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Reference
Comprehensive performance and incertitude analysis of multi-energy portfolios: Combination of multi-variante portfolio theory and multi-criteria diversity analysis for energy generation planning
TRUTNEVYTE, Evelina, KIENZLE, Florian, ANDRESSON, Göran
TRUTNEVYTE, Evelina, KIENZLE, Florian, ANDRESSON, Göran. Comprehensive performance and incertitude analysis of multi-energy portfolios: Combination of multi-variante portfolio theory and multi-criteria diversity analysis for energy generation planning. 2008
Available at:
http://archive-ouverte.unige.ch/unige:104613
Disclaimer: layout of this document may differ from the published version.
Comprehensive Performance and Incertitude Analysis of Multi-Energy Portfolios
Combination of Multi-Variance Portfolio Theory and Multi-Criteria Diversity Analysis for Energy Generation Planning
Prepared by:
Evelina Trutnevyte,
Vilnius Gediminas Technical University
Under supervision of:
Florian Kienzle,
Prof. Dr. Göran Andersson,
Swiss Federal Institute of Technology, Zurich
Zürich, 2008
Index
1 Introduction ... 3
2 Mean-Variance Portfolio Theory ... 4
2.1 General Overview: Benefits and Limitations ... 4
2.2 Analysis of the Available Data to Form Multi-Energy Portfolios and their Modification 5 2.2.1 Short Description of Available Data ... 5
2.2.2 Modification and Extension of Available Data ... 6
2.2.3 Example of Applying Mean-Variance Portfolio Theory ... 8
3 Multi-Criteria Diversity Analysis ... 9
3.1 General Overview: Benefits and Limitations ... 9
3.2 Mathematical Fundamentals ... 10
3.3 Example of Applying Multi-Criteria Diversity Analysis ... 11
3.3.1 Initial data ... 11
3.3.2 Results of Applying Multi-Criteria Diversity Analysis ... 14
4 Combining Mean-Variance Portfolio Theory and Multi-Criteria Diversity Analysis16 4.1 Combination based on the Mean - Variance Portfolio Efficient Frontier ... 17
4.1.1 Mathematical Fundamentals ... 17
4.1.2 Example of Combination on the Basis of the Mean-Variance Portfolio Efficient Frontier 18 4.2 Comprehensive Performance and Incertitude Analysis ... 21
4.2.1 Mathematical Fundamentals ... 21
4.2.2 Example of Applying Comprehensive Performance and Incertitude Analysis ... 23
4.3 Comparison of Results Provided by Mean-Variance Portfolio Theory, Multi-Criteria Diversity Analysis and Comprehensive Performance and Incertitude Analysis ... 25
4.4 Generation of Efficient Portfolios to Meet Certain Greenhouse Gas Emissions ... 27
5 Conclusion ... 28
6 Suggestions for Further Research ... 29
7 References ... 31
8 Appendix ... 32
8.1 Mathematical Formalization of Estimating Efficient MDA Frontier ... 32
8.2 Mathematical Formalization of Combining MDA and MVP as a Multi-Objective Optimization Task ... 34
1 Introduction
This project is an extension to a current research at Power System Laboratory, Swiss Federal Institute of Technology in Zurich. The research is a part of the project “Vision of Future Energy Networks” and pretends to develop a methodology to form efficient multi-energy generation portfolios for the future [5].
Scenario-based mean variance portfolio (MVP) theory to generate efficient portfolios with multi- energy outputs (electricity, heat and cooling power) was already developed and presented in [5].
As MVP theory analyses the return and risk of investment into energy generation, it presents only a limited investor’s viewpoint. Viewpoint of society is reflected only to some extent as the higher return of investment results in lower prices of final energy services. MVP theory is beneficial as it is a mathematical basis to diversify energy generation technologies in order to form portfolios of high return and low risk. Risk is measured as a standard deviation of a set of scenario-based portfolio return and presents a spread of return values around the expected mean return. This evaluation includes information only about well-defined outcomes and their known probabilities. Thus, MVP theory ignores importance of unpredicted outcomes and unknown probabilities of some outcomes, even though they should be also included in planning of energy systems.
As a criterion of sustainability is recently set for energy generation systems, it is important to include environmental, social or other aspects into decision making process. Moreover, diversification according to economical, technological, organizational and other aspects could build a hedge against possible negative effects of unknown outcome with known or unknown probabilities. Thus, multi-criteria diversity analysis (MDA) is very helpful [2]. Firstly, MDA assesses multi-criteria performance of energy generation technologies, so environmental and other performance indicators could be included in addition to pure economic assessment.
Secondly, MDA includes a method to diversify energy generation technologies according to technological and other chosen aspects. In this way MDA presents a broader viewpoint of society. However, as MDA does not cover any information about effectiveness of investment, it might form efficient portfolios that are not beneficial to investors and nobody will be willing to invest.
Application of pure MVP theory and pure MDA for energy generation planning has both advantages and disadvantages. One presents investor’s viewpoint, while the other one – society’s perspective. Each of the two methods cover different spectrums of “total incertitude”. But if these methods would be combined to cover each other’s disadvantages, then a tool to generate efficient multi-energy portfolios would be developed. That is the aim of this research.
The research was started by overtaking and modifying data of previously conducted MVP analysis of electricity and heat generation technologies [5]. At first, MVP analysis and MDA were conducted in isolation to analyze their possibilities. Integration of both methods was then carried out on the basis of efficient MVP portfolios as presented in [1]. Finally, a methodology of so-called comprehensive performance and incertitude analysis (CPIA) of multi-energy portfolios is developed by combining MVP theory and MDA as a multi-objective optimization task. For illustration CPIA is applied to two cases in this paper: to form efficient portfolios for energy system that produces 90% of electricity and 10% of heat, and to form efficient portfolios with the same energy production balance, but also satisfying an additional constraint of limited greenhouse gas emissions.
As this research pretends to develop a mathematical basis to conduct CPIA, the reader should prudently consider the obtained results about technology shares in efficient portfolios. When
CPIA is applied to a real case, more detailed analysis of initial data about energy generation technologies is important.
2 Mean-Variance Portfolio Theory
Benefits and limitations of applying MVP theory to form efficient energy generation portfolios are presented in this chapter. In chapter 2.2 the application of the theory for multi-energy portfolios is also discussed, regarding results presented in [5], and modified for further analysis.
2.1 General Overview: Benefits and Limitations
MVP theory was developed as a method to find optimal economic portfolios that would guarantee the highest economic performance at a certain level of risk [1, 6]. Assuming that the past performances could reliably help to evaluate future effects, statistical analysis of historical data by calculating mean as performance and standard deviation (variability) as risk is a basis for estimating a set of different portfolios. As historical fluctuations cover both random and systematic variations in performance, standard deviation of historical data is assumed to reliably reflect the potential risk in the future. When performance is maximized at a chosen level of risk (or risk is minimized at a chosen level of performance), a so called efficient frontier is determined. Portfolios forming the efficient frontier ensure highest performance at a certain level of risk. Mathematical fundamentals of the theory and examples could be found in [1, 5, 6].
This theory has been also successfully applied to planning of energy generation [5, 6, 10, 11, 12, 13, 14]. In this context, economic performance is expressed either as costs of energy generation [10, 11] or as return – inverse of energy generation costs expressed in “amount of produced energy per monetary unit”, e.g. kWh/USD [5, 6]. Presentation [14] also introduces a concept of performance as benefits of electricity production (measured by income for sold electricity minus electricity generation costs).
Risk is mostly evaluated as a standard deviation of historical fluctuations in costs of energy generation. In [5] MVP theory was applied differently than usual. Pretending to extend planning horizon, scenario analysis is conducted instead of statistical analysis and it leads to calculating mean performance and its risk on the basis of officially accepted future scenarios of energy generation. In [14] Monte Carlo Simulation is used to obtain fluctuations of annual return of each technology.
MVP theory is applied not only for preplanning of investment into energy generation technologies. As efficient portfolios should be maintained over time responding to ongoing changes, in [12] MVP theory is also used as a tool for portfolio management. In [13] effects of adding new technologies to current portfolios are assessed. MVP theory could be used not only for planning energy generation capacities, but also for designing operation strategies in short- time periods [9].
Limitations
However, the application of MVP theory leads to relatively limited results in energy planning as it provides the decision making process only with information about economic performance of energy generation [1]. As a criterion of sustainability for energy systems is set recently, energy generation technologies should be assessed in a wider perspective that includes an analysis of environmental, social, and security aspects in addition to economic performance. Results of
MVP analysis should not be used as the only basis to make a decision, because other external effects of energy generation should be taken into account too.
In addition, uncertainty, ambiguity and ignorance are also of great importance in planning of energy generation [1]. Even analysis of historical data could be replaced by scenario analysis, this cannot totally cover other types of incertitude – uncertainty, ambiguity and ignorance.
Concept of different types of incertitude – risk, ambiguity, uncertainty and ignorance – is presented Figure 1.
Figure 1. Concept of different types of incertitude [1]
It could be seen that total incertitude of energy generation technologies is not only risk, when outcomes and their probabilities are defined. Other cases of poorly defined outcomes with either well-defined probabilities or poorly-defined probabilities should also be included when assessing incertitude of energy generation portfolios. Thus, MVP theory covers only one part of total incertitude and this limits the reliability of MVP results.
MVP theory is also criticized for its concept of risk [15, 16]. When risk is measured as a variance or a standard deviation, it includes fluctuations both above and beneath the mean value.
However, the case when return of portfolio is higher than the mean value is beneficial (in energy generation case – more electricity is produced for the same costs) and should not be considered as risk. It is possible to assess risk as a semivariance that includes only fluctuations beneath the mean value [15]. Unfortunately, this makes the optimization task much more complicated and sets higher computational requirements. As a rule only approximate methods are used to evaluate portfolio semivariance [16]. There are only minor differences in efficient frontier based on variance and semivariance though [16]. In this research MVP theory will be combined with another method and this already will result in more complicated calculations. Thus, it is assumed that variance reliably presents the portfolio risk and semivariance is not assessed.
2.2 Analysis of the Available Data to Form Multi-Energy Portfolios and their Modification
2.2.1 Short Description of Available Data
In [5] MVP theory is applied to a system with multi-energy outputs to form efficient so-called multi-energy portfolios. This research contributes to previously developed MVP models by two aspects: firstly, scenario analysis is used to determine expected performance and its risk;
secondly, it applies the MVP theory to an energy system with several energy outputs (electricity and heat).
Wind power plants, photovoltaics (PV), biogas engines, natural gas fired engines, solar (thermal) elements and gas boilers are taken into analysis. These groups of technologies mainly represent different fuels and fundaments of technologies. In [5] the performance and risk of energy generation technologies are evaluated in terms of energy production per monetary unit (MWh/USD).
Aiming at a long-time horizon of energy generation planning, in [5] performance and risk of portfolios are assessed by conducting a scenario analysis. Different scenarios are developed including various external factors (external drivers) that influence energy generation costs. In [5]
these external drivers are concerns about climate change (with effect on price of CO2 emissions), energy-related research efforts (with effect on improvement in efficiency) and geopolitical tension (with effect on price of primary energy resources).
Portfolio is defined as a set of shares of energy generation technologies in the final mix.
Portfolios with the lowest risk at chosen levels of performance form the efficient frontier that presents efficient multi-energy portfolios. It must be noticed that in [5] a certain wanted ratio of electricity and heat production is not set. The output of electricity and heat is only later derived from the chosen generation mix.
2.2.2 Modification and Extension of Available Data
Available data for efficient multi-energy generation portfolios are modified and supplemented in the following ways:
1. ratio of desired electricity and heat production is set as initial data, 2. more energy generation technologies are included,
3. risk of economic performance is additionally expressed in percentage.
1. Ratio between desired electricity and heat production
In the present application of MVP theory to multi-energy portfolios, a desired ratio of heat and electricity production is not set. However, if applying MVP theory to a real energy system, the exact demands of electricity and heat (or ratio between desired electricity and heat production) are known. Regarding Figures 6 and 7 in [5], analysis with a predefined ratio of electricity and heat production results in only one point on the efficient frontier. In analyzed case, several portfolios with the same energy production balance, but different performance and risk could be found only for the case when share of electricity is 0 % and share of heat is 100 %.
However, this considerably limits the possibility to explore other portfolios generating the desired ratio of electricity and heat. Moreover, in this case the application of MDA based on [1]
would be impossible because, with the present formalization, a set of efficient portfolios provided by the MVP analysis is needed.
In further research a predefinition of ratio of electricity and heat production is advised. Firstly, it would make this theory directly (without any modifications) applicable to real systems where electricity and heat demand is exactly known. Integrated and overall analysis of energy systems should include all elements of energy generation (centralized/decentralized/individual, various fuels and technologies), and not only a part of them, so that the best option to provide final consumers with the desired amount of heat and electricity would be found.
Thus, the application of MVP theory to multi-energy portfolios is based on [5] and extended here by one more constraint in the formulation of the optimization task. The efficient frontier is calculated as a set of portfolios with minimum risk at chosen levels of performance when an exact ratio of heat and electricity production is defined. Portfolios of efficient frontier are then estimated by solving this optimization task (regarding symbols used in [5]):
2
1 1
1
1
1
min min cov
1
,..., 0
n n
p i j i j ij
n
i i
el
n i
i tot i i n
x x x
x b
x x
(1.1)Where bis a predefined share of electricity that should be generated (from 0 to 1). The share of heat is automatically defined then as
1b
.The second constraint in Formula (1.1) defines a desired ratio of electricity in the total energy production balance.
According to statistical information gathered by the International Energy Agency, in Switzerland 17300 TJ (4806 GWh) of heat and 59612 GWh of electricity were produced in 2005 and that results in an electricity share of 92 %. Accordingly, in these calculations it is assumed that 90 % share of electricity and 10 % share of heat should be produced.
2. Additional energy generation technologies
In addition to the present analysis, two other electricity generation technologies – small hydro power plants and fuel cells using natural gas – are added to the calculation. Economic performance for different scenarios was calculated in previous stages of this project (Table 1).
Table 1. Performance of new energy generation technologies, USD/MWh
S1 S2 S3 S4 S5 S6 S7 S8
Small Hydro 39.7 39.7 40.5 40.5 39.7 39.7 40.5 40.5
Fuel Cell (fuelled with natural gas) 86.8 67.5 74.5 96.4 70.7 51.4 58.4 80.3
On a broader analysis even more technologies should be considered and they could be grouped including other aspects too (e.g. centralized/decentralized generation). For instance, heat generation in natural gas boilers is presently described without defining the size of these boilers.
However, decentralized heat generation in individual boilers would lead to higher costs of heat generation, lower efficiency etc. than in a centralized boiler plant. Thus, this MDA theory application could suggest how much heat should be produced in centralized and individual plants.
3. Measurement of risk in percentage
In [5] risk was evaluated as a standard deviation of portfolio return, measured in MWh/USD.
Conceptually, standard deviation shows the spread of performance values around the mean performance [8]. E.g. if analyzing one point of the efficient MVP frontier in Figure 6 of [5], the efficient portfolio with a return of 74 kWh/USD results in a risk (or standard deviation) of 5 kWh/USD as a negative or positive change in the value of mean performance.
For illustrative purpose, the level of risk could be also expressed in percentage and this would lead to a clearer first-sight understanding of these results. In the mentioned case, the efficient portfolio with a return of 74 kWh/USD results in a risk of 6.8 %, which means that a positive or negative deviation of 6.8 % from the mean return is expected in the average.
However, it must be noticed that when MVP theory and MDA results are combined using criteria of total performance and total incertitude [1], MVP performance and risk, measured by kWh/USD, should be evaluated in calculations.
2.2.3 Example of Applying Mean-Variance Portfolio Theory
Regarding the available data from [5] and its modifications discussed in the previous chapter, MVP analysis for multi-energy portfolios was carried out. The analysis covers four electricity generation technologies (wind power plants, small hydro plants, solar PV cells, and fuel cells using natural gas), two CHP technologies (biogas and natural gas CHP plants) and two heat generation technologies (solar collectors and natural gas boilers). When ratio of energy outputs is premised to be 90 % of electricity and 10 % of heat, the resulting MVP efficient frontier is presented in Figure 2.
Figure 2. MVP efficient frontier
The shares of energy generation technologies along the efficient MVP frontier are presented in Figure 3.
Figure 3. Shares of technologies along MVP efficient frontier
Considering these results of MVP analysis, additional limitations of applying MVP theory (especially scenario-based analysis) to energy systems could be observed. In this case scenario- based economic performance of different electricity generation technologies correlates well, because effects of so-called external drivers are similar for all electricity technologies. Moreover, use of CHP technologies in this case is limited by predefined ratio of electricity and heat production. Thus, the electricity technology with the highest performance (in this case – small hydro power plants) covers the major part in the balance of electricity generation. Even though results could be made more realistic by defining potential of hydro power, it must be emphasized that a pure MVP analysis might not properly evaluate importance of diversification and might not prevent from one technology considerably leading in the balance. E.g. even according to the scenario analysis small hydro power plants guarantee relatively low economic risk, having 85 % of one technology in the total balance does not ensure stable and reliable supply of energy. Here MDA could help in diversifying energy generation technologies towards more diverse and secure portfolios.
3 Multi-Criteria Diversity Analysis
3.1 General Overview: Benefits and Limitations
The very profound origin of diversity analysis is a need for scientific basis to answer a question why do we generally need to diversify [2]. When benefits of different options are divergent, then a diversification of these options could lead to more beneficial effects than in the case where every option is used in isolation. Diversity is described as variety, disparity and balance, where variety expresses a number of different options, disparity describes differences between these different options and balance describes a contribution of these different options in the overall mix [1, 2].
Therefore, diversification as a means to increase security and reliability of energy systems is commonly used in energy planning, especially when determining a mix of energy generation technologies. In these cases diversity analysis is important as a mathematical basis for the decision-making process. Diversity analysis is also used in many other fields, e.g. ecology, economy, computer science, engineering etc. [2].
As a type of diversity analysis, multi-criteria diversity analysis (MDA), was applied to energy generation planning in [1]. In this case MDA helps to determine mixes of energy generation technologies according to different performance indicators (e.g. economic, environmental etc.).
Moreover, in comparison to MVP theory, MDA does not use any historical data to predict future effects and it tends by diversifying technologies to take unknown effects and outputs into account. Thus, MDA serves as a method to evaluate and minimize full incertitude that covers uncertainty, ambiguity and ignorance [1]. Uncertainty could be described as a lack of basis for assigning probabilities, ignorance – as a lack of knowledge about possible outcomes, and ambiguity – as a lack of agreement how to describe outcomes [3].
In energy generation planning the performance of different options that form a portfolio is evaluated by already taking different viewpoints into account and, with the help of MDA, this performance is maximized at chosen levels of incertitude. Thus, MDA is a method to evaluate performance of energy generation technologies in different points of view that might include presently important factors like environmental or social performance, increase in security etc.
Minimization of incertitude (or maximization of diversity) would minimize possible negative
effects of unknown, in other words – would ‘hedge against uncertainty, accommodate ambiguity and build resilience against ignorance’ [1].
Limitations
However, used in isolation, MDA does not explore any information about risk; therefore, it does not comprehensively represent incertitude [1].
Moreover, economic performance of energy generation portfolios is a very important factor as it defines the price of an energy unit and the efficiency of investment. Pure MDA results would lead to more secure and society-oriented energy generation portfolios, but it must be emphasized that these portfolios should also match the needs of investors (so that they would be interested in investing) and of final consumers (so that they would be able and willing to pay for energy services).
3.2 Mathematical Fundamentals
Performance in MDA is conceptualized as a set of multi-dimensional properties of energy generation technologies, e.g. economic efficiency or environmental performance [1].
Mathematically it can be expressed as:
,
port i i i j
P
X W r (1.2)Where
i is the number of particular technology,
j is the number of performance factor (criterion), Xi is the fraction of technology i in the portfolio,
Wi is the weighting multiplier reflecting priorities of different performance factors (
W 1),,
ri j is the performance rank for technology i and performance factor j. Performance rank is normalized on the basis of assigned scores si j, for technology i for a criterion j in this way:
, ,
i j i j
s MIN s r MAX s MIN s
(1.3)
Portfolio incertitude is calculated as:
,
1 1
port
port i k i k
U D X X d
(1.4)Where
Dport is diversity of portfolio,
Xi, Xk are fractions of technology i and technology k in the portfolio,
,
di k is the measure of disparity between technology i and technology k. Disparity is measured as the n-dimensional Euclidian distance between the disparity attributes of the alternative technologies:
2, , ,
i k n i n k n
d SQRT
w a a (1.5)Where
,
ai n and ak n, are disparity attributes for technology i and strategic attribute n,
wn - weight of strategic attribute n.
For convenience MDA performance, diversity and uncertainty could be also expressed as normalized criteria, where ‘1’ presents the highest value and ‘0’ – the lowest.
According to the Formula 1.4, higher level of diversity leads to lower incertitude. Minimization of portfolio incertitude (or maximization of portfolio diversity) at chosen levels of performance would form an efficient frontier in MDA viewpoint. However, regarding the Formula 1.4, optimization task to find MDA efficient frontier becomes nonlinear and more complicated to solve.
3.3 Example of Applying Multi-Criteria Diversity Analysis
In this chapter MDA is carried out in isolation on the modified data of [5] as presented in 2.2.1 and 2.2.2 chapters.
3.3.1 Initial data
MDA performance factors and their weightings in addition to disparity attributes and their weightings should be evaluated for each technology.
Regarding a criterion of sustainability that is recently set for energy generation systems, these three performance factors are included into calculation:
1. Economic performance, calculated as investment and O&M costs per unit of energy production. In evaluation of economic performance prices of CO2 emissions are not included in order to prevent double counting of these emissions.
2. Environmental performance, calculated as greenhouse gas emissions per unit of energy production.
3. Import dependency, calculated as a share of imported fuels in the total fuel consumption in the region under consideration (in this case – Switzerland [7]). The criterion of import dependency could also include dependency on equipment and labor import. However, in this analysis import dependency only reflects security of primary energy supply and, therefore, import of equipment and labor is neglected.
It must be noticed that MDA could include an arbitrary number of performance factors depending on which effects of energy generation technologies are emphasized. These performance factors could be wider environmental effects, increase in employment, reliability of technologies, public acceptability, influence on national budget, etc.
For the chosen three factors, weightings are given as follows: economic performance – 35 %, environmental performance – 35 %, import dependency – 30 %.
As this analysis is developed on the basis of the scenario-based MVP application, the performance factors are evaluated for each technology and each scenario. As every scenario is assumed to have an equal probability to occur [5], the overall performance factor of each technology is evaluated as a mean performance. The performance rank is evaluated according to Formula 1.3, where a performance rank of ‘1’ indicates the best performance and performance rank of ‘0’ – the worst performance. The resulting performance factors and ranks for each technology are presented in Table 2, Table 3, and Table 4.
It must be noticed that in the MDA application, contrarily to the MVP case presented in [5], characteristics of CHP technologies are not split into separate characteristics of heat and electricity generation of the same technology. This is done intentionally: in MDA performance ranks and disparity attributes of different technologies should be compared according to Formulas 1.3 and 1.5 for different technologies, but not for different energy outputs of one particular technology. Further, when a methodology of combined MDA and MVP theory is developed and applied to multi-energy systems, non-separation of heat and electricity generation data for CHP technologies could be considered also when conducting MVP analysis.
Table 2. Economic performance (investment and O&M costs per unit of energy production)
Electricity
Costs, USD/MWh Inverted
costs, kWh/USD
Performance rank S1 S2 S3 S4 S5 S6 S7 S8 Average
Wind 40.7 40.7 44.2 44.2 40.7 40.7 44.2 44.2 42.4 23.6 0.186 Small Hydro 39.7 39.7 40.5 40.5 39.7 39.7 40.5 40.5 40.1 24.9 0.198 Solar PV 253.3 253.3 287.8 287.8 253.3 253.3 287.8 287.8 270.5 3.7 0.000 Fuel Cell (NG) 70.7 51.4 58.4 80.3 70.7 51.4 58.4 80.3 65.2 15.3 0.109
CHP
S1 S2 S3 S4 S5 S6 S7 S8 Average
Inverted costs, kWh/USD
Performance rank Biogas/engine 30.4 30.4 32.0 32.0 30.4 30.4 32.0 32.0 31.2 32.1 0.265 Gas/engine 52.1 41.1 42.0 53.2 52.1 41.1 42.0 53.2 47.1 21.2 0.164
Heat
S1 S2 S3 S4 S5 S6 S7 S8 Average
Inverted costs, kWh/USD
Performance rank Solar (thermal) 13.2 13.2 14.7 14.7 13.2 13.2 14.7 14.7 14.0 71.6 0.634 Gas boiler 10.4 7.5 7.6 10.6 10.4 7.5 7.6 10.6 9.0 110.8 1.000 Table 3. Environmental performance (emissions of CO2 per unit of energy production)
Electricity
Emissions, tCO2/MWh Performance
rank S1 S2 S3 S4 S5 S6 S7 S8 Average
Wind 0 0 0 0 0 0 0 0 0.000 1.000
Small Hydro 0 0 0 0 0 0 0 0 0.000 1.000
Solar PV 0 0 0 0 0 0 0 0 0.000 1.000
Fuel Cell (NG) 0.473 0.473 0.537 0.537 0.473 0.473 0.537 0.537 0.505 0.000
CHP S1 S2 S3 S4 S5 S6 S7 S8 Average Performance
rank Biogas/engine 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.000 1.000 Gas/engine 0.113 0.113 0.115 0.115 0.113 0.113 0.115 0.115 0.114 0.774
Heat S1 S2 S3 S4 S5 S6 S7 S8 Average Performance
rank
Solar (thermal) 0 0 0 0 0 0 0 0 0.000 1.000
Gas boiler 0.208 0.208 0.213 0.213 0.208 0.208 0.213 0.213 0.210 0.584 Table 4. Import dependency (share of fuel import)
Electricity
Share of imported fuel, % Performance rank S1 S2 S3 S4 S5 S6 S7 S8 Average
Wind 0 0 0 0 0 0 0 0 0% 1.000
Small Hydro 0 0 0 0 0 0 0 0 0% 1.000
Solar PV 0 0 0 0 0 0 0 0 0% 1.000
Fuel Cell (NG) 1 1 1 1 1 1 1 1 100% 0.000
CHP S1 S2 S3 S4 S5 S6 S7 S8 Average Performance
rank
Biogas/engine 0 0 0 0 0 0 0 0 0% 1.000
Gas/engine 1 1 1 1 1 1 1 1 100% 0.000
Heat S1 S2 S3 S4 S5 S6 S7 S8 Average Performance
rank
Solar (thermal) 0 0 0 0 0 0 0 0 0% 1.000
Gas boiler 1 1 1 1 1 1 1 1 100% 0.000
Divergent disparity attributes of different energy generation technologies present differences in these technologies and they are the foundation for diversification. Disparity attributes of different technologies are compared and their effect is evaluated by two factors – diversity and uncertainty (Formula 1.4). Thus, disparity attributes should be chosen carefully to present important differences in energy generation technologies as energy generation technologies will be diversified on the basis of chosen attributes (Table 5).
Table 5. Description of chosen disparity attributes Disparity
attribute
Weighting
factor Values Explanation
Origin of fuel
[2] 0.1
fossil fuel – 1, renewable sources – 0
This feature helps to diversify energy generation technologies according to primary energy resources so that both cheaper fossil fuel technologies and more expensive renewable energy sources would be included. Both technologies differ in their cost allocation – renewable technologies require high investment costs, while fossil fuel technologies have higher yearly operation costs.
Integration to households [2]
0.2 yes – 1, no – 0
By this feature energy generation technologies are diversified according to their ability to be integrated into households as individual energy generators so that both small capacity energy generation units and centralized ones would be included in final efficient portfolios.
Energy
polygeneration 0.2 yes – 1, no – 0
Technology shares in final energy generation portfolios are diversified so that it would include energy polygeneration technologies (CHP plants) in addition to electricity or heat plants.
Technology
development 0.3
under intensive development – 1, well-established – 0
It is important that both well-established and emerging technologies would be included in efficient portfolios. Well- established technologies usually guarantee better economic performance and secure energy generation. However, emerging technologies should be also included in final portfolios because only in this way technological development and rapid cost decrease can be achieved.
Availability of
supply 0.1
flexible – 1, less
controllable – 0
This criterion describes the ability of energy generation
technologies to follow changes in energy demand and the ability to supply the needed amount of energy at a certain moment.
Regarding matters of supply side and demand side management, heat and electricity storage etc., final energy generation
portfolios should include technologies of flexible and reliable generation in addition to less easily controllable ones.
Greenhouse gas emission regulation
0.1 yes – 1, no – 0.
This feature helps to diversify energy technologies in the final portfolio according to the importance of greenhouse gas emission regulation for their performance.
Set of disparity attributes that are used in calculation is presented in the Table 6.
Table 6. Disparity attributes for different technologies Disparity attributes Origin
of fuel
Integration to households
Energy polygeneration
Technology development
Availability of supply
Emission regulation
Weightings of disparity attributes 0.1 0.2 0.2 0.3 0.1 0.1
Electricity
Wind 0 0.5 0 0.5 0 0
Small Hydro 0 0 0 0 0 0
0 1 0 1 0 0
Fuel Cell (NG) 1 1 0 1 1 1
CHP
Biogas/engine 0 0 1 0.5 1 0
Gas/engine 1 1 1 0 1 1
Heat
Solar (thermal) 0 1 0 0.5 0 0
Gas boiler 1 1 0 0 1 1
Initial data presented in this chapter are used for MDA analysis on the basis of present application of MVP theory to multi-energy systems.
3.3.2 Results of Applying Multi-Criteria Diversity Analysis
For the previously described system of energy generation technologies with multi-energy outputs (chapter 2.2) MDA is carried out. Portfolios with minimum uncertainty (maximum diversity) at different levels of multi-criteria performance are presented in Figure 4.
Figure 4. Portfolios with minimum uncertainty at different levels of performance
It must be noticed that for some values of uncertainty there are two portfolios with different multi-criteria performance: portfolios with relatively low or relatively high performance can result in the same level of uncertainty. This implies that portfolios with low performance could be replaced by more efficient portfolios that guarantee higher multi-criteria performance with the same level of uncertainty, so these ineffective portfolios should not belong to MDA efficient frontier. Therefore, MDA efficient frontier is estimated on the basis of Figure 4, eliminating portfolios with lower level of performance at the same level of uncertainty. MDA efficient frontier and corresponding shares of technologies are presented in Figure 5 and Figure 6.
Figure 5. MDA efficient frontier
Figure 6. Shares of technologies along MDA efficient frontier
While analyzing MDA efficient frontier, it must be firstly emphasized that increase in multi- criteria performance on the efficient frontier results in increase of uncertainty (decrease of diversity). But for efficient portfolios with high multi-criteria performance a small increase in performance results in relatively high increase in uncertainty. And for portfolios with value of performance from app. 0.45 to 0.70, the higher performance can be achieved at the expense of slight increase in uncertainty.
However, even MDA could help to estimate portfolios that guarantee lowest level of uncertainty for chosen levels of multi-criteria performance, it does not fully present the investor’s viewpoint.
Economic performance of chosen portfolio and its risk should match the expectations of investor so that it would be willing to invest. For illustrative purpose MDA characteristics of the points of MDA efficient frontier and MVP efficient frontier are presented in Figure 7.
Figure 7. Comparison of MDA and MVP efficient portfolios
In Figure 7 it could be seen that MDA characteristics of some MVP efficient portfolios are very close to MDA efficient frontier. Even the MVP efficient portfolios guarantee relatively high multi-criteria performance too, they result in higher uncertainty (lower diversity) though. Thus, there is a potential to lower the uncertainty at the same level of multi-criteria performance. When a different portfolio with lower uncertainty and the same multi-criteria performance is built, then the economic performance of portfolio decreases, but it is compensated by increase in either environmental performance or security.
These observations lead to a conclusion that both MDA and MVP theory include important aspects to assess portfolios of energy generation technologies. Each method can offer what the other lacks. Therefore, their combination could be developed into a powerful tool for forming efficient energy generation portfolios as suggested in [1].
4 Combining Mean-Variance Portfolio Theory and Multi- Criteria Diversity Analysis
MVP theory helps to determine efficient portfolios with maximum economic performance at chosen levels of risk. Results provided by MVP are very important to energy companies as investors because it presents a rate of investment return and its risk. MDA leads to the formation of portfolios with the best performance in several viewpoints that covers economic, environmental efficiency and other chosen factors. Thus, MDA represents a broader society- oriented view to energy systems. Economic solutions provided by MVP theory are limited because they do not cover external effects of energy generation portfolios and do not assess their uncertainty (in MVP theory boundaries are set for possible outcomes [1]). MDA might provide a decision-making process with solutions including optimal performance in several viewpoints and its incertitude. However, used in isolation MDA partly hides the very important effects of economic performance and its risk.
Therefore, a combination of both methods, a so-called full spectrum risk analysis, is suggested in [1]. Firstly, it combines pure economic performance of MVP theory and broader performance in several viewpoints offered by MDA. Then, it leads to an overall evaluation of incertitude because analysis of both historical variations in MVP theory and incertitude in MDA is conducted.
In this work integration of MVP theory and MDA is carried out relying on the available data of MVP theory application to generate efficient multi-energy generation portfolios as presented in [5]. The combination is carried out in two different ways:
combination based on the efficient MVP frontier, using a method presented in [1],
combination based on maximization of total performance at chosen levels of total incertitude (multi-objective optimization). Here this method is called comprehensive performance and incertitude analysis (CPIA).
Description and limitations of these two methods and suggestions are presented in corresponding chapters.
4.1 Combination based on the Mean - Variance Portfolio Efficient Frontier
Firstly, the combination of both methods is carried out according to the concept presented in [1].
Fundamentals of the methodology and its application for multi-energy systems are presented in the following chapters.
4.1.1 Mathematical Fundamentals
The combination of MDA and MVP theory in [1] is carried out by defining two new factors that combine performance and risk/uncertainty calculated by both methods. These factors are estimated in a flexible manner by allowing the user to choose the weighting factors for importance of MDA and MVP theory. The total performance and total incertitude of a portfolio are calculated as follows:
1
MVP MDA
port port port
P P P (1.6)
1
MVP MDA
port port port
U U (1.7)
Where Pport- total performance (value from 0 to 1);
Uport- total incertitude (value from 0 to 1);
MVP
Pport and PportMDA - normalized MVP and MDA performance of the portfolio,
MDA
Uport - normalized MDA uncertainty of the portfolio,
MVP
port - normalized risk of the portfolio calculated by MVP theory,
- the weighting factor. When the value of is 0, then the stress is put only on MDA and vice versa.
Efficient frontiers can be found by minimizing Uport at chosen levels of performance Pport and predefined weighting factor . Actually, this combination is expected to provide an optimal portfolio from the viewpoint of both MVP and MDA. However, MDA uncertainty of a portfolio is not linear or even quadratic function of shares of technologies. Thus, the optimization process is much more complicated. In [1] this integration is planned to be made as a multi-objective optimization task.
Currently, in [1] the combination is conducted by firstly calculating the efficient frontier using MVP theory and then evaluating these optimal portfolios by MDA. This concept is similar to pre-emptive ordered goal programming when the solution satisfies one objective and then the
other optimal solution is looked for [1]. In this project MDA is firstly carried out for the efficient MVP portfolios (along MVP efficient frontier). In this way, results of MDA are important in decision making process because portfolios of MVP efficient frontier can be additionally examined according to their multi-criteria performance and level of diversification.
4.1.2 Example of Combination on the Basis of the Mean-Variance Portfolio Efficient Frontier
Results of modified and extended application of MVP theory for multi-energy systems (chapter 2.2.3) are taken for further analysis. At first, MDA performance and diversity are calculated for one hundred portfolios of MVP efficient frontier and presented in Figure 8.
.
Figure 8. MDA performance and diversity for portfolios of MVP efficient frontier
As MDA calculations are performed for MVP efficient portfolios (from Figure 2), this causes an unusual shape of a curve in Figure 8. Moreover, some portfolios of MVP efficient frontier might have the same level of multi-criteria performance, but different levels of diversity, or the same level of diversity, but different levels of multi-criteria performance. Both MVP and MDA characteristics along MVP efficient frontier are presented in Figure 9 and Figure 10.
Figure 9. MDA and MVP characteristics along MVP efficient frontier, when risk is plotted on the x-axis
Figure 10. MDA and MVP characteristics along efficient MVP frontier, when number of a certain portfolio is plotted on the x-axis
While analyzing Figure 9, it should be firstly noticed that along the MVP efficient frontier performance and diversity of MDA highly vary. Higher diversity does not guarantee higher MDA performance – even controversially, the highest MDA performance is achieved for the portfolio with comparatively low diversity. Maximum diversity in most cases is achieved for portfolios including more technologies. However, when differences in disparity attributes of some technologies are relatively low, then even some technologies in the portfolio could lead to its low diversity. E.g. in analyzed case technologies of small hydro plants and solar (thermal) units have relatively low differences in their disparity attributes (Table 6). Thus, in Figure 9 for the portfolios from no. 84 to no.100 higher share of small hydro plants and solar (thermal) units together leads to lower diversity than for portfolios including other, more divergent technologies.
Although Figure 9 is expected to be an important illustration for decision making process, it should be noticed that when more different technologies and more different criteria are included into calculation, it is getting very complicated to draw a conclusion about efficient MDA and MVP portfolios.
Thus, two integrated criteria are introduced in [1] to evaluate both MVP and MDA performance and risk/uncertainty. These two criteria – total performance and total incertitude (chapter 4.1.1) – are calculated for every of one hundred efficient MVP portfolios for five values of. Then, for every value of the portfolio with the lowest total incertitude is found. When is equal to 0, then all emphasis is put on MDA. When is equal to 1, then MVP performance and risk reflects the total performance and uncertainty. These MVP and MDA efficient portfolios and their characteristics are presented in Figure 11. and Figure 12.
Figure 11. Portfolios of MVP efficient frontier with lowest total incertitude for different values of Phi
Figure 12. Total characteristics of MVP efficient portfolios with lowest total incertitude for different values of Phi (regarding Figure 10).
It can be seen in Figure 11. that when the value of is equal to 0 (when MDA results are emphasized), the portfolio of energy generation technologies is most diversified1. The higher stress is put on MDA, the more diversified efficient portfolio is.
Figure 13 illustrates the change in total performance and total incertitude along the MVP efficient frontier when a value of =0.5 is chosen.
1 From the Figure 3 it might look like that there are no such diversified portfolios on the MVP efficient frontier as in Figure 11. But in Figure 9, when a number of each portfolio is presented on the x-axis, these more diversified portfolios could be observed.
Figure 13. Total incertitude and total performance along MVP efficient frontier, when Phi=0.5
In Figure 13 portfolio no. 10 has the lowest total incertitude and, therefore, it is presented in Figure 11. as efficient portfolio, when value of is equal to 0.5.
It should be noticed that when some emphasis is put on MDA (when is equal to 0, 0.25, 0.5 or 0.75 in Figure 11.), the portfolios with the lowest total incertitude do not differ much from each other. This observation makes the decision-making process easier, but it is not necessarily an often case.
However, all portfolios with minimum total incertitude presented in Figure 11. are portfolios lying on the MVP efficient frontier. This methodology of integration is a task of pre-emptive ordered goal programming as only efficient MVP portfolios are analyzed to pick portfolios with best MDA characteristics. Though it should not be neglected that there might be other portfolios (not on the MVP efficient portfolio) that might guarantee higher total performance with lower total incertitude. Thus, integration of MVP theory and MDA could lead to deeper analysis if it would be conducted as a multi-objective optimization task.
4.2 Comprehensive Performance and Incertitude Analysis
The portfolio analysis combining MVP theory and MDA as a multi-objective optimization task here named as comprehensive performance and incertitude analysis (CPIA) of Multi-Energy Portfolios. The concept of this method is based on suggestions in [1].
4.2.1 Mathematical Fundamentals
Integration of MVP theory and MDA is carried out here as a task of multi-objective optimization, using a weighted sum method. Total performance and uncertainty of combined MVP and MDA are expressed using weighting factor as shown in Formulas 1.6 and 1.7. The total incertitude then is minimized for different levels of total performance and in this way efficient frontier of combined MVP and MDA is formed. The steps of whole calculation procedure are shortly described in Table 7. Development of formulas to use in the MatLab calculation tool is presented in Appendix 8.2.
Table 7. Steps of conducting CPIA
Description Mathematical basis
I. Calculation of minimum and maximum MVP performance
As MVP and MDA performances have different measuring systems, they must be normalized for calculating the total performance.
1
1
1
1
min min
1
,..., 0
MVP n
i i i
n
i i
el
n i
i tot i i n
P R x
x
x b
x x
II. Calculation of minimum and
maximum MDA performance
1 ,
1
1
1
min min
1
,..., 0
MDA n
i i i i j
n
i i
el
n i
i tot i i n
P x w r
x
x b
x x
III. Calculation of minimum and
maximum MVP risk
1 1
1
1
1
min min cov
1
,..., 0
n n
p i j i j ij
n
i i
el
n i
i tot i i n
x x x
x b
x x
IV. Calculation of minimum and maximum MDA uncertainty It must be noticed that disparity attributes ai n, (Table 6) should not be equal for two technologies in order to prevent division from zero.
1 1
1
1
1
min min 1
1
,..., 0
MDA
n n
i k i j ik
n
i i
el
n i
i tot i i n
U x x d
x
x b
x x
V. Calculation of minimum and maximum total performance in order to define different levels of total performance
When minimum and maximum total performance is known, then different levels PdefinedTOT of total performance are calculated.
1 min
max min
1 , min
max min
1
1
1
min min
1 1
,..., 0
n MVP
i i i
TOT
MVP MVP
n MDA
i i i i j
MDA MDA
n
i i
el
n i
i tot i i n
R x P
P P P
x w r P
P P
x
x b
x x
