dissipative processes in electrical engineering: a multi-scale approach
TRANSCRIPT
2nd International Conference on Engineering OptimizationSeptember 6-9, 2010, Lisbon, Portugal
Dissipative processes in electrical engineering: a multi-scale approach
Vincent Mazauric and Nadia Maızi
Mines ParisTech, Center for Applied Mathematics, BP 207, 06904, Sophia-Antipolis cedex, France
Abstract
In order to address the abysmal lack of efficiency of the electrical system (73% of losses, 45% of CO2emissions worldwide!), we propose to overcome the classical presentation of electromagnetism througha physical description of the electrical energy workflow, formulated as an optimal problem. In thisframework, the laws of electromagnetism are derived from a thermodynamic viewpoint. Particularly, theso-called Maxwell-Faradays law of induction results from an optimal path toward reversibility. Takingadvantage of the quadratic property of the thermodynamic functionals along with the well-splitted scalesof the current density, this approach is shown supporting a multi-scale analysis where the brute mini-mization condition is replaced by embedded minimizations on the various scales excited by the powerelectrical system. Following the previous thermodynamic viewpoint, these various scales are reviewedfrom deep within the material to the whole electrical system. Attention is paid on the dissipative pro-cesses and on the weakness of the modeling to properly consolidate them at the design and/or powermanagement levels: (i) In ferromagnetic materials, the dynamic behaviour should take anomalous lossesinto account. Recent improvements argue for a deviated Ohms law at the macroscopic scale while themagnetic law is kept from the static regime; (ii) At the design scale, eddy currents are at the originof losses and signal distortion. In order to improve the finite element modelling of macroscopic eddycurrents, a quadratic energy-based criterion is expressed from the numerical deviation observed on theelectric power conservation. However, further calculations of the electric field within the dielectric regionare expected to make the most of the criterion; (iii) At the integration level, the signal integrity issue inElectro-Magnetic Compatibility may be viewed as a competition between the electrostatic and the mag-netic couplings in dielectric regions. Galilean invariance enforcing that one regime dominates the otheraccording to the frequency; abrupt transition is expected between the two regimes. Some parentage withthe phase transition theory could be explored; (iv) At the network management level, the electromagneticenergy coupling acts as a stock to ensure the stability of the electrical system under load fluctuations.Hence the appropriate level of reliability may be faced to the investment. Finally, the interplay betweenenergy efficient issues should be discussed within a long-term planning exercise dedicated to sustainabledevelopment.Keywords: Thermodynamics, variational principles, electromagnetism, power system, losses
1. Introduction
Future investments in the electricity sectors over the next three decades are estimated at about US$10 Trillion [1], which is equivalent to the 2/3 of total worldwide energy investment needs, in all areas ofthe energy-supply chain, and is three times higher in real terms than investments made in the electricitysectors over the past three decades. Such significant growth in the electricity sector is mainly based on thefollowing assumptions: increased domestic energy demand in emerging countries and the refurbishmentof ageing facilities in developing countries. Moreover, the move to electricity is expected to promotetechnologies with lower environmental impacts in order to combat global warming due to greenhousegas emissions and to counterbalance fossil energy depletion. While most scenarios on the future of theelectricity sector assess a mix through available primary energy sources (hydro, nuclear, renewables, fossil,etc.), they usually neglect the actual energy efficiency of the electrical supply chain as a whole.
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Figure 1: Energy supply-chain (compiled from [2])
Indeed, electrical energy is severely disadvantaged by the efficiency of the Carnot cycle, transmission lossesand the low performance of applications. If we analyze the chain linking the producer to the consumer,shown in figure 1 in the conversion of the primary energy source to the final (commercial) energy supply,we find that electricity drains around 32% of the primary energy source. Further transformation of finalenergy to useful energy (the energy actually needed in line with available technologies) provides the overall“efficiency” of the “all-fuel” energy chain at around 37%, whereas it only reaches 27% in the specific caseof electricity [3].
Figure 2: US Residential (up) and Industrial (down) Total Energy Consumption, Major Sources, from1950 to 2005 [5]: electricity use (and related losses) expanded dramatically.
Moreover, this abysmal lack of efficiency may be exacerbated by the fact that the optimization of theelectrical power system is a long way from being achieved, as illustrated by the electrical losses in theUSA given in figure 2: for both sectors represented, the level of losses induced by the electrical gen-
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eration follows a rise in electricity demand.Ultimately, electricity generation contributes 40% of globalCO2 emissions, before deforestation and transportation [4]. This could infer that the state-of-the-art ofelectrical engineering does not allow energy efficiency of the whole electrical supply chain to be achieved.In contrast to this, this paper provides an energy-efficient description of electromagnetism based on areversible interpretation of Faraday’s Law, first within a global description of the electrical system (seeSect. ) then through a local focus (see Sect. ). This type of physics-oriented framework appears suitablefor the consolidation (i) in space of all scales involved in the conversion process of the electromagneticenergy; (ii) in time, over the whole life-cycle of the electrical supply chain. Whereas the latter appearsto be related to network management (see Sect. 0.1), the former is linked to developments in the designof devices (see Sect. 0.1). Lastly, the interplay between energy-efficient issues should be discussed withina long-term planning exercise dedicated to sustainable development (see Sect. ??). As far as electricityphenomena are derived through electromagnetism laws, we propose a variational approach leading fromthermostatics to Maxwell’s equation [16].
2. Variational principles in electromagnetism
To circumvent the impossibility of providing any degree of deterministic evolution of a complex system,a statistical description is adopted. Given the probability distribution p, the lack of information on thesystem is given by the Shannon’s entropy [6]
S = S (p) = −k∑
i
p (xi) ln p (xi) (k > 0) (1)
where xii describes the set of configurations assumed by the system. 2.1. Steady State
The only unbiased assignment on uncertainty of the system consists in maximizing its entropy withrespect of the macroscopic knowledge [7]. In other words, to use any other assignment would amount toan arbitrary assumption of information which, by hypothesis, we do not have. Hence, for each macroscopicexpected value Ar of the random value Ar
Ar =∑
i
p (xi)Ar (xi) (2)
the Lagrangian multipliers’ λrr are introduced to form the partition function
Z (λ1, · · · , λr, · · · ) =∑
i
exp
(−∑
r
λrAr (xi)
)(3)
Then, the so-called Boltzmann-Gibbs’ statistic law p gives the maximum-entropy probability with respectto the macroscopic averages (2)
p (xi) =
exp
(−∑
r
λrAr (xi)
)Z (λ1, · · · , λr, · · · )
(4)
Furthermore, the values of the Lagrangian multipliers λr completely define the statistic law (4). Theentropy of the system at the steady state S (p) (denoted by S) follows
S ≤ S = k
(lnZ +
∑r
λrAr
)(5)
where Ar denotes the average provided by the maximum-entropy probability (4)
Ar =∑
i
p (xi)Ar (xi) (6)
It follows from (3) that
Ar = −∂ lnZ
∂λr(7)
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Given that framework, the laws of thermostatistics are restored by considering that steady state of anyphysical system is at least known by its energy average, commonly called its internal energy and denotedU . Introducing the hamiltonian H of the system [8], U is given by
U =∑
i
p (xi)H (xi) (8)
The related Lagrangian multiplier defines the temperature T through the relation
λ1 =1
kT(9)
where k is the so-called Boltzmann’s constant. To ensure the convergence of (3), notice that the temper-ature T is always positive if the energy levels are positively unbounded. Hence, the other macroscopicaverages are defined through a set of (chemical) potentials
αr = − (kT)λr (r ≥ 2) (10)
so that the inequality (5) may be replaced by
U −∑r≥2
αrAr − TS ≥ −kT ln Z = G (T, · · · , αr, · · · ) (11)
(T, · · · , αr, · · · ) are also known as the system state variables. The left-hand side of (11) defines theGibbs’ potential as a functional on the set of probability distributions p. Its minimum, obtained forthe Boltzmann’s-Gibbs statistic law, provides the state function of the system: the so-called Gibbs’ freeenergy G. Thus, the steady state may be viewed as a competition between (see figure 3 in the context ofelectromagnetism):
• an ordered energy due to the internal interactions of the system U and couplings −αrAr with somereservoirs ensuring the prescribed averages Ar through relevant values of the potentials αr;
• a disordered energy −TS induced by the maximum entropy assignment and enforced by the contactwith a thermostat at the temperature T.
If the hamiltonian of the system H is parametrized by the boundary conditions X, differential calculationsobtained from (6) yield the variation of the Gibbs’ free energy with respect to its state variables
dG = −∑r≥2
Ar dαr − S dT +∑
F · dX (12)
where the force F applied by the actuator X to the system is derived from the virtual works principle
F = +gradG (13)
Conversely, for isothermal processes, derivatives of the Gibbs’ free energy provide state equations
Ar = − ∂G
∂αr(14)
As an experimental evidence, the energy couplings in the context of electromagnetism refer to threecontributions (figure 3):
• the boundary conditions prescribed by the positions of the actuators’ moving bodies X;
• a magnetic term (φI) where the magnetic flux φ is created by the electrical current I exciting thegenerator;
• an electrostatic term (QV0) where Q is the electric charge squeezed from the earth at the voltageV0.
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Figure 3: Chart of the energy exchanges between the various subsystems involved in the thermodynamic frame-
work. (µ, σ) and (µ, ε) denote respectively the behavior laws of conductors region C and dielectrics region Dsoaked with the electromagnetic field. While current generators and moving parts can only exchange work with
the electromagnetic field (respectively through current variation and a modification of the boundary conditions
on the moving parts), the thermostat, at temperature T, can only receive heat from the other sub-systems.
Whereas some improvements are summarized in [9] for steady current flow regimes, classical thermo-dynamic approaches of electromagnetism do not consider any extensions toward time-varying regimes[10][11][12][13]. In the following, a thermodynamic interpretation of Faraday’s Law within the quasi-static approximation is given, firstly in the context of an overall description of the electrical system (seeSect. ), and then by adopting a local viewpoint (see Sect. 0.1).
2.2. Time-varying regimes
Statistical assignment does also include an explanation of non-equilibrium conditions and irreversibleprocesses, since equilibrium is merely an ideal limiting case of the behavior of matter. In order to becoherent with maximum-entropy principle checked by an isolated system, two cases may be considered:
1. the evolution does not modify the entropy S +Sth of the isolated system. Such an evolution is saidto be reversible and involves mainly work exchanges, through sufficiently smooth variations in theenergy couplings, namely:
• the mechanical power exchanged between actuators Pmech =∑
F ·V where V =dXdt
isthe velocity of the moving body X experiencing the external force F; and in the context ofelectromagnetism
• the electromagnetic couplingd(φI+QV0)
dt;
2. the evolution increases the entropy of the isolated system so that some of the work exchangedbetween actuators is lowered in heat during the process: such an evolution is said to be irreversible.
The first principle conveys the energy conservation and reads
dU
dt= Pmech − T
dSth
dt(15)
where TdSth
dtdenotes the heat produced by the field, i.e. received by the thermostat. Hence, the
irreversibility experienced by the system coupled with its thermostat may be inferred from the Helmoltzfree-energy
F = U − TS (16)
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by expressing
Pmech −dF
dt= T
(dS
dt+
dSth
dt
)(17)
where the right-hand side matches the power lowered in heat by the whole system, commonly known asthe Joule losses PJoule. According to the second principle, this term is always positive, and the lowerthe Joule losses, the more reversible the evolution. In order to explicitly take into account the inertialbehavior of the electromagnetic coupling (Lenz’ Law), it is convenient to use the Gibbs free-energy
G = F − φI−QV0 (18)
on which another reversible assignment may be expressed
Pmech −dGdt
= min(
PJoule +d(φI)
dt+
d(QV0)dt
)(19)
In the process of the quasi-static approximation, Faraday’s Law of induction is achieved by forcing thecondition of reversibility expressed by the Gibbs free-energy, whereas the condition obtained from theHelmoltz free-energy only restores the flow behavior of the direct current. In order to derive the Maxwell-Faraday equation from (19), it is convenient to describe electromagnetic effects through macroscopic fieldslinked to the sources.
3. Where electromagnetism is recovered
0.1 Source fields
After spatial averaging on the microscopic distribution of charge to discard short time and space- varia-tions with respect to excitation provided by generators, the conservation of the electric charge, reads, inthe bulk of materials [14]
divJ + ∂tρ = 0 (20)
where ρ is the average charge density and J denotes the free current density, i.e. involving charges ableto move on a large scale with respect to atomic structure. Conductors may provide such non-vanishingsources. To replace the span of the electromagnetic interaction in an unbound way, it is convenient todefine the so-called electric displacement field D and magnetic field H. They satisfy respectively
• the Maxwell-Gauss equationdiv D = ρ +
∑S
n ·[D]δS (21)
where [D] denotes the interfacial discontinuities of D occuring at conductors/dielectric and dielec-tric/dielectric interfaces S oriented by n;
• and the Maxwell-Ampere equation
curlH = J + ∂tH +∑∂C
n×[H]δ∂C (22)
where [H] denotes the interfacial discontinuities of H occuring at conductors/dielectric and conduc-tors/conductors interfaces ∂C . Within the admissible couple of fields (D,H), optimality principlesderived from thermodynamics will enforce those (D,H) matching the complementary Maxwellequations.
The quasi-static approximation assumes no coexistence between the free current density and the dis-placement terms in the Maxwell-Ampere equation (22). Thus, the current density J remains divergence-free and only a static averaged charge density ρ can exist in conductors. Hence, considering invariance ofJoule losses under Galilean transformations, such a charge density does vanish in the bulk of conductors(i.e. D ≡ 0 therein). As a result, only free currents carried by conductors can yield heat losses, and freecharges can only exist on the surface of conductors where the source fields check
n× ([H]−V×[D]) = 0 (23)
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3.1. Statics
For magnetostatics, the H field is simply created by a static current density J, which is actually givenby the minimization of the Joule losses (see below Sect. 0.1 or [9]). Hence, equilibrium is obtainedby considering variations δH of the field H which change neither the Gibbs free energy (18), nor thecurrent density. Introducing (−B) as a conjugate field of H related to the variation of the magnetostaticcontribution to Gibbs’ free-energy density with respect to H, the stationary condition of
Gmag (T, I,X) =∫ (∫ H
0
(−B) · δ h
)d3r (24)
under admissible variations δH = −grad δφ keeping J constant in the conductors, yields, after anintegration by part throughout the whole space
div B = 0 (25)
along with a condition that falls on B.Thus, B appears as the magnetic flux density. Note that the numerical resolution of (24) requires a
magnetic behavior field-to-field relationship B (H). For permanent, local, non-dispersive and homoge-neous media, this behavior law has to check
δ B ·δH > 0 (26)
to enforce the stationary point of the Gibbs free-energy to its minimum.The same kind of thermodynamic procedure describes the electrostatic behavior of a dielectric region
D polarized by a surface charge distribution spread over the surface of conductors connected to voltagegenerators. The electrostatic Gibbs free-energy minimum enforces a behavior law D (E) of the dielectricmedia for which δ D ·δE > 0.
3.2. Time-stepping evolution within the quasi-static approximation
Expressing the weak reversibility condition (19) from continuous fields, the mechanical power suppliedby the actuators checks
Pmech −dGdt
= minH,E
(∫C
σ−1 (curlH)2 d3r +ddt
∫(B (H) ·H + D (E) ·E) d3r
)(27)
where:
1. the first term in the right-hand side corresponds to Joule losses monitored in the conductors C(σ > 0). This term is even to respect the invariance of losses with inversion of time;
2. the second term in the right-hand side is related to the variation with time of the energy couplingthe current generators and the electromagnetic field.
After calculations on the convective derivative of the coupling energy [15], the contribution of con-ducting regions C to the minimization of (27) reads
minH
∫C
(σ−1 (curlH)2 + ∂t (B ·H)
)d3r (28)
Some tedious variational calculations and a partial integration by part over the entire space yield theMaxwell-Faraday equation and result in Ohm’s Law in the stationary frame of each conductor [16]. Then,considering invariance over time of any magnetic flux under Galilean transformation, it follows that theelectric field E in conductors [17]:
• is given thanks to the so-called Ohm’s Law with motion
J = σ (E+V×B) (29)
where V is the local velocity of the conductor in the stationary frame and σ its conductivity; and
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• checks the Maxwell-Faraday equation
curlE = −∂t B (30)
Hence, the contribution of the dielectric region D to the minimization of (27) reads
minH,E
ddt
∫D
(B ·H + D ·E) d3r (31)
and enforces a reversible exchange between magnetic and electrostatic energies. Some calculations yield:
• the Maxwell-Faraday equation (30) in the dielectric region as well;
• a matching condition[B] · n = 0 (32)
on the normal component of the magnetic flux density B across the dielectric/conductor and di-electric/dielectric interfaces which therefore remains valid from static regimes;
• a matching condition on the tangential component of the electric field E across the dielectric/conductorand dielectric/dielectric interfaces
n× ([E] + V×[B]) = 0 (33)
3.2. Quasi-static field sketch
To summarize, stationary condition (24) is related to the determination of the magnetic field according tocurrent flow density, whereas (28) yields the eddy current distribution according to magnetic flux varia-tions. At each time-step, the surface charges spread over the conductors to minimize the electromagneticcoupling variations in the dielectrics (31), but with respect to the current flowing on ∂C according to(33). Such an approach:
• enforces the best reversible evolution of the electromagnetic field;
• corresponds to the best just-in-time workflow between actuators over a time-period;
• sketches the quasi-static evolution of the electromagnetic field, providing a thermodynamic-orientedinterpretation of the variational theory of electromagnetism [18].
Taking advantage of their quadratic property, thermodynamic functionals (24),(27) support spatialfiltering operations and mean field assumptions to provide a fully multi-scale framework.
4. Multi-scale modeling
So far, the electrical network has been considered within a global description. For modeling purposes ornetwork management, it is convenient to derive the voltage drop UΩi
of a dipolar component from thePoynting’s vector S = E×H. If Ii denotes the net current flowing therein, and Ωi a surrounding volumeof the component, the voltage drop UΩi is defined by
UΩi= − 1
Ii
∮∂Ωi
(E×H) · nd2r (34)
As the Maxwell-Faraday equation and Ohm’s Law with motion are satisfied in each subspace Ωi , theelectrical power Pelec =
∑i
UΩiIi and the mechanical power Pmech within the partition ∪iΩi surrounding
all the conductors Ci ⊂ Ωi , check the energy conservation equation
Pmech + Pelec =∫
C
σ−1 (curlH)2 d3r +ddt
∫ (∫ B
0
H ·δ b+∫ D
0
E ·δ d
)d3r (35)
Notice that:
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• Pmech gathers the Laplace’s force J×B experienced by the conductors and the switching reluctanceeffects acting on materials;
• the last term in the right-hand side denotes the electromagnetic powerdF
dtderived from the Hel-
moltz free energy (16) over the whole space.
Note that the last term in the right-hand side denotes electromagnetic power over the whole space,corresponding to the reactive power in a time-harmonic regime. In order to highlight the multi-scaleability of the former approach, we consider successively: the management of a power network, in order tounderline the role of energy coupling in the electromagnetic field and excitation in the reliability of theinstallation; and the design of devices, in order to put the relevant method in practice. A deeper scale isaddressed in [19].4.1. Power network management
The conservation equation (35) addresses two distinct behavior patterns in the network.The cancellation of electrical power in (35) over the partition ∪iΩi expresses the conservation of the
energy supplied by the field. The fields obtained in each subspace Ωi may be composed to check thecrossing conditions at the interfaces ∂Ωi , and the right-hand side of (35) is simply the mechanical powerreceived by the field from the actuators. Furthermore, the definition (34) becomes intrinsic in the sensethat:
• two parallel connected branches i and j have the same voltage drop as a third one
UΩi= UΩj
= UΩi∪Ωj(36)
which consequently does not depend on Ωi ⊃ Ci ;
• it matches the usual expression of the voltage drop obtained from AV formulation [20]; and
• it enforces a reversible – i.e. adapted – electrical coupling between the electromagnetic field in eachsubspace Ωi and the remaining electrical network.
Conversely, exhibiting a difference in the voltage drop (34) between two parallel-connected branchesof the network means that the power network experiences an irreversible process at the time scale dtand the length scale ∼ 3
√|Ωi | on which it is monitored. In other words, sudden change occurs either
outside the partition (i.e. extra connection such as switching, earth leakage or lightning) or within it(i.e. transition in behavior laws due to ageing), leading to unbalanced behavior in the network. Only arush to adapt the (mechanical) power production or a disconnection of loads (black-out) may lead to itsrecovery. As it is illustrated in figure 4 where two events experienced by the network are represented:it is shown that while an admissible load fluctuation is smoothed out, at the beginning, by the couplingenergy (φI) and the generation realignment (left in figure 4), the coupling energy is completely reducedin heat by a short circuit, leading to a collapse in power transmission (right in figure 4).
4.2. Electrical design
The previous approach addresses a fuller justification of the Finite Element Method (FEM), which consistsin building an approximation of the variational formulations given in equations (24) and (28), but witha finite number of degrees of freedom chosen on a mesh (figure 5) [21]. Assuming a balanced network,the design can be partitioned corresponding to each device, the behavioral composition of which providesthe global design. Following (27), the contribution of Ωi to the global mechanical power checks
Pmech (Ωi)−dG (Ωi)
dt+∑j(i)
Uj Ij
= min(H,E)|Ωi
(∫Ci
σ−1 (curlH)2 d3r +ddt
∫Ωi
(B ·H + D ·E) d3r
) (37)
where j (i) indexes the electrical circuits connected to Ωi , Pmech (Ωi) is the mechanical power acting onΩi , and G (Ωi) is the restriction of the Gibbs free energy to Ωi .
Several kinds of devices can be highlighted.
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Figure 4: Two events experienced by the network
Motors, generators and transformers While the optimal behavior of electromechanical devices achievesover any thermodynamic cycle
min∮ Pmech (Ωi) +
∑j(i)
Uj Ij
dt (38)
to ensure the highest efficiency, in this case, the transformer requires Pmech (Ωi) ∼ 0 as an additionalconstraint to avoid mechanical power being trapped within it. As efficiency is expressed directlyfrom the energy conversion balance (37), the FEM remains the obvious method for designing suchdevices. To achieve a sufficiently low level of integration, vanishing boundary conditions on thefields are prescribed on ∂Ωi , taken far enough away from the device. Thus coupling between de-vices is only performed thanks to electrical circuits, the ideal behavior of which is achieved when thevoltage does not drop. For higher levels of integration, a robust design should ensure the device’slack of sensitivity to the outer electromagnetic field acting on ∂Ωi .
Couplings Transmission lines, distribution busways, cablings, PCBs, etc. belong to this category ofcomponents. Their optimal behavior achieves
∀j ∈ Ωi min (Uj Ij ) with Pmech (Ωi) ∼ 0 (39)
to express that neither Joule losses nor time-varying-induced reactive power occur therein. Whilesuch a condition enforces signal integrity flowing through any electrical coupling, it involves:
• opened domains, where power transmission prevents vanishing fields;
• extended domains, where air volume could lead to significant problems;
• little-known boundary conditions in between the conductors of the same circuit loop.
As a result, the FEM appears ill suited to designing electrical couplings. In time-harmonic design,the Partial Element Equivalent Circuit (PEEC) method [22][23][26] uses knowledge of the currentflow orientation and the localization of the electromagnetic energy in the conductors to provide analternative to the FEM.
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Figure 5: Flowchart of the thermodynamic approach of electromagnetism. The full lines show the main steps
of the approach, and the dashed lines indicate its numerical description. The mixed lines provide the usual
introduction of the Finite Element Method.
5. Conclusion
The previous point of view consisted in starting with variational principles discussed from thermody-namics and showing that they yield functionals containing Maxwell’s equations. Thus, electromagneticconversion achieves the best reversible evolution just counteracted by the dispersive properties of the ma-terials. Therefore, optimization entails applying the FEM for power conversion purposes, and applyingPEEC to couplings. Both methods are available in the same CAD environment [29].
Unfortunately, in spite of this ideal design, a real network undergoes irreversible processes over thelength and time scales during which it is monitored. Usually, this lack of information is counteractedfirst by the reactive energy (stock), and then by the reserves for admissible load fluctuations. Hence,for a given level of reliability, network management results from a compromise between: (i) the stock ofreactive energy, and subsequent oversizing and extra losses; and (ii) investments in monitoring and rushproduction capabilities.
In long-cycle industries, scientific uncertainties must be reduced in order to develop the most sustain-able technologies and anticipate market trends. In order to center eco-design methodology on medium-and long-term prospective thinking, Life Cycle Assessment should be sufficiently consolidated to matchthe technological breakdown required by technical and economic energy long-term planning tools, suchas the MARKAL model [30]. Hence, the method will provide a complete energy-related description ofthe energy management chain [19], from deep within the structure of materials to the acknowledgmentof demand fluctuations, based on an optimal description of energy transfers, therefore conducive to:
• optimization of architectures subject to operating and environmental constraints;
• discrimination of technologies and research initiatives benefiting from favorable environmental lever-age;
• analysis of the sensitivity of prospective scenarios to determining operating factors, in particularthe flexibility of demand;
• arbitration between the availability of electrical power and the investments to be made, betweenthe production mix and transport, monitoring and reserve capacities, as well as the sensitivity ofsuch arbitration.
Such an approach should improve the Life Cycle Assessment of the electrical supply chain, provide aninsight into achieving power electrical engineering dedicated to energy efficiency, and ultimately enablekey features of the electricity production sector to achieve sustainable long-term planning [31].
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AcknowledgementsThis work was supported by the Chair Modeling for sustainable development, driven by MINES ParisTech,Ecole des Ponts ParisTech, AgroParisTech, and ParisTech, supported by ADEME, EDF, RENAULT,SCHNEIDER ELECTRIC and TOTAL.
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