Maximum Entropy Econometric approach

*<sup>m</sup>* in the corresponding parameterized model

in a situation of limited sample data where

, where 1 *Pp p* : ( , , )' *<sup>m</sup>* is a m-

 as ˆ ˆ *P* .

C*eteris paribus*, a positive effect of this variable on electricity prices is expected. *Gross Domestic Product per capita*, GDP per capita: This variable aims to study the effect of the general economic activity on electricity prices. A positive effect of this variable on electricity prices is expected by keeping constant both known and unknown factors that may also influence the relationship between household electricity price and the GDP

*Electricity generated from renewable sources- % Total gross electricity generation.* 

 *Electricity generated from nuclear- % Total gross electricity generation Electricity generated from natural gas- % Total gross electricity generation Electricity generated from petroleum - % Total gross electricity generation Electricity generated from hard coal- % Total gross electricity generation.* 

Moreover, the following variables are also used:

such international energy commodities prices.

independent variables.

allows to recover the estimates of 1 2

**4.1 Maximum entropy econometric approach**  Consider a regression-based method: *y X u*

minimal distributional assumptions and therefore recover

The estimation of

model *yX u*

problem for recovering

By solving the maximization problem it is possible to obtain the estimated probabilities <sup>1</sup> <sup>ˆ</sup> ˆ ˆ , , *Pp p <sup>n</sup>* . The maximum entropy distribution does not have a closed-form solution and therefore numerical optimization techniques must be used to compute the probabilities.

Working towards a criterion for recovering the parameters of the regression model related to electricity price in the general inverse problem *y X u XP u* , if there is no evidence that a specific independent variable is more significant than others, the related probability distribution (P) would be the uniform (according to Laplace Indifference Principle). However, the principle of maximum entropy provides a basis for using the sample information in a probability distribution P that reflects the uncertainty about the individual independent variable. Therefore, the problem consists of estimating a nonnegative distribution P by

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 253

For the solution of the optimization the GAMS program version 21.3 (*General Algebraic Modeling System*) is used. This is a programming language which allows diverse

Firstly, it is necessary to establish an a priori range for the possible values that may be assumed by u error in the model, which may be employed to assume certain characteristics of its distribution: V. Since this decision is arbitrary, a support vector for the errors (-v, -v/2, 0, v/2, v) for v>0 it is assigned. It guarantees error´s symmetry around zero. The decision regarding the amplitude of the range of values which it may assume is arbitrary. According with Pukelsheim (1994) support vector v can be assessed if the variability presented on y was knonwn and it would be possible to use the *three standard deviation rule* as estimation for v. In fact, the proposal of Golan et al (1997) who use the sample variance of y as an estimate for v is used. In sample data used the variance of *y* is 9,6 (euros/MWh) and then v=16,15. However, as a widening of the error bound by increasing v the estimated weights converge on the uniform distribution (the difference between the weights of the variables is reduced),

also established. Thus, the support space Z has to be chosen, and then use the data to

through Z should reflect the prior knowledge about the unknown parameters. However, such knowledge is not available as the estimated models are scarce, and a variety plausible

However, a vector support symmetrical and centered on zero is considered according with the value ranking that the independent variables may take. Moreover, as an initial

Z= (-z, 0, z) was considered (z>0 which guarantees its symmetry around zero). The same z for all coefficients (z=0.6) was located. It implies to be very cautious in the interpretation of

The reported estimated coefficients for the model correspond with highest R obtained. The

The estimated information index R=0,67 indicates a reduction of the uncertainty by using the maximization entropy approach, however, the findings yield that the variable Electricity generated from renewable energies (RES-E) have sense to explain electricity prices as ˆ 0,38 *S pi* . Electricity from RES contributes a reduce prices as negative sign is found.

Also natural gas and energy dependency contributes to explain the increasing in electricity

. As *Sp*ˆ*i* are reported and ˆ 1 *S pi* implies 0

identification of the information content of a given *<sup>i</sup> x* is just the normalized entropy.

. The restrictions imposed on the parameter space

in the model is

was finding. So,

*<sup>i</sup>* a natural criterion for

) under the reparameterized

A general maximum entropy model with a reparametrized error is considered.

the most reduced v that makes the solution feasible (v=18) is used.

Moreover, a priori range for the possible values that may be assumed by

approximation, a covariate matrix was calculated and negative values in

**4.2 Maximum entropy econometric estimated model** 

optimization problems to be solved.

estimate the P which in turn yields

may want to be entertained.

Table 5 shows estimated weights for the electricity price (

results are those obtained under the narrowest V vector.

bound on

the estimated ˆ

system.

prices.

maximizing the value of the entropy H(P) subject to the available information. By solving the optimization problem the estimated probability distribution ˆ P is obtained.

A general inverse problem *y X u XP u* it is considered where the goal is to determine the unknown and unobservable frequencies *P p p* <sup>1</sup> ,..., ' *<sup>m</sup>* , representing the data generating process. Then, within the possible sets of probabilities fulfilling 1 1 *m i i p* , <sup>0</sup> *<sup>i</sup> <sup>p</sup>* , a single vector must be chosen. Through the application of the principle of maximum entropy H(P) is maximized under the restrictions of information consistency *yX u* , and the adding up-normalization constraint for P: *P*' 1 .

If the vector of disturbances, u, is assumed to be a random vector with finite location and scale parameters, it is possible to represent the uncertainty about it by treating each *ut* (t=1, ..., T) as a finite and discrete random variable with 2 *J* possible outcomes.

Thus, it is assumed that each *ut* is limited by an interval ( *<sup>t</sup>*<sup>1</sup> *v* , *tJ v* ), whose probability, Pr( <sup>1</sup> ) *t t tJ v uv* , can become as small as it is wanted. For example, for J=2, the error can be defined as: 1 (1 ) *u wv w v t tt t tJ* where each *wt* 0,1 is a vector of error weights. Furthermore, 2 *J* can be used to assume certain characteristics of symmetry and kurtosis about the error distribution.

Because there may be different levels of uncertainty underlying each *<sup>i</sup>* , for more general inferential purposes, point estimates may be limiting and unrealistic. Consequently, it is possible to generalize the maximum entropy problem to permit a discrete probability distribution to be specified and obtained for each *<sup>i</sup>* . Rather than search for the point estimates of , each *<sup>i</sup>* is viewed as the mean value of some well defined random variable z.

Then, for each *<sup>i</sup>* , it is assumed there exists a discrete probability distribution that is defined over a parameter space *<sup>K</sup>* by a set of equally distanced discrete points *zz z i K* <sup>1</sup> ,, ' with corresponding probabilities *pp p i i iK* <sup>1</sup> ,, ' and with *K* 2 . Therefore: *<sup>i</sup> i Pi E z* or *E z <sup>P</sup>* .

Using the Maximum entropy econometric approach, one investigates how "far" the data pull the estimates away from a state of complete ignorance (uniform distribution). In order to measure the reduction in the initial uncertainty, the information index entropy measure R is defined (Golan, 1994; Soofi 1992, 1994) and where *R* 0,1 . Higher is the value of R better is the estimated model.

Moreover, some measures are defined to evaluate the information in each one of the variables i = 1,2,..., m as the normalized entropy: *Sp*ˆ*i* . These variable-specific information measures reflect the relative contribution (of explaining the dependent variable) to the independent variable. Where *S p* ˆ*i*0,1 , zero reflects no uncertainty while one reflects total uncertainty in the sense that P is uniformly distributed.

#### **4.2 Maximum entropy econometric estimated model**

252 Modeling and Optimization of Renewable Energy Systems

maximizing the value of the entropy H(P) subject to the available information. By solving the

determine the unknown and unobservable frequencies *P p p* <sup>1</sup> ,..., ' *<sup>m</sup>* , representing the data generating process. Then, within the possible sets of probabilities fulfilling

, <sup>0</sup> *<sup>i</sup> <sup>p</sup>* , a single vector must be chosen. Through the application of the principle of

maximum entropy H(P) is maximized under the restrictions of information consistency

If the vector of disturbances, u, is assumed to be a random vector with finite location and scale parameters, it is possible to represent the uncertainty about it by treating each *ut* (t=1,

Thus, it is assumed that each *ut* is limited by an interval ( *<sup>t</sup>*<sup>1</sup> *v* , *tJ v* ), whose probability, Pr( <sup>1</sup> ) *t t tJ v uv* , can become as small as it is wanted. For example, for J=2, the error can be defined as: 1 (1 ) *u wv w v t tt t tJ* where each *wt* 0,1 is a vector of error weights. Furthermore, 2 *J* can be used to assume certain characteristics of symmetry and kurtosis

inferential purposes, point estimates may be limiting and unrealistic. Consequently, it is possible to generalize the maximum entropy problem to permit a discrete probability

over a parameter space *<sup>K</sup>* by a set of equally distanced discrete points *zz z i K* <sup>1</sup> ,, ' with corresponding probabilities *pp p i i iK* <sup>1</sup> ,, ' and with *K* 2 . Therefore: *<sup>i</sup>*

Using the Maximum entropy econometric approach, one investigates how "far" the data pull the estimates away from a state of complete ignorance (uniform distribution). In order to measure the reduction in the initial uncertainty, the information index entropy measure R is defined (Golan, 1994; Soofi 1992, 1994) and where *R* 0,1 . Higher is the value of R better

Moreover, some measures are defined to evaluate the information in each one of the variables i = 1,2,..., m as the normalized entropy: *Sp*ˆ*i* . These variable-specific information measures reflect the relative contribution (of explaining the dependent variable) to the independent variable. Where *S p* ˆ*i*0,1 , zero reflects no uncertainty while one reflects

*<sup>i</sup>* , it is assumed there exists a discrete probability distribution that is defined

*<sup>i</sup>* is viewed as the mean value of some well defined random variable z.

P is obtained.

*<sup>i</sup>* . Rather than search for the point

*<sup>i</sup>* , for more general

*i Pi E z*

it is considered where the goal is to

optimization problem the estimated probability distribution ˆ

, and the adding up-normalization constraint for P: *P*' 1 .

..., T) as a finite and discrete random variable with 2 *J* possible outcomes.

Because there may be different levels of uncertainty underlying each

distribution to be specified and obtained for each

total uncertainty in the sense that P is uniformly distributed.

A general inverse problem *y X u XP u*

1

*m i i p* 

1

about the error distribution.

, each

is the estimated model.

*yX u* 

estimates of

or *E z <sup>P</sup>* .

Then, for each

For the solution of the optimization the GAMS program version 21.3 (*General Algebraic Modeling System*) is used. This is a programming language which allows diverse optimization problems to be solved.

A general maximum entropy model with a reparametrized error is considered.

Firstly, it is necessary to establish an a priori range for the possible values that may be assumed by u error in the model, which may be employed to assume certain characteristics of its distribution: V. Since this decision is arbitrary, a support vector for the errors (-v, -v/2, 0, v/2, v) for v>0 it is assigned. It guarantees error´s symmetry around zero. The decision regarding the amplitude of the range of values which it may assume is arbitrary. According with Pukelsheim (1994) support vector v can be assessed if the variability presented on y was knonwn and it would be possible to use the *three standard deviation rule* as estimation for v. In fact, the proposal of Golan et al (1997) who use the sample variance of y as an estimate for v is used. In sample data used the variance of *y* is 9,6 (euros/MWh) and then v=16,15. However, as a widening of the error bound by increasing v the estimated weights converge on the uniform distribution (the difference between the weights of the variables is reduced), the most reduced v that makes the solution feasible (v=18) is used.

Moreover, a priori range for the possible values that may be assumed by in the model is also established. Thus, the support space Z has to be chosen, and then use the data to estimate the P which in turn yields . The restrictions imposed on the parameter space through Z should reflect the prior knowledge about the unknown parameters. However, such knowledge is not available as the estimated models are scarce, and a variety plausible bound on may want to be entertained.

However, a vector support symmetrical and centered on zero is considered according with the value ranking that the independent variables may take. Moreover, as an initial approximation, a covariate matrix was calculated and negative values in was finding. So, Z= (-z, 0, z) was considered (z>0 which guarantees its symmetry around zero). The same z for all coefficients (z=0.6) was located. It implies to be very cautious in the interpretation of the estimated ˆ . As *Sp*ˆ*i* are reported and ˆ 1 *S pi* implies 0 *<sup>i</sup>* a natural criterion for identification of the information content of a given *<sup>i</sup> x* is just the normalized entropy.

Table 5 shows estimated weights for the electricity price ( ) under the reparameterized system.

The reported estimated coefficients for the model correspond with highest R obtained. The results are those obtained under the narrowest V vector.

The estimated information index R=0,67 indicates a reduction of the uncertainty by using the maximization entropy approach, however, the findings yield that the variable Electricity generated from renewable energies (RES-E) have sense to explain electricity prices as ˆ 0,38 *S pi* . Electricity from RES contributes a reduce prices as negative sign is found.

Also natural gas and energy dependency contributes to explain the increasing in electricity prices.

An Analysis of the Effect of Renewable Energies on Spanish Electricity Market Efficiency 255

An electricity generation technology based on renewable energies produce a least-cost merit order in the wholesale electricity market and its associated efficiency gains should also lead to lower electricity prices. However, there is not clear evidence about the effect of renewable

A large share of RES-E power generally gives lower electricity prices reducing the profitability of investing in new electricity capacity. If RES-E generators are exposed to

In this context, the participation of the governments is necessary in the initial phase of the introduction of the renewable electricity generation technologies for securing their development and protecting them of the direct competition that suppose the conventional

In Spain, the public support in electricity generation using renewable energies is the feed-in tariff which guarantees a price higher than that existing in the wholesale market for the renewable technology employed. This cost increment is financed by electricity

Therefore, although large share of RES-E power generally gives lower wholesale electricity prices, a controversial debate has arisen about the RES-E effects on final

In order to contribute to this debate, in this chapter we propose a maximum entropy econometric model with the aim of explaining the electricity prices as a function of variables

The sample data regarding the introduction of renewables in the wholesale is limited and when trying to estimate models through regression procedures a dimensionality problem arises. As an alternative to estimate the model, when a dimensionality problem arises, we

The obtained results show that electricity generated by reneawable energies contributes to increase final electricity prices. But also, the most important variables affecting prices is energy dependency. Spain has a high dependence (around 80% levels) on imported resources such as crude oil and natural gas (100%), therefore electricity prices are linked to such international energy commodities prices and introduces some risk to energy generation

Del Río, P. (2008). Ten years of renewable electricity policies in Spain: An analysis of

Deloitte-Appa (2009). *Estudio macroeconómico del impacto de las energías renovables* 

European Commission (2009*). Directive 2009/28/EC of the European Parliament and of the* 

successive feed-in tariff reforms. *Energy Policy*, Vol. 36, (August, 2008), pp. (2917-

*en la economía española*, Asociación de Productores de Energía Renovable,

*Council of 23 April 2009 on the promotion of the use of energy from renewable sources* 

related to renewable energy sources and other electricity generation sources.

energies on the final electricity prices.

technologies.

electricity prices.

**6. References** 

tariffs.

market prices, it directly affects their market revenues.

propose a Maximum Entropy Econometric approach.

related to volatility of international market prices.

2929), ISSN 0301-4215.

Madrid.


Table 5. Estimated Household electricity price model by Maximum entropy econometric approach

Energy dependency has also an important effect. Spain has a high rate of energy dependency due to the scant presence of primary fossil fuel deposits. That great dependence introduces some risk to energy generation related to volatility of international market prices.
