Fuzzy Logic Modeling of Energy Systems

Fuzzy Logic Modeling of Energy Systems

Fuzzy Logic Modeling of Energy Systems BJØRN LUDWIG Technowledgement Consulting Go¨ttingen, Germany Technical University of Clausthal Clausthal, Germa...

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Fuzzy Logic Modeling of Energy Systems BJØRN LUDWIG Technowledgement Consulting Go¨ttingen, Germany Technical University of Clausthal Clausthal, Germany

1. 2. 3. 4. 5.

Introduction Methodical Aspects Modeling Approach with Soft Computing Application to the Energy Supply Results and Discussion

Glossary fitness function A mathematical expression transforming a certain combination of several input values into one output value, representing the evaluation of the input, often in relation to the optimum. fuzzy logic A soft computing method to carry out logic operations; uses verbal expressions described by fuzzy sets generated from membership functions. genetic algorithms A soft computing method that uses biological processes of evolution for optimization. linguistic variable A variable for which the values are linguistic terms (for instance, ‘‘small’’) instead of numbers. Fuzzy sets generated by membership functions build the connection between the linguistic expression and the numerical information. membership function A mathematical expression that represents to what degree an element belongs to a mathematical set, considering that these memberships do not have to be total memberships. operator A mathematical manipulation on numbers or terms. Classical examples are addition, multiplication, differentiation, etc. In soft computing, other operators are needed, e.g., fuzzy or genetic operators such as aggregation, accumulation, crossover, mutation, and reproduction. soft computing Mathematical methods of calculating, developed mainly within the past 30 years, and not necessarily operating on classical functions and numbers, but taking into account the human kind of information processing and principles of biological operating.

Encyclopedia of Energy, Volume 2. r 2004 Elsevier Inc. All rights reserved.

sustainable development A goal for the future development of mankind under intragenerational and intergenerational justice. The negotiated road map for future economical, ecological, and societal sound development is the Agenda 21, from the United Nations Conference for Environment and Development held in Rio de Janeiro in 1992. technology assessment An interdisciplinary scientific and societal approach to the questions of what impacts technology application and the technological progress have on mankind, and what technologies are needed and preferred.

The energy supply system is crucial to the further development of humans. Evaluations concerning the impact of energy conversion technologies on this development are difficult due to the complexity of the problem. This leads to different conclusions about how to proceed. Using soft computing techniques, both technically measurable quantities and nontechnical evaluation criteria can be taken into account to obtain an objective evaluation, to discover hidden evaluation criteria, and to formulate a goal for a future sustainable development.

1. INTRODUCTION Current global development increasingly calls attention to the realities of population growth, extended mobility, and increasing energy demands and their contribution to environmental problems. This collision between civilization and the ecosystem illuminates two dilemmas: (1) The goals of maintaining society’s prosperity and protecting the environment are not independent. Rather, they are competing goals, because technological applications increasingly

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Fuzzy Logic Modeling of Energy Systems

require extensive use of nonrenewable resources. (2) The globally increasing demand for nonrenewable resources necessitates even more technology application and development. Thus, guided by the vision of sustainability, the concept of technology assessment (TA) must deal with the question of technology’s ability to enable sustainable development. To avoid increasing both resource use and risk, new technologies should be appropriate to address developmental needs, yet ecologically ‘‘reasonable.’’ Designing such technologies requires improved information on and knowledge of technical systems. This knowledge encompasses development and application of technical systems and the connections among economic, social, and political systems, as well as impacts on the environment. Such knowledge should be used when evaluating both existing and new technologies within decision support systems. Problems characteristically encountered in making such evaluations involve multicriteria decision making, aggregation of facts, weighting of goals, uncertain information, and integrating people into the decision-making process. The overall problem is not a lack of data, but the difficulty in generating knowledge from a vast amount of partially uncertain, partially fuzzy data and information, often structured in such a way as to be rather useless. The global energy supply system, for example, is structured to consist of different primary energy carriers. This leads inherently to different evaluations of the system (for instance, due to the different ranges or durations of the primary energy carriers, or to the differences in the dangers of operating different conversion technologies), thus the present structure cannot be the most desirable one. An essential question for the future, concerning technology assessment studies, is how to achieve a sustainable energy supply. Under the present energy supply structure, evaluations for each applied energy conversion technology are needed. Approaches to this entire problem are possible using a hybrid method of the soft computing techniques, fuzzy logic and genetic algorithms, first introduced by the author. This approach of recursive optimization considers different technical and nontechnical evaluation criteria to achieve optimum primary energy percentages that form the best energy mix. The aim is to reduce the multidimensional decision-making problem to a comparison of aggregated one-dimensional numbers. This should be achieved by combining the advantages of human thought-process structures and processes of biological development with the logical and analytical accuracy of computers.

2. METHODICAL ASPECTS Examining such problems analytically requires cognitive information processing. The desired systematic procedures require application of methods to organize and to extend our knowledge on complex systems. Generally, problem-specific combinations of methods are applied. Methodical problems and deficits usually emerge due to the complexity of the processes or of the impacts. Problems in describing complex dynamical systems are due to nonlinear impacts and boundary conditions, to boundary conditions that are time dependent and/or solution dependent, and to different time constants within the subsystems.

2.1 Deficit Analysis Knowledge about technical systems is commonly gained by analyzing the underlying interactions. This is mainly characterized by the building of models. These are mainly mathematical models that allow systematic simulations to identify potential reactions of the system. In the case of a holistic and systemic view, the system border is extended to all participating and impacting areas. Thus, interactions and impacts have to be considered, but often resist analysis using a description with classical mathematics––say, a formalistic handling. These entities have one of the following properties: unique basic variables are not identifiable, unique causalities are not recognizable among them, or measurable indicators cannot be determined in case of known interactions. This plays a considerable role in the discussion of sustainability, because the entities could be impacts such as risk, quality of life, or diversity. These are hardly measurable quantities. Mainly they are considered in models by boundary conditions or verbal discussion. Neglecting these entities or disaggregating them to other available variables leads, in any case, to an information loss, so the complexity of the system is not represented completely by the model.

2.2 Deficit Synthesis Complex systems in practice are handled, or at least are made more accessible, by composite numbers or indicators. This is familiar, for instance, in process or chemical engineering. In contrast to the information content of single-measurement data, composite indicators generate new knowledge by aggregating single data and information. The procedure of

Fuzzy Logic Modeling of Energy Systems

aggregation varies depending on the indicators. Nevertheless, indicators of different aggregation levels always provide the same information content. Indicators are especially important in environmental discussions. Assessing or evaluating a situation or guiding a decision requires differently aggregated indicators that characterize a process or a state. The aggregation problem points out the following dilemma: the methodical deficits of assessments are due to problems of comparison because of the transformation of indicators to highly aggregated assessment measures. On the other hand, this procedure is necessary, which becomes obvious, for instance, if entities such as environmental pollution are to be quantified. This property allows less aggregated indicators to be used at the beginning of systems analysis (for instance, within an inventory analysis of modern environmental management approaches) and enhances transparent aggregation procedures.

2.3 Deficit Assessment A multidimensional assessment problem can be considered as a logical measurement operation. The assessment problem then is the optimal arrangement of a set of possible alternatives, with respect to the relevant aims and to the preferences of the decision makers, considering given restrictions. The aim of an assessment is to determine a particular value for any alternative so that its relative advantages are represented in only a single expression. The solution of this selection problem will be especially difficult if the problem structure is complex (that is, when many objectives are to be considered, different assessment scales emerge, objectives are weighted differently, information is at least partially uncertain, the problem is time-dependent, many participants are to be involved in the decision-making process, or no unique criterion exists for decision making). Contributing variables could be factors such as risk or diversity, which represent heterogeneous fuzzy expressions that cannot be broken down any further, often with many different impacts. This makes comparability of these variables difficult, especially when factors of different aggregation degree are involved, and makes any evaluation subjective. Nevertheless, to describe real systems, these influences are indispensable quantities.

2.4 Goal Designing innovative technologies, considering widespread aspects such as sustainable development,

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requires environmental information for design, assessment, and decision processes. The difficulties mainly emerge during recording and assessing of entities. The problem is caused by aggregated system variables that are incomparable. More adequate methodical approaches maintain the complexity of these variables. Necessary subjective inputs during a system analysis then have to be made absolutely clear to maintain transparency, and they have to be introduced as early as possible in an algorithmic procedure to avoid a conscious manipulation of the solution.

3. MODELING APPROACH WITH SOFT COMPUTING At a level of thinking that does not have to be conscious, humans use a kind of knowledge that is not sharp and is hard to formulate explicitly. This fuzziness is not a property of reality, but of perception. This kind of knowledge is characterized by the handling of incomplete and not well-structured patterns and information. Here the question emerges as to how and with what language we can model such uncertainties. The methodical deficits mentioned previously can be softened by application of the techniques of soft computing. Apart from artificial neural nets, these are primarily fuzzy logic and genetic algorithms; their usefulness is due to their potential to integrate complexity into calculation.

3.1 Fuzzy Logic Within the context of this discussion, only fuzzy set theory, one branch of fuzzy logic, is of interest for application to technology assessment, because the important questions of TA do not deal with maintaining states of operation, the subject of control theory, but with fuzzy information processing. Fuzzy logic enables the handling of verbal and indeterminate information using exact mathematics. Traditional binary logic is part of fuzzy logic, as a special case operating only with two values of interpretation. A set is fuzzy limited if not all the members of the set are assigned a value of 1, or 100%, which means total membership. A fuzzy set is defined by the generalized characteristic function, called the membership function m. This real function can take on any values, but usually it is normalized into the interval [0, 1]. The key notion when modeling with fuzzy logic is the linguistic variable. In contrast to mathematically precise quantifiable equations, verbal descriptions of

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Fuzzy Logic Modeling of Energy Systems

activities contain terms of properties with smooth gradation transitions. Linguistically formulated variables are generally more intelligible, but have a higher aggregated information content. These entities are therefore more difficult to quantify without an information loss. The values of a linguistic variable are verbal expressions, called linguistic terms (for instance, the term ‘‘small’’). The content of each linguistic term is identified with one fuzzy set and is assigned to the related numerical scale of the basic variable by a particular membership function. Thus, the fuzzy sets build the connection between the linguistic expression and the numerical information. To process fuzzy formulated knowledge, several linguistic variables must be linked by linguistic operators. The connecting rules represent the knowledge, which is stored in a rule base or knowledge base, similar to expert systems. The procedure consists of the sequential steps fuzzification, inference, and defuzzification.

3.2 An Assessment Method Using Fuzzy Logic Along with the preceding remarks, a general assessment method using fuzzy logic has been developed. An assessment is a determination of the goodness of a considered issue. This will be made by a function that systematically integrates knowledge into the procedure of modeling and also handles the aggregation procedure. The assessment procedure is based on a multidimensional vector of criteria containing all knowledge input, which characterizes the matter to be evaluated related to the chosen degree of aggregation. The aim is to get statements of the form ‘‘sth is good, fair, satisfying, to introduce, desirable, etc.’’ This kind of statement can be understood as a result of logic operations. The concept of linguistic variables enables the formulation of scales of values with linguistic terms as gradations. The essential and critical process is the formalistic assignment of human knowledge to fuzzy sets. However, through this procedure, existing subjective preferences are integrated transparently at each stage, so a conscious manipulation has only a limited impact on the final result. The general five-step top-down procedure is as follows: 1. Definition of a scale for evaluation. 2. Definition of subordinate criteria for ranking onto this scale. The question concerning which of the possibly independent criteria and impacts will influence the superior scale from step 1 has to be answered.

3. Stepwise disaggregation of these basic criteria. The subordinate impacts are disaggregated to further arbitrary substages, depending on the given knowledge. The result is an aggregation hierarchy of the different impacts. Generally, steps 2 and 3 answer the question of which of the less aggregated and possibly independent impacts are impacting on the next higher aggregation level. Depending on the character of an impact, it can be described by a measurable quantity on a less aggregated level. Otherwise, it keeps the character of an aggregated entity. Both kinds of characters can be considered simultaneously. 4. Fixing the specifications of the fuzzy system parameters. This includes fixing the numbers of linguistic terms and the verbal connotations of all attendant variables, the shape of the membership functions (default: triangular), and the logic operators for aggregation (default: minimum) and for accumulation (default: maximum), as well as the defuzzification method (default: center of gravity). It is recommended to choose an odd number of linguistic gradations for each input variable (default: three) to ensure a certain resolution, with the additional possibility of naming a medium term. The number of linguistic terms of all superior aggregated variables then can be taken as determined in order to get an objective rule base (see step 5). 5. Formulation of the rule base. The phenomenological dependencies between the variables are stored in the rule base. The rules state which combination of linguistic input terms is connected with which linguistic output term. To achieve a definite assignment in the rule base, the number of linguistic terms of a superior aggregated variable then is determined by all possible combinations of input linguistic terms of all related subordinate variables. This is carried out by a numerical codification. The N linguistic terms of a subordinate variable are assigned to numbers in the interval [(N1)/ 2,...,1, 0, 1, ..., (N1)/2] (N ¼ 3, 5, 7, ...), where the medium term is assigned to zero. For instance, a variable with NIn ¼ 3 linguistic terms (small, medium, large) would be assigned to (1, 0, 1). Thus, we can determine the number NOut of linguistic terms of an output variable from its M input variables (M ¼ 1, 2, 3, ...) with their NIn,1, ..., NIn,M linguistic terms as follows [Eq. (1)]: NOut ¼ 1 þ 2

M X NIn;i1 i¼1

2

ð1Þ

For instance, an output variable aggregated from M ¼ 2 input variables with NIn,1 ¼ 3 and NIn,2 ¼ 5

Fuzzy Logic Modeling of Energy Systems

linguistic terms would then have NOut ¼ 1 þ 2(1 þ 2) ¼ 7 linguistic terms. The rules can be derived from the following procedure. The conditions of all rules that are the permutations of all linguistic input terms can be written numerically codified. The conclusion is then represented by the linguistic term of the output variable (numerical codification is assigned by the sum of the numerical input codifications). For instance, considering one three-term input variable with the terms (small, medium, large) codified as (1, 0, 1), and a second input variable with five terms (very small, small, medium, large, very large) codified as (2, 1, 0, 1, 2), the particular rule may be written as follows: IF Input1 ¼ small AND Input2 ¼ medium THEN Output ¼ ?

ð2Þ

which is codified as IF Input1 ¼ 1 AND Input2 ¼ 0 THEN Output ¼ 1 þ 0 ¼ 1

ð3Þ

According to Eq. (1), the output variable has seven terms codified as (3, 2, 1, 0, 1, 2, 3). Therefore, the conclusion Output ¼ 1 means that the result is the linguistic output term that is located one term smaller than the medium term, whatever the verbal connotation may be.

3.3 Genetic Algorithms The idea of technology assessment to point out alternatives to any problem solution can be taken as an optimization problem, where optimization of an application is aimed at, for instance, its energy efficiency, its material intensity, or its sustainability. The application of usual optimization procedures is limited. Restraints can be nonlinear problems or problems with many parameters or many restrictions due to the programming effort and memory demand. Genetic algorithms, instead, refer to a behavior that would be similar to a biological process of evolution. These techniques do not have requirements for linearity, differentiation, or a certain structure, and often require less programming effort. Problem solutions are kept or changed due to the principle of survival of the fittest. The point is that the search for more capable solutions takes place with a high probability at sites of the solution space, where an enhancement higher than average is required. These features are used in genetic algorithms to evaluate systematically created solutions. A genetic algorithm can be defined as an iterative

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procedure maintaining a population of structures that are candidate solutions to specific domain challenges. Each candidate solution is called a chromosome. As in biological systems, a chromosome can make a copy of itself. The copy might be slightly different from its parent. During each temporal increment, called a generation, the structures in the current population are rated for their effectiveness as domain solutions, and on the basis of these evaluations, a new population of candidate solutions is formed using specific ‘‘genetic operators,’’ which are used by most genetic algorithms, such as reproduction, crossover, and mutation: 1. Reproduction: Through reproduction, genetic algorithms produce new generations of improved solutions by selecting parents with higher fitness ratings or by giving such parents greater probability to be contributors. 2. Crossover: Usually genetic algorithms use binary strings to represent solutions. Crossover means choosing a random position on the binary string (that is, after a certain digit) and exchanging the segments either to the right or to the left of this point with another string partitioned similarly. 3. Mutation: This genetic operator is an arbitrary change of a single bit of a chromosome. It is needed to keep the algorithm from getting stuck. This operator changes a 1 to 0 or a 0 to 1. The natural optimization techniques can be distinguished by their main operators. In case of genetic algorithms, the main operator is crossover, which is chosen with a probability of 80–100%. Mutation, in contrast, is a very subordinate operator. Such a bit change occurs at a low probability of 1– 10%. Genetic algorithms use a three-step iterative process: (1) test a solution to see how good it is, (2) select the best parents, and (3) generate offspring. Genetic algorithms provide a set of efficient, domainindependent search heuristics. Information that enables rejection of inferior solutions and accumulation of good ones is received. Only genetic algorithms and their derivatives provide methods powerful enough for optimizing complex problems. Traditional optimizing tools are not applicable.

4. APPLICATION TO THE ENERGY SUPPLY Decreasing resources of fossil primary energy carriers and globally increasing mean per capita energy demand and population growth bring to view the

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question how and by which technologies the future energy supply can be guaranteed. Of primary concern here is the evaluation of the sustainability of the energy supply system. An energy supply system provides energy via different conversion technologies. In the present global energy supply system, 90% of the demand is still provided by fossil fuel-based primary energy carriers. Increasing energy demands and increasingly urgent global impacts require development of a new supply system, keeping in mind future generations and threshold countries with vastly growing economies; this is referred to as intergenerational and intragenerational justice. Due to the requirement for sustainability of a new energy supply system, one of the most important questions to be answered concerns which primary energy carriers can be and should be used, and to what extent in consideration of limited resources and potential risks. This question concerns the optimal structure of percentages of the different primary energy carriers, of the different energy conversion technologies, or the optimal energy mix. The applied recursive optimization is a two-step iterative procedure. First, a given energy supply structure must be evaluated, considering the technologies involved, with regard to certain evaluation criteria. Second, the input structure will be varied to find the best rated energy mix.

4.1 Assessment: The Fuzzy Fitness Function To evaluate a given primary energy structure, the fuzzy logic assessment method described previously is used. The assessment procedure is based on a multidimensional vector of criteria that characterizes a conversion technology. The method is applied as follows, according to the previous description: 1. Definition of a scale for evaluation. The interesting entity is the sustainability of an energy conversion technology, here reduced to the related primary energy carrier. 2. Definition of subordinate criteria for the ranking onto the scale sustainability. In case of energy conversion technologies, the independent criteria duration of the primary energy carrier and ecological relevance of the technology are chosen. 3. Stepwise disaggregation of these subordinate criteria. In this case (Fig. 1), the ecological relevance could be considered as being made up from the basic criteria climate impact, area demand, and risk. Area demand is composed of the value and the size of the

Value (3)

Size (3)

Small, medium, large Small, medium, large

Risk (3) Small, medium, large

Area demand (5)

Ecological relevance (9)

Climate impact (3) Small, medium, large

Duration (3) Small, medium, large

Sustainability (11)

FIGURE 1 Aggregation hierarchy of the assessment function. The number of linguistic terms of each variable determined with Eq. (1) is in parentheses. Reproduced with permission from the Society of Automotive Engineers, SAE 1999-01-2708. Copyright 1999, SAE.

used area. Duration, for instance, may depend on the level of technology, on the reserve of the primary energy carrier considered, and on social factors. The climate impact can be specified with technical quantities such as emissions and global warming potential. However, both are not differentiated further here. Risk in this approach should specify the potential danger in case of an accident. 4. Fixing the specifications of the fuzzy system parameters. Here, for all specifications, the default standard parameters are chosen. Each input variable has been aligned to three linguistic terms (small, medium, large). According to Eq. (1) and the hierarchical aggregation structure (Fig. 1), the numbers of linguistic terms for the superior variables given in Fig. 1 are obtained. All fuzzy sets at the edges of the intervals of the basic variables are performed in such a way that defuzzification of the minimum (maximum) linguistic term would provide the value of 0 (1) of the basic variable. Therefore, the membership functions of these linguistic terms of the input basic variables have to be defined symmetrical to 0 (1). 5. Formulation of the rule base. With the preceding procedure, the objective rule bases for the three aggregation levels (for area demand with 9 rules, for ecological demand with 45 rules, and for sustainability with 27 rules) are obtained. For example, the aggregation levels would have the following formulations: First aggregation level: IF Value¼ medium AND Size ¼ small THEN Area Demand ¼ small

ð4Þ

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Fuzzy Logic Modeling of Energy Systems

Second aggregation level: IF Risk ¼ small AND

Input vector

Area Demand ¼ small AND Climate Impact ¼ small THEN Ecological Relevance ¼ very very small

ð5Þ

α

Genetic algorithm optimization

Fuzzy fitness function

β γ

Varied input vector

FIGURE 2 Recursive optimization, where the contributions of primary energy carriers are a, fossil fuels; b, regenerative (renewable) energy; and g, nuclear energy. Reproduced with permission from the Society of Automotive Engineers, SAE 1999-01-2708. Copyright 1999, SAE.

Third aggregation level: IF Ecological Relevance ¼ very very small AND Duration ¼ large THEN Sustainability ¼ very very very large: ð6Þ

5. RESULTS AND DISCUSSION Each primary energy carrier being considered can now be characterized by the vector of criteria (elements are the values of the inputs, each on a specific aggregation stage). The assessment result can be a verbal or a numerical value of the highest aggregated output variable, ‘‘sustainability.’’ In case of an energy mix, different technologies are considered with their relative percentages. An input value for the mix then has to be built by the sum of the values of the single technologies weighted with the percentage within the mix. This assessment procedure operates as the fitness function of the genetic algorithm.

4.2 The Optimization Procedure

TABLE I

The optimization problem consists of determining the percentages of each considered technology that participates in supplying the energy demand. During the optimization procedure, the following fitness function F has to be optimized: F ¼ Fða; b; gÞ - max

Vectors of criteria for different technologies and sustainability results are shown in Table I. The current energy mix has been determined twice, first considering the nuclear percentage based on uranium-235 (mix 1) and second based on the breeder technology (mix 2). A separate control calculation with the analytical hierarchy process method has resulted in the same ranking tendency. Although fossil energy carriers use oil and gas in addition to coal, oil and gas have not been differentiated here, which should be done for more precise results. Also, in reality, hydro energy is the most significant

ð7Þ

where a represents the contribution of fossil-fuelbased primary energy carriers, b represents that one of the regenerative (renewable)-energy-based primary energy carriers, and g represents the contribution of nuclear-energy-based primary energy carriers. The boundary condition, of course, is a þ b þ g ¼ 1, or 100%. First, the aggregated inputs to the fuzzy block are determined (Fig. 2). Therefore, each single, specific evaluation of a technology taken from Table I is weighted by its percentage and then summarized. This resulting vector of criteria for a considered energy structure serves as input vector for the fuzzy fitness function that determines the total goodness of the structure. The genetic algorithm now varies the input vector of percentages of a given energy structure until the maximum fitness is achieved (Fig. 2).

Vectors of Criteria for Different Primary Energy Carriers Criteriaa Vector

CI

Value

Size

Risk

D

S

Test casesb Max

0

0

0

0

1

1

Min

1

1

1

1

0

0

Fossil (coal)

1.0

1.0

0.01

0.01

0.3

0.439

Nuclear (uranium-235)

Technologiesc

0.1

1.0

0.01

0.95

0.1

0.400

Nuclear (breeder) 0.1

1.0

0.01

1.0

0.8

0.510

Regenerative (solar)

0.01

1.0

0.01

1.0

0.780

0.1

Mix 1995d (1) (2) a

0.906 0.968 0.042 0.079 0.308 0.444 0.082 0.359

Data from SAE 1999-01-2708. CI, climate impact; D, duration; S, sustainability. b The two test cases verified the algorithm. c The single values for the technologies were determined from the literature. d Evaluation of the 1995 energy structure in Germany: (1), standard nuclear energy; (2), fast breeder technology.

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Fuzzy Logic Modeling of Energy Systems

contributor to regenerative energy supplies, but the percentage in Table I has been taken as consisting entirely of solar energy (photovoltaics). Therefore, the values in Table I for climate impact, value, and size of area are not calculated properly. Nevertheless, the results allow the following statements: 1. The assessment procedure provides correct results for given vectors of criteria. 2. Maximum sustainability cannot be achieved. 3. The traditional nuclear energy conversion technology has been ranked worst because of the high risk and the simultaneous short duration. 4. The current energy mix is debatable, considering this evaluation. 5. The best sustainability is achieved for regenerative energies, and is strongly dependent on the value of the size of area used. Not surprisingly, the optimization procedure provides 100% regenerative energy carriers as the best solution if risk is considered, and 100% nuclear energy if not. Far more interesting are results evaluated as being a little worse than the pure regenerative solution, but better than today’s mix. In this case, the dominant regenerative part decreases with decreasing evaluation values, which also start to oscillate. Simultaneously, the percentages of fossil-fuel-based and nuclear-based primary energy carriers increase alternately, also with increasing oscillations. However, there are obviously several solutions with the same evaluation result. Using a linguistic notion of arrangement, fuzzy logic allows simultaneous consideration of entities and technically measurable quantities by a verbal description of interactions. The evaluation results for the current energy mix and for certain other possibilities with respect to meeting the worldwide future energy demand show that this kind of consideration is useful

to discover hidden evaluation criteria and to formulate a goal for a future sustainable development.

SEE ALSO THE FOLLOWING ARTICLES Computer Modeling of Renewable Power Systems  Modeling Energy Supply and Demand: A Comparison of Approaches  Neural Network Modeling of Energy Systems  Population Growth and Energy

Further Reading Gore, A. (1993). ‘‘Earth in the Balance.’’ Plume, New York. Grefenstette, J. (1982). Optimization of control parameters for genetic algorithms. IEEE Transact. Syst. Mgmt. Cybernet. 16(1). International Standards Organization (ISO). (1997). ‘‘Environmental Management––Life Cycle Assessment.’’ ISO 14040. Beuth, Berlin. Ludwig, B. (1995). ‘‘Methoden zur Modellbildung in der Technikbewertung.’’ Papierflieger, Clausthal-Zellerfeld, Germany. Ludwig, B. (1997). On the sustainability of future energy systems. Energy Convers. Mgmt. 38(15–17), 1765–1776. Ludwig, B. (1997). The concept of technology assessment––An entire process to sustainable development. Sus. Dev. 5(3), 111–117. Ludwig, B. (1997). ‘‘Optimization of Energy Systems Under the Aspect of Sustainability.’’ Proceedings, 32nd Intersociety Energy Conversion Engineering Conference (IECEC), Honolulu, July 27–August 1, 1997, pp. 2070–2075. Ludwig, B. (1998). Fuzzy logic applications in technology assessment studies. J. Intelligent Fuzzy Syst. 6, 375–388. Mitchell, G. (1996). Problems and fundamentals of sustainable development indicators. Sus. Dev. 5, 1–11. Saaty, T. L. (1980). ‘‘The Analytic Hierarchy Process.’’ McGrawHill, New York. United Nations. (1997). ‘‘Yearbook of World Energy Statistics.’’ United Nations, New York. Zadeh, L. A. (1965). Fuzzy sets. Inform. Control 8, 338–353. Zimmermann, H.-J. (1991). ‘‘Fuzzy Set Theory and Its Applications.’’ Kluwer Academic Publ., Boston.