A four objectives optimization for an energy system considered in the environment

M. Blaise 0 1 M. Feidt 0 1 0 M. Blaise Agence de l'Environnement et de la Matrise de l'Energie (ADEME), 20, avenue du Gresille- BP 90406, 49004 Angers Cedex 01, France 1 M. Blaise (&) M. Feidt Laboratoire d'Energetique et de Mecanique Theorique et Appliquee (LEMTA-ENSEM) , 2, avenue de la Foret de Haye, 54516 Vanduvre-Le`s-Nancy, France To optimize a system, the two main points to consider are a precise system definition (boundary between the system and the environment) and a precise choice of criteria to fulfill. Various energy, environmental and economical goals imply various criteria to minimize energy consumption, rejected nonrecoverable heat or entropy production as well as a maximization of the useful effect (even the efficiency). We propose a generalized criterion that takes into account the main four objectives of an energy system. Weighting factors are allocated to the different objectives. To illustrate this point, we consider a Carnot engine in contact with two infinite heat reservoirs (thermostats). The heat transfer model considered uses linear transfer law (constant heat transfer conductance). Thermal losses are considered at various scales. The converter entropy production is supposed to be independent of its hot and cold temperatures that are the two optimization variables. Numerical optimization is performed with various weighting factor distributions. Limit cases are recovered and correspond to optimization results existing in the literature (mainly one-criterion optimizations). However, we distinguish clearly three fundamental assemblies that could be optimized: the converter in contact with the source and sink, the system composed of converter, source and sink and the same system placed in the environment. - Superscripts * Optimum value Many criteria exist for optimizing a given energy system. Figure 1 shows the principal ones, under the form of a triangular scheme. The three basic criteria are energy consumption (EC), rejected heat R (or heat pollution P) and more frequently the energy useful effect (UE). Classically, equilibrium thermodynamics considers the first law efficiency gI. More recently, the minimum of entropy production was considered. The first types of optimization performed in the past considered the maximization of the first law efficiency according to the Carnot equilibrium thermodynamics for the appraisal of an energy system. During the second part of the last century, maximization of the mechanical power of engines was considered. This appraisal was initiated by Chambadal [13] and Novikov [4] and renewed by Curzon and Alhborn [5]. These papers mark the beginning of the finite time thermodynamics, FTT. Today, the simultaneous consideration of energy, environment and economy concern implies a multicriterion analysis. Multicriterion optimization of thermal systems design considering simultaneously energy, economy and environment as objectives is developed [6]. These studies are developed for engines and plants, as well as for refrigeration systems [7] and for combined cooling, heating and power systems [8], even more at a national level [9]. New heuristic optimization methods (evolutionary algorithms) are also applied, for example to heat pump [10] or solar heat engine [11]. These algorithms appear strongly connected to the second law. A criterion combining the minimization of the entropy production and the maximization of the mechanical power output is proposed by Angulo-Brown [12] and completed by Yan [13]. It is valuable for the global system. The utilization of this kind of criteria is a means to obtain a compromise between power, cost and pollution objectives. In this paper, a new mixed objective is proposed, taking account of four aspects, which are power, cost, heat rejection and entropy production. A compromise between all of the criteria is difficult to consider. For example, the multiobjective Pareto approach can be used and defines an area of optimal trade-offs among all the objectives [6]. The four aspects considered in this paper are related through weighting factors. The compromise between each objective is therefore flexible. The weighting factor variation permits the observation of physically acceptable areas. Modeling of NovikovCurzonAhlborn thermomechanical engine The model proposed here is an extended one (Fig. 2a) of the thermomechanical engine proposed initially by NovikovCurzonAhlborn. This extended model takes Fig. 1 The triangle of optimization criteria Fig. 2 a Scheme and b entropy diagram of a NovikovCurzon Ahlborn converter with infinite source and sink Table 1 Expression of the first and second laws at different scales, the converter, the system and the system in the environment Converter with its thermal contact System System in the environment Input and output rate the converter internal irreversibilities and various thermal losses into account. The steady state hypothesis is used. The cycled fluid is in contact with two infinite thermal capacities, the hot source temperature THS and the cold sink temperature TLS. The corresponding cycle is represented in Fig. 2b. We see on this (T, S) diagram that exchangers of heat occur at constant temperatures, as it is the rule for a Carnot cycle. The cycled fluid is at TH at the hot side and at TL at the cold side. Consequently, the converter alone is subject to an entering heat flow rate Q_ HC, an outgoing heat flow rate Q_ LC and an outgoing mechanical power W_ . If the system considered here integrates not only the converter, but also two heat exchangers with uniform external fluid temperatures equal now to THS and TLS, the entering heat flow rate is Q_ H and the outgoing flow rate is Q_ L. These two rates are affected by the converter internal heat loss Q_ li, depending on TH, TL and Kli, the internal heat loss conductance. If we consider the thermomechanical system (converter ? heat source ? heat sink), the entering heat flow becomes Q_ HS and the outgoing heat flow Q_ LS. The thermal heat losses occurring between the heat source and the cold sink is Q_ lS and depend on THS, TLS and KlS. The environment is characterized by temperature T0, that could be different from TLS. This allows to define a new assembly consisting of the system in the environment. The corresponding heat rate becomes Q_ HS0 and Q_ LS0. The complementary heat losses Q_ lS0 are dependent on THS, T0 and the corresponding heat loss conductance KlSO. The hypothesis of infinite heat source and cold sink is consistent with the modeling of heat transfer using lumped thermal conductances KH and KL, such as: The thermodynamics sign convention implies that Q_ H is positive and Q_ L is negative. The different heat losses are represented as positive quantities (see Table 1 line 1) whatever the control volume. The input and output heat flow rates are given in Table 1, for the various control volumes (line 2). Applying the first law at different scales (Table 1 line 3) shows that t (...truncated)