Revista Facultad de Ingeniería Universidad de Antioquia ISSN: Universidad de Antioquia Colombia - PDF

Revista Facultad de Ingeniería Universidad de Antioquia ISSN: Universidad de Antioquia Colombia Muela, Edgar; Secue, Janneth Environmental economic dispatch with

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Revista Facultad de Ingeniería Universidad de Antioquia ISSN: Universidad de Antioquia Colombia Muela, Edgar; Secue, Janneth Environmental economic dispatch with fuzzy and possibilistic entities Revista Facultad de Ingeniería Universidad de Antioquia, núm. 59, junio, 2011, pp Universidad de Antioquia Medellín, Colombia Available in: How to cite Complete issue More information about this article Journal's homepage in Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative Rev. Fac. Ing. Univ. Antioquia N. 59 pp Junio, 2011 Environmental economic dispatch with fuzzy and possibilistic entities Despacho económico ambiental con variables difusas y posibilitas Edgar Muela 1*, Janneth Secue 2 1 Grupo Calposalle, Facultad de Ingeniería, Universidad de La Salle, Carrera 2 N , Bloque C séptimo piso, Bogotá, Colombia 2 División Eléctrica, Ingetec S.A. Ingenieros Consultores, Carrera 6 N. 30 A - 30, Bogotá, Colombia (Recibido el 18 de abril de Aceptado el 25 de abril de 2011) Abstract In this paper a fuzzy possibilistic model for Environmental Economic Dispatch is presented, in order to consider adequately some involved uncertain variables. The developed model can be viewed as an integrative focus where uncertainty is not considered like a simple decision making parameter but it is analyzed as a criterion decision. Here, a fuzzy possibilistic model looks for reflecting the imprecision, ambiguity, and vagueness present in the analyzed problem. Fuzzy sets and possibility theory are an alternative to do this, because they allow including this sort of imperfect information into the problem Keywords: Decision-Making, Environmental, Economic Dispatch, Fuzzy sets, Possibility theory Resumen En este documento se presenta un modelo difuso posibilista para el problema del Despacho Económico Medioambiental con el propósito de considerar adecuadamente algunas formas de incertidumbre. El modelo desarrollado puede ser visto como un enfoque integrador, donde la incertidumbre se considera no solo como un simple parámetro en la toma de decisiones, sino fundamentalmente como un criterio de decisión. El modelo propuesto busca presentar adecuadamente la imprecisión, la ambigüedad, y la vaguedad presente en el problema analizado sobre todo la proveniente de la variable ambiental. Los conjuntos difusos y la teoría de posibilidad son una alternativa * Autor de correspondencia: teléfono: , fax: ext , correo electrónico: (E. Muela) 227 Rev. Fac. Ing. Univ. Antioquia N. 59. Junio 2011 para lograr este propósito, porque permiten incluir esta clase de información imperfecta en el problema Palabras clave: Despacho económico, medio ambiente, conjuntos difusos, teoría de posibilidad, toma de decisión Introduction As the principles of free market have been applied to the power system, an increasing emphasis in development and utilization of optimization tools (decision tools) have occurred which on the one hand, should model the power system elements appropriately, and also, consider the planner s needs adequately, in relation to the problem representation and the interpretation of solution. Inside these tools, the economic dispatch (ED) is found, which in the two last decades has received a lot of attention. This tool has particular interest for the electric generation companies and systems operators, which regard it as one of their more important needs for activities of operation and planning. Economic dispatch is the economic optimization process that deteres a combination of generators and levels of electricity output to meet demand at the lowest cost, given the operational constraints of the generation fleet and the transmission system. Such as constraints are represented by a set of equality and inequality restrictions that are satisfied by means of adjustment of the control variables of power system. The equality restrictions are the equations of active and reactive power flow at each bus, and the inequality restrictions are the limits that exist on the control variables, in addition to the operating limits of the dependent variables of the power system. On the other hand, nowadays there is a growing recognition that the current growth of human activity cannot continue without significantly affecting the environmental quality. Then, new instruments are required to handle sustainability issues in different areas of human activity. All over the world, electricity remains to be a fundamental element of national development. Obviously, energy production and consumption is connected to environmental pressure in many aspects. Particularly, environmental damage of power generation can be quite significant. In electricity generation, the emissions, discharges, and other effects of power production affect the health of nearby and sometimes distant populations, as well as the natural environment. Although these impacts have a direct bearing on individuals well-being, the impacts are not usually factored into any of the decisions to generate or consume electric power [1]. Additionally, the increase of electrical demand raises concerns about the environment ability to sustain this development without harm to itself; therefore, it is indispensable to develop adequate decision-making tools in order to face the environmental problems from the perspective of power systems [2, 3]. When the environmental variable is included in economic dispatch, it is necessary to deeply analyze the attached characteristics and uncertainty of this variable, and consequently, to exae the current mechanisms for dealing with environmental issues. As a consequence of previous analysis, the necessity for alternative focuses as well as the use of nontraditional techniques which could strengthen the decisionmaking process will be justified. In this context, structures like: multi-criteria paradigm, fuzzy logic and possibility theory are proposed as tools which provide a greater quantity of information for a better decision-making [4]. The traditional ED basically characterizes for two fundamental aspects: It is problem of great dimension, strongly restricted, nonlinear and nonconvex. The limited and vague knowledge on the performance of power system conform these have evolved. 228 Environmental economic dispatch with fuzzy and possibilistic entities When environmental criterion is included in ED, both features are accentuated. The first limitation is managed through numerical optimization procedures based on the successive linearization, specifically, the first and second derivate of the objective function and its restrictions, that are used as a direction search (steepest descent method), or by methods of linear programg for imprecise models. The advantages of such methods are found in their mathematical bases. The second limitation, related with the incomplete knowledge of the problem, preclude the reliable use of expert systems where structuring a complete, coherent and closed system of rules, is not possible. Therefore it is necessary to make use of adequate mathematical foundations for the development of appropriate tools which allow not only modeling such aspects but besides, interpreting in that context the obtained solutions. This article seeks to contribute in this second limitation to the improvement of ED solution when an environmental criterion is added. Methodology Traditional formulation Mathematically, the traditional ED is shown in (1) as follow: Minimize F( x, u ) subject to hi ( x, u ) = 0 i = 1... n ( equality constrains) g ( x, u ) 0 j = 1... m ( inequality constrains) j (1) Where x is the vector of control variables (the generator active powers, the generator bus voltages, the transformer tap settings, and the reactive power of switchable VAR sources); and, u is the vector of dependent variables (slack bus power, load bus voltages, generator reactive power outputs, and transmission line flows). Additionally, it is necessary to consider a set of non control variables such as active and reactive power demand of load bus. Generally in the ED, the objective function is imization of total cost of the active power as shown in (2). It is assumed that individual costs of each generator are only dependent on the generated active power, and that is represented by second-rate curves. The objective function for the entire system can be written as the sum of quadratic costs of each generator, that way: NG 2 i i * i i * i (2) i= 1 F = a + b Pg + c Pg Where NG is the number of generators included the slack generator. Pg i is the active generation of the ith generator; and a i, b i, c i are the coefficients of the cost curves of ith generator. While the objective function is imized, it is necessary to insure that the total generation should satisfy the demand system plus the losses on transmission lines. The power flow equations are defined in (3) Pk Pk ( V, θ ) ( Pgk Pdk ) 0 = Q = k Qk ( V, θ ) ( Qgk Qdk ) 0 (3) With k=1 NB Where NB is the number of system buses, Pd k and Qd k are active and reactive demand of load bus respectively, and the net injections in the bus k of active and reactive power, Pk and Qk, are defined in (4): NB P ( V, θ ) = V V ( g cos( θ ) + b sin( θ )) k k m km km km km m= 1 NB Q ( V, θ ) = V V ( g sin( θ ) + b cos( θ )) k k m km km km km m= 1 (4) Where V k is the voltage magnitude of bus k, θ k the voltage phase of bus k, g km and b km are the conductance and susceptance of the line between buses k y m, and θ km =θ k -θ m On the other hand, the involved inequality constraints reflect the laws governing the power generation-transmission systems and the operating limitations of the equipment, in order to make sure the system security, these restrictions include: 229 Rev. Fac. Ing. Univ. Antioquia N. 59. Junio 2011 The generation capacity of each generator has some limits. They can be expressed by (5), (6) and (7). Pg Pg Pg (5) i i i Qg Qg Qg (6) i i i Vg Vg Vg (7) i i i With i=1.. NG, where: Where NG is the number of power station on the system; Pg i, Pg i : Lower and upper limit of active power of generator i; Qg i, Qg i : Lower and upper limit of reactive power of generator i; Vg i, Vg i : Lower and upper limit of generator voltage i; Security constraints include the restrictions on magnitude voltages at buses, and transmission line loadings as shown in (8), (9) and (10). V V V (8) k k k θ θ θ (9) k k k Fl km Fl (10) km With k=1 NB, and m=1.nb Fl km : Maximum power flow through line between nodes k and m; Transformer tap settings are restricted by the lower and upper limits, as expressed in (11). With j=1 NT T T T (11) j j j Where NT is the number of transformers in the system. If they exist, it is possible modeling environmental restrictions, e.g., imum limit of total emissions, which can be expressed by (12). NG Ec ( Pg i ) ETc (12) i= 1 Where c represent a set of pollutants, e.g., SO 2, CO 2 General analysis of the optimization problems As it is mentioned in previous paragraphs, in general, a planner confronts optimization problems that are nonlinear and non-convex. Particularly, ED for real system is a large dimension problem, which increases enormously according to the size of analyzed power system. Then, for conventional optimization techniques based on gradient, it proves to be difficult not only to solve them, but at times even; it is very problematical to find feasible solutions. In addition, the traditional solution of optimization problems is based in the supposition that it is known with certainty and precision the variables implicated in the decision model. However, some uncertain parameters, whose definition can come from forecasting models, include some imprecision degree (error), and therefore for the planner or operator the more important issue is not the realization of such forecasting, but the manner how the variability of those parameters can affect the decisions, and how modeling such uncertainty in the decision models. As expressed in [5], most decisions in the real world are carried out in situations where objectives, constraints, possible actions (solutions space) and their consequences are not known accurately. The fuzzy set theory provides a natural structure to model imprecise relationships or concepts such as: big, polluting, economical, satisfactory, suitable, and so on. 230 Environmental economic dispatch with fuzzy and possibilistic entities Fuzzy sets and possibility theory Basic definitions on fuzzy sets are introduced in [6]. Let X be a collection of objects generally denoted by x, then a fuzzy set à in X corresponds to the group of ordered pairs, as expressed in (13). {(, µ % ( )) / } A% = x x x X A (13) In reference [7] the possibility theory on the basis of fuzzy sets theory is developed. Zadeh proposed the idea of representing an incomplete state of knowledge by means of a fuzzy set. If for instance, we only know about quantity X that X is large, it means that the possible values for X are those compatible with the meaning of large in the considered context. Such a label is represented by the membership function of a fuzzy set, i.e., π x = m large, where π x denotes the possibility distribution describing the more or less possible values for X according to what is known. A possibility distribution π x on U is a mapping from U to the unit interval [0,1] attached to the single-valued variable X. The function π x represents a flexible restriction which constrains the possible values of X according to the available information, with the following conventions: π x (u) = 0 means that X = u is definitely impossible, π x (u) = 1 means that absolutely nothing prevents that X=u From a possibility distribution, seen as a repository of knowledge about a variable X, one can build different uncertainty measures to characterize what can be said about any event (i.e. a subset of the domain of X). The most commonly used ones are the possibility and necessity measures, denoted by π and N respectively, that are defined in (14) and (15) as follows [8]: Π K ( A) = sup [ µ ( u), π ( )] A x u (14) u A c Ν K ( A) = 1 Π K ( A ) = inf [ µ A ( u),1 π x ( u)] (15) u A Fuzzy vs. Possibilistic entities Many applications have been developed in the scope of fuzzy sets theory without taking a lot of attention on the semantics represented by these sets. As a result, a very friendly and solid mathematical structure to combine fuzzy sets has been created, but frequently, this is shown without assigning it to any interpretative structure. When doing it that way the risk of depriving, in guidelines about how and what situations should be used, is suffered by users of this technique. Irretrievably, this situation produces the simple development of senseless operations semantically, leaving to the obtained results without an interpretative structure. In this point, it is necessary to make a clarification. There is often confusion between fuzzy and possibilistic optimization. Fuzzy and possibilistic entities have different meanings/semantics. Fuzzy and possibility model different entities and the associated solution methods are different. Fuzzy entities are sets with nonsharp boundaries in which there is a transition between elements that belong and elements that do not belong to the set. In this situation there is no uncertainty [9]. Possibilistic entities are obtained from sets that are classical sets (crisp), but the evidence associated with whether a particular element belongs to the (crisp) set or not, is incomplete or hard to obtain. Then, there is uncertainty [8]. Fuzzy sets can be used to represent constraints in optimization problems, these may relate to two basic semantic: plausibility (uncertainty) and preference (flexibility) [7]. The case of uncertainty is related to uncontrollable or unknown variables, where there is no complete or consistent information about the value that these variables might take within the constraint. In this context, reference to the plausibility that the variable under consideration takes some specific value is done. For this case the restrictions are observed in an unfavourable direction for the objectives, this situation takes place because attention is focused on the occurrence of adverse situations of uncertain variable, and therefore 231 Rev. Fac. Ing. Univ. Antioquia N. 59. Junio 2011 the planner will seek to protect the system from such situations. On the other hand, the flexible case refers to controllable variables, in which it is considered that it is possible to take advantage of the relaxation in the requests of a restriction based in the control on the implicated variable. For this situation, it is emphasized on the preference degrees of the values that it is possible to assign to the variable. Note than converse to the previous semantics, in these circumstances the restrictions are satisfied in the favourable direction to the objectives (relaxation), because the planner s concern is the expansion of the area of feasible solutions by means of a light violation of the requests. At times, this situation even allows getting bigger levels from fulfillment in the attributes of decision (optimization). In short, when a fuzzy restriction represents the imprecise knowledge about a no-controllable variable, the satisfaction degree of restriction describes necessity degrees (Ν) caused by the desire of getting a feasible and robust solution in the bigger quantity of possible states of the implicated variable. The previous situation is in total contrast to the case of a fuzzy restriction which contains a controllable decision variable, where the satisfaction degree of restriction describes possibility degrees (π), since it is always possible to select the best suitable value of variable. In this case, it is sufficient that there is a variable value that satisfies the restriction on the best possible way; hence the feasibility in this situation is only expressed in terms of possibility [8]. Results and discussion Formulation of Fuzzy-Possibilistic Economic DispatchIn the following, a model for ED with fuzzy and possibilistic variables, is presented. For the objective F a triangular membership function has been selected as defined in (16). µ F ( x) = 1 if F( x) F F( x) F µ x = 1 if F F x F F ( ) ( ) F F µ ( x) = 0 if F( x) F F (16) Where, F and F represent the imum and imum admissible values for the objective function, such it is presented in the figure 1. Figure 1 Representation of fuzzy objective In what follows, it will be presented a flexible form for restrictions associated with controllable variables such as limits of imum and imal voltage in the system buses, as well as for the imum power flow for transmission lines. It is considered that these restrictions are susceptible of some loosening, since in power system real operation, at times the operators recur to relaxation manual of such restrictions in order to achieve the convergence in the power flow. Of course, such relaxation and its representation by means of membership functions of fuzzy sets, it will be based fundamentally in the knowledge of the operator on the system behavior. The equations for imum and imum case are and, respectively; and the graphic representation is shown in the figure 2. µ ( x) = 1 if G1( x) G1 G1 ( G1 + dg1 ) G1( x) µ G1( x ) = dg1 if G1 G1( x) G1 + dg1 G1 µ ( x) = 0 if G1( x) G1 + dg1 (17) Where G1 can represent for example: V k, Fl km, θ k, Ec among others; and dg1 denote the imum admissible variation of function G1. 232 Environmental economic dispatch with fuzzy and possibilistic entities µ ( x) = 1 if G2( x) G2 G2 G2( x) ( G2 dg2 ) µ G2( x ) = dg2 if G2 G2( x) G2 dg2 G2 µ ( x) = 0 if G2( x) G2 dg2 (18) Where G2 can represent V k, θ k, among others; and dg2 denote the imum admissible variation of function G2.
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