A Kaissoine
email: abdou.kaissoine@telecom-paristech.frhuyart@telecom-paristech.fr

B Huyart

A Mbaye

K Mabrouk
email: kais.mabrouk@esigetel.fr

Demodulation of aggregated RF Signals with a Unique Rx Chain

In the next generation of the mobile radio systems, the information will be distributed on discontinuous frequency bands. To demodulate the RF signal, a possible architecture is to use a number of receivers Rx equal to that of the discontinuous RF frequency bands. This paper demonstrates that this operation can be made with a single receiver Rx. The method is based on the mixing of the RF signal with a local oscillator (LO) signal constituted by several Continuous Wave (CW) signals. The technique is studied theoretically and then applied to the case of RF signal constituted of two frequency bands. The CW frequencies of the LO signal are selected to convert both bands of the RF signal in Zero Intermediate Frequency (ZIF) and LOW Intermediate Frequency (LIF).

I.

INTRODUCTION

The International Telecommunication Union of Radiocommunication (ITU-R) recommended that the 4th generation network must reach the data rate of 1 Gbit/s for the downlink (base station to mobile station) and 500 Mbit/s for the uplink (mobile station to base station) [START_REF] Shen | Overview of 3GPP LTEAdvanced Carrier Aggregation for 4G Wireless Communications[END_REF], [START_REF] Yuan | Carrier aggregation for LTE advanced mobile communication systems[END_REF]. This data rate requires a frequency bandwidth between 70 MHz and 100 MHz. The availability of this frequency range is not always guaranteed, especially for RF carrier frequency below 3 GHz. The solution is spectrums aggregation which consists in distributing the information on multiple discontinuous frequency bands. Nowadays, the proposed architecture is to use a number of Rx receiver equal to the number of frequency bands of the RF signal. In the case of 2 bands, another way is to use the architecture of the Weaver demodulator and a LO frequency equals approximately to the half of both carrier frequencies [START_REF] Wu | A 900-MHz/1[END_REF], [START_REF] Kitayabu | Implementation of Concurrent Dual-Band Receiver Using IF Undersampling[END_REF]. This last technique is not applicable for the demodulation of more than 2 frequency bands.

In parallel with our research works, a team of NTT published in [START_REF] Kaho | A simultaneous receiving multi-band mixer with independent gain control[END_REF] practical results on the transposition by a mixer of a RF signal constituted of 2 frequency bands. The used technique is similar to that developed in this paper and consists in using a LO signal constituted of 2 frequency tones. It is shown in [START_REF] Kaho | A simultaneous receiving multi-band mixer with independent gain control[END_REF] that power levels of the IF signals may be identical even if the RF signal is composed of two frequency bands of different power levels. This property is obtained by monitoring the relative power of CW signals of the LO. However, that involves a degradation of the signal to noise ratio for one of the bands by increasing the conversion loss of the mixer.

In the work presented in this paper this technique is applied directly for the demodulation of RF signal aggregated in frequency and without any control of the power level of the LO signal.

The principle of transposition in baseband of a RF signal constituted of discontinuous frequency bands is demonstrated in section II. Finally the section III gives a validation of the theoretical results by the measure. The variation of the quality of the In phase and Quadrature (IQ) constellation of the demodulated signals is given according to the relative power level of both frequency bands of the RF signal.

II. THEORY AND TECHNIQUE OF THE DOWN CONVERSION OF THE RF SIGNAL

The transposition of a RF signal to baseband is done by the mixing with a LO signal. Assuming the RF signals is constituted of 2 discontinuous frequency bands, respectively centered at the frequencies f 1 and f 2 and of bandwidth BW 1 and BW 2 ; the expression of the aggregated RF signal is expressed thus:

1 2 2 2 1 2 ( ) R e[ ( ) ( ) ] j f t j f t RF v t Z t e Z t e    (1) with (
) ( ) ( ) j jt j j Z t a t e  
, j=1,2 the complex envelope of 2 carrier frequencies f 1 and f 2 .

In our case the objective is to lower the frequency bandwidth of the converted signals to relax the required performances of the Analog to Digital Converter (ADC). The strategy is consequently to convert both frequency bands of the RF signal to ZIF for one band and LIF at frequency Δf 2 for the second one. In that way, both CW frequencies of the LO signal are equal to f 1 and f 2 -Δf 2 with Δf 2 > BW 1 /2. The expression of the LO signal is defined as follows:

1 1 2 2 2 ( 2 ) ( 2 ( ) ) ( ) R e[ ( )] j f t j f f t LO LO v t V e e           (2)
with V LO the amplitude of the CW signals and angles i  (i=1,2) representing phase shifts between RF and LO signals.

Assuming that 21 ff  so that we obtain the following condition:

21 LPF f f f  (3)
where

2 2 /2 LPF f f BW    (4)
is the cutoff frequency of the low-pass filter.

The expression to the output signal of the low-pass filter is expressed as follows:

1 1 1 1 2 2 2 2 2 cos( ( ) ) () cos( ( ) 2 ) k k L O k A a t v t V A a t f t                (5)
where A kj , j=1,2 are gains that include the conversion gain of the mixer and the losses of splitters, phase shifters and filters. And k represents the output path of the mixer.

The equation [START_REF] Kaho | A simultaneous receiving multi-band mixer with independent gain control[END_REF] shows that the first frequency band is transposed directly to baseband around zero, it is a ZIF demodulation. The second band is transposed to the intermediate frequency Δf 2 , it is a LIF demodulation.

Moreover the separation between both baseband spectrums is possible if:

2 2 1 / 2 / 2 f BW BW    ( 6
)
This technique is applied to the Three Phase Demodulator (TPD) instead of the Classical IQ (CIQ) demodulator. Indeed, a previous paper [START_REF] Mabrouk | Architectural solution for second-order intermodulation intercept point improvement in direct down-conversion receivers[END_REF] has shown TPD gives a gain of 20 dB in second-order inter-modulation distortion (IMD2) rejection in comparison with CIQ. The operation mode of TPD differs from the CIQ. The RF and LO signals are divided into three paths instead of two. The phase shift between the LO signals is equal to  120° for the TPD instead of 90 ° for the CIQ. The CIQ delivers directly the IQ baseband signals. In the case of the TPD, the output signals are three low frequency signals, named v 1 (t), v 2 (t) and v 3 (t). The TPD (Fig. 1) was essentially realized using COTS components. The phase shifters were made with microstrip propagation lines. The phase shift varies from 60 ° to 150 ° in function of frequency in the range 2 to 4.2 GHz. The splitters insertion loss is 5dB. The mixers present the following characteristics: RF/LO inputs (DC=2-7 GHz), FI output (DC-1.3 GHz), LO power (7dBm), Conversion Loss (7dB) and intercept point of order 2 (IIP2 = 27dBm).

[7], [START_REF] Mabrouk | 3-D Aspect in the Five-Port Technique for Zero-IF Receivers and a New Blind Calibration Method[END_REF] demonstrate that if v 2 (t) output is symmetric in comparison with 2 others, the three output signals of the TPD can be added by an analog circuit of IQ regeneration performing the following operations: The mismatch between the paths of the demodulator depends upon the deviation in symmetric conditions of amplitude and phase. In [START_REF] De La Morena-Álvarez-Palencia | Direct Baseband I-Q Regeneration Method for Five-Port Receivers Improving DC-offset and Second-Order Intermodulation Distortion Rejection[END_REF], the simulated results demonstrate that a phase asymmetry at port3 of value Δ introduces an amplitude error on IQ output. For example, a phase imbalance, Δ=10° gives  3 dB imbalance IQ output as shown in the Fig. 3 of [START_REF] De La Morena-Álvarez-Palencia | Direct Baseband I-Q Regeneration Method for Five-Port Receivers Improving DC-offset and Second-Order Intermodulation Distortion Rejection[END_REF].

1 2 3 ( ) [ ( ) 2 ( ) ( )] out I I t V t V t V t        (7) 13 ( ) [ ( ) ( )] out Q Q t V t V t     (

III. MEASUREMENT RESULTS

The Fig. 2 Due to components limitations (data acquisition card, Low Frequency amplifiers, PC…) the measurements were made with data rate of 100 kS/s. The frequency conversion of the aggregated RF signal is performed by the mixing with a LO signal constituted of two CW signals. According to section II, the first CW frequency is 2 GHz to convert the first band to ZIF and the second CW frequency is 2.2998 GHz to convert the second band to LIF. The synchronization of the generators is performed by a common clock signal of frequency 10 MHz (Fig. 2).

The RF signal is demodulated by the TPD. The three outputs signals of the TPD are combined in the IQregeneration circuit (Fig. 2).

The spectrum of the baseband Iout signal shows two distincts bands centered respectively between 0-67kHz (ZIF configuration) and 133-267 kHz (LIF configuration) (Fig. 4). The sampling frequency of the data acquisition card (MI3033) is equal to 800 KHz. The Iout and Qout signals are oversampled with a factor equal to 8. Then equalization of I out and Q out is performed using the training sequence. Fig. 5 shows that the two QPSK constellations are recovered and centred. The red points show that the synchronization with the CAZAC sequence functions correctly. These results validate the theoretical assumptions developed in II.

The performances of this receiver are evaluated using the Error Vector Magnitude (EVM). We call ZIF signal (resp. LIF), the signal of carrier frequency 2 GHz (resp. 2.3 GHz), which is transposed to ZIF (resp. LIF). The results show a gain of 2 to 3 dB in sensitivity (defined here for EVM=-15dB), for the LIF signal in comparison with the ZIF signal (Fig. 6). The results of the EVM presented in the Fig. 6 were performed by varying simultaneously the input power level of the ZIF and LIF signals. The Fig. 7 and the Fig. 8 show the scenarios where one of the power level of a signal (ZIF or LIF) is fixed to -47 dBm and the other varies. By comparing the EVM results of ZIF signal in Fig. 6 and Fig. 7, there is a degradation of about 5 dB when the fixed (-47 dBm) power level of the LIF signal is greater than the power level of the ZIF signal. For the same limit value of -47 dBm, there is only a drop of about 2.5 dB for the EVM of the LIF signal in Fig. 6 and Fig. 8. Thus the LIF signal influences more the quality of the converted ZIF signal. This can be explained by second-order inter-modulation distortion (IMD2), which is more important in the ZIF converted band than in the LIF one. This effect has been checked by theoretical study and by spectrum measurement but is not described here due to space limit of the paper.

8 )

 8 Both constants μ I and μ Q are determined by a classical equalization procedure. In this work this operation is done thanks to a training sequence. Finally the transmitted I(t) and Q(t) signals are recovered.

Fig. 1 .

 1 Fig. 1. Three Phase Demodulator

Fig. 2 .

 2 Fig. 2. Test Bench for the demodulation of RF signal aggregated in frequency

Fig. 3 .

 3 Fig. 3. Power Spectral Density in dBm/Hz of V3 signal in function of frequency for bandwidth to 20 MHz.

Fig. 4 .

 4 Fig. 4. Power Spectral Density in dBm/Hz of Iout signal in function of frequency for bandwidth to 100 KHz.

Fig. 5 .

 5 Fig. 5. Phase constellation of 2 discontinuous frequency bands RF signal translated Zero-IF and 200 KHz LIF

Fig. 6 .

 6 Fig. 6. EVM in dB in function of RF input signal in dBm constituted of 2 carriers modulated by QPSK signals.

Fig. 7 .

 7 Fig. 7. EVM in dB in function of RF input signal in dBm constituted of 2 carriers modulated by QPSK signals, so the power of signal RF to frequency 2.3 GHz fixed to -47 dBm.

Fig. 8 .

 8 Fig. 8. EVM in dB in function of RF input signal in dBm constituted of 2 carriers modulated by QPSK signals, so power of signal RF to frequency 2 GHz fixed to -47 dBm. VI. CONCLUSION This paper demonstrates that it is possible to demodulate a RF signal constituted of two discontinuous frequency bands with a single Rx chain. However, this technique can be extended to a greater number of bands. The used technique is based on a LO signal composed of several CW signals. The used method is demonstrated theoretically and confirmed by measurements results. The demonstration has been done for two useful signals centered at 2 GHz and 2.3 GHz and QPSK modulated with a symbol rate of 100Ks/s. The signals have been respectively converted to ZIF and LIF in such a way the ADC has to convert low frequencies signals of frequency bandwidth 300 kHz.


Mireille Bossy
email: mireille.bossy@inria.fr

Odile Pourtallier
email: odile.pourtallier@inria.fr

Nadia Maïzi
email: nadia.maizi@mines-paristech.fr

Game theory analysis for carbon auction market through electricity market coupling

In this paper, we analyze Nash equilibria between electricity producers selling their production on an electricity market and buying CO 2 emission allowances on an auction carbon market. The producers' strategies integrate the coupling of the two markets via the cost functions of the electricity production. We set out a clear Nash equilibrium on the power market that can be used to compute equilibrium prices on both markets as well as the related electricity produced and CO 2 emissions released.

Introduction

The aim of this paper is to develop analytic tools in order to design a relevant mechanism for carbon markets, where relevant refers to emissions reduction. For this purpose, we focus on electricity producers in a power market linked to a carbon market. The link between markets is established through a market microstructure approach. In this context, where the number of agents is limited, a standard game theory applies. The producers are considered as players behaving on the two financial markets represented here by carbon and electricity. We establish a Nash equilibrium for this non-cooperative J-player game through a coupling mechanism between the two markets.

The original idea comes from the French electricity sector, where the spot electricity market is often used to satisfy peak demand. Producers' behavior is demand driven and linked to the maximum level of electricity production. Each producer strives to maximize its market share. In the meantime, it has to manage the environmental burden associated with its electricity production through a mechanism inspired by the EU ETS1 framework: each producer emission level must be counterbalanced by a permit or through the payment of a penalty. Emission permit allocations are simulated through a carbon market that allows the producers to buy allowances at an auction. Our focus on the electricity sector is motivated by its introduction in phase III of the EU ETS, and its prevalence in the emission share. In the present paper, the design assumptions made on the carbon market aim to foster emissions reduction in the entire electricity sector.

Based on a static elastic demand curve (referring to the time stages in an organized electricity market, mainly day-ahead and intra-day), we solve the local problem of establishing a non-cooperative Nash equilibrium for the two coupled markets.

While literature mainly addresses profit maximization, our share maximization approach deals with profit by making specific assumptions, i.e. no-loss sales, and a balance struck between the purchase of allowances and the carbon footprint of the electricity generated. Here the market is driven through demand dynamics rather than the electricity spot price dynamics used in recent works (see [START_REF] Carmona | The valuation of clean spread options: Linking electricity, emissions and fuels[END_REF][4] [START_REF] Carmona | Singular forwardbackward stochastic differential equations and emissions derivatives[END_REF]).

In Section 2, we formalize the market (carbon and electricity) rules and the associated admissible set of players' coupled strategies.

We start by studying (in section 3.2) the set of Nash equilibria on the electricity market alone (see Proposition 1). This set constitutes an equivalence class (same prices and market shares) from which we exhibit a dominant strategy.

Section 3.3 is devoted to the analysis of coupled markets equilibria: given a specific carbon market design (in terms of penalty level and allowances), we compute the bounds of the interval where carbon prices (derived from the previous dominant strategy) evolve. We specify the properties of the associated equilibria.

Coupling markets mechanism

Electricity market

In the electricity market, demand is aggregated and summarized by a function p → D(p), where D(p) is the quantity of electricity that buyers are ready to obtain at maximal unit price p. We assume the following: Assumption 1. The demand function D(•) : R + → R + is decreasing, left continuous, and such that D(0) > 0.

Each producer j ∈ {1, . . . , J} is characterized by a finite production capacity κ j and a bounded and increasing function c j : [0, κ j ] -→ R + that associates a marginal production cost to any quantity q of electricity. These marginal production costs (a) delivery 9-10 am (b) delivery 3-4 pm Fig. 1 The orange curve is the function q → D -1 (q) on the EPEX market. The evolution of the spot market confirms the relevance of Assumption 1 on the Demand function p → D(p).

depend on several exogenous parameters reflecting the technical costs associated with electricity production e.g. energy prices, O&M costs, taxes, carbon penalties etc. This parameter dependency makes possible to build different market coupling mechanisms. In the following we use it to link the carbon and electricity markets.

The merit order ranking features marginal cost functions sorted according to their production costs. These are therefore increasing staircase functions whereby each stair refers to the marginal production cost of a specific unit owned by the producer.

The producers trade their electricity on a dedicated market. For a given producer j, the strategy consists in a function that makes it possible to establish an asking price on the electricity market, defined as s j :C j × R + -→ R + (c j (•), q) -→ s j (c j (•), q), where C j the set of marginal production cost functions are explicitly given in the following (see (13)). s j (c j (•), q) is the unit price at which the producer is ready to sell quantity q of electricity. An admissible strategy carries out the following sell at no loss constraint

s j (c j (•), q) ≥ c j (q), ∀q ∈ Dom(c j ). (1)
For example we can take s j (c j (•), q) = c j (q) or s j (c j (•), q) = c j (q) + λ (q), where λ (q) stands for any additional profit.

As mentioned in the introduction, the constraint (1) guarantees profitable trade as much as the equilibrium established through this class of strategy will benefit each producer. This establishes a link between market share maximization and profit maximization paradigms. Let us denote S as the class of admissible strategy profiles on the electricity market. We have

S =    s = (s 1 , . . . , s J ); s j : C j × R + -→ R + (c j (•), q) -→ s j (c j (•), q) such that s j (c j (•), q) ≥ c j (q), ∀q ∈ Dom(c j )    . (2)
As a function of q, s j (c j (•), q) is bounded on Dom(c j ). For the sake of clarity, we define for each q ∈ Dom(c j ), s j (c j (•), q) = p lolc , where p lolc is the loss of load cost, chosen as any overestimation of the maximal production costs. For producer j's strategy s j , we define the associated asking size at price p as

O(c j (•), s j ; p) := sup{q, s j (c j (•), q) < p}. ( 3
)
Hence O(c j (•), s j ; p) is the maximum quantity of electricity at unit price p supplied by producer j on the market.

Remark 1.

(i) The asking size function p → O(c j (•), s j ; p) is, with respect to p, an increasing surjection from [0, +∞) to [0, κ j ], right continuous and such that O(c j (•), s j ; 0) = 0.

For an increasing strategy s j , O(c j (•), s j ; .) is its generalized inverse function with respect to q.

(ii) Given two strategies q → s j (c j (•), q) and q → s ′ j (c j (•), q) such that s j (c j (•), q) ≤ s ′ j (c j (•), q), for all q ∈ Dom(c j ) we have for any positive p

O(c j (•), s j ; p) ≥ O(c j (•), s ′ j ; p). Indeed, if p 1 ≥ p 2 then {q, s j (c j (•), q) ≤ p 2 } ⊂ {q, s j (c j (•), q) ≤ p 1 } from which we deduce that O(c j (•), s j ; •) is increasing. Next, if s j (c j (•), •) ≤ s ′ j (c j (•), •)
, for any fixed p, we have {q, s ′ j (c j (•), q) ≤ p} ⊂ {q, s j (c j (•), q) ≤ p} from which the reverse order follows for the requests.

We shall now describe the electricity market clearing. Note that from a market view point, the dependency of the supply with respect to the marginal cost does not need to be explicit. For the sake of clarity, we write s j (q) and O(s j ; p) instead of s j (c j (•), q), O(c j (•), s j ; p). The dependency will be expressed explicitly whenever needed.

By aggregating the J asking size functions, we can define the overall asking function p → O O(s; p) a producer strategy profile s = (s 1 , . . . , s J ) as:

O O(s; p) = J ∑ j=1 O(s j ; p). ( 4
)
Hence, for any producer strategy profile s, O O(s; p) is the quantity of electricity that can be sold on the market at unit price p. The overall supply function p → O O(s; p) is an increasing surjection defined from [0, +∞) to [0, ∑ J j=1 κ j ], such that O O(s; 0) = 0.

Electricity market clearing

Taking producer strategy profile s = (s 1 (•), . . . , s J (•)) the market sets the electricity market price p elec (s) together with the quantities (ϕ 1 (s), . . . , ϕ J (s)) of electricity sold by each producer. The market clearing price p elec (s) is the unit price paid to each producer for the quantities ϕ j (s) of electricity. The price p(s) may be defined as a price whereby supply satisfies demand. As we are working with a general non-increasing demand curve (possibly locally inelastic), the price that satisfies the demand is not necessarily unique. We thus define the clearing price generically with the following definition.

Definition 1 (The clearing electricity price). Let us define

p(s) = inf {p > 0; O O(s; p) > D(p)} and p(s) = sup {p ∈ [p(s), p lolc ]; D(p) = D(p(s))} (5)
with the convention that inf / 0 = p lolc . The clearing price may then be established as any p elec (s) ∈ [p(s), p(s)] as an output of a specific market clearing rule. To keep the price consistency, the market rule must be such that for any two strategy profiles s and s

′ , if p(s) < p(s ′ ) then p elec (s) < p elec (s ′ ), if p(s) = p(s ′ ) then p elec (s) = p elec (s ′ ). (6)
Note that p(s) = p(s) only if the demand curve p → D(p) is constant on some intervals [p(s), p(s) + ε]. Note also that price p(s) is well defined in the case where demand does not strictly decrease. This includes the case where demand is constant. In such case, p(s) = p lolc only if the demand curve never crosses the supply.

0 price quantity • • • • • Total offer p → O O(p) • • Demand p → D(p) • • • • • • p(s) • p(s) • quantity sold
Next, we define the quantity of electricity sold at price p elec (s). When O O(s; p elec (s)) ≤ D(p elec (s)), each producer sells O(s j ; p elec (s)), but cases where O O(s; p elec (s)) > D(p elec (s)) may occur, requiring the introduction of an auxiliary rule to share D(p elec (s)) among the producers that propose O O(s; p elec (s)). Note that in this last case, due to the clearing property (6) on p elec (•), we have

O O(s; p(s)) ≥ D(p elec (s)) = D(p(s)).
Hence the D(p elec (s)) is totally provided by producers with non null offer at price p(s). The rule of the market is to share D(p elec (s)) among these producers only. This gives an explicit priority to the best offer prices p(s).

Let us break down supply as follows:

O O(s; p(s)) = J ∑ j=1 O(s j ; p(s) -) + J ∑ j=1 ∆ -O(s j ; p(s)),
where ∆ -O(s j ; p(s)) := O(s j ; p(s)) -O(s j ; p(s) -). The market's choice is to fully accept the asking size of producers with continuous asking size curve at point p(s). For producers with discontinuous asking size curve at p(s), a market rule based on proportionality that favors abundance is used to share the remaining part of the supply. We resume the market rule on quantities as follows.

Definition 2 (Clearing electricity quantities). The quantity ϕ j (s) of electricity sold by Producer j on the electricity market is

ϕ j (s) =                O(s j ; p elec (s)), if D(p elec (s)) ≥ O O(s; p elec (s)), O(s j ; p(s) -) + ∆ -O(s j ; p(s)) D(p(s)) -O O(s; p(s) -) ∆ -O O(s; p(s)) , if D(p elec (s)) < O O(s; p(s)), (7)
where

∆ -O O(s; p(s)) := ∑ J j=1 ∆ -O(s j ; p(s)) > 0.
Note that, when D(p(s)) < O O(s; p(s)), we have ∆ -O O(s; p(s)) > 0. Note also that we always have

J ∑ i=1 ϕ j (s) = D(p elec (s)) ∧ O O(s; p elec (s)). (8)

Carbon market

Let us recall the CO 2 regulation principle on which we base our analysis. Producers are penalized according to their emission level if they do not own allowances. Hence, in parallel to their position on the electricity market, producers buy CO 2 emission allowances on a separate CO 2 auction market.

In the following, we formalize producer strategy on the CO 2 market only.

Assumption 2 (Capped carbon market).

(i) The carbon market is capped and has a finite known quantity Ω of CO 2 emission allowances available.

(ii) Each producer j can buy a capped number of allowances E j , related to its own CO 2 emission capacity. (iii) Emissions that are not covered by allowances are penalized at a unit rate p.

On this market, producers adopt a strategy that consists in an offer function τ → A j (τ) defined from [0, p] to [0, E j ]. Quantity A j (τ) is the quantity of allowances that producer j is ready to buy at price τ. This offer may not be a monotonic function.

We denote ❆ the strategy profile set on the CO 2 market,

❆ := {A = (A 1 , . . . , A J ); s.t. A k : [0, p] → [0, E j ]}.
The CO 2 market reacts by aggregating the J offers by

A A (τ) := J ∑ j=1 A j (τ),
and the clearing market price is established following a second item auction2 as:

p CO 2 (A) := sup{τ; A A (τ) > Ω }, with the convention sup / 0 = 0. ( 9
)
Note that p CO 2 (A) = 0 indicates that there are too many allowances to sell. It is worth a reminder here that the aim of allowances is to decrease emissions. In section 3.3, we discuss a design hypothesis (Assumption 6) that guarantees an equilibrium price p CO 2 (A) > 0. Therefore, in the following, we assume that the overall quantity Ω of allowances, is such that p CO 2 (A) > 0.

Next, we define the amount of allowances bought at price p CO 2 (A) by the producers. By Definition (9), we have A A (p CO 2 (A)) ≥ Ω and A A (p CO 2 (A) + ) ≤ Ω . When A A (p CO 2 (A)) > Ω , the CO 2 market must decide between the producers with an additional rule. We define

∆ (A i ) := A i (p CO 2 (A) + ) -A i (p CO 2 (A)).
For a producer i, ∆ (A i ) ≥ 0 means that its CO 2 demand does not decrease if the price increases. It is therefore ready to pay more to obtain the quantity of allowances it is asking for at price p CO 2 (A). The CO 2 market gives priority to this kind of producer, which will be fully served. The producers such that ∆ (A j ) < 0 share the remaining allowances. This can be written as follows.

Each producers with A j (p CO 2 (A)) > 0 obtains the following quantity δ j (A) of allowances

δ j (A) :=                  A j (p CO 2 (A)), if ∆ (A j ) ≥ 0, A j (p CO 2 (A) + ) + (-∆ (A j )) + J ∑ i=1 (-∆ (A i )) + Ω - J ∑ i=1 A i (p CO 2 (A))✶ {∆ (A i )≥0} ,
otherwise.

(10)

Carbon and electricity market coupling

In the following, we formalize the coordination of a producer's strategy on the CO 2 and electricity markets.

As mentioned earlier, for each producer, the marginal cost function is parametrized by the positions A of the producers on the carbon market. Indeed, producer j can obtain CO 2 emission allowances on the market to avoid penalization for (some of) its emissions. Those emissions that are not covered by allowances are penalized at a unit rate p.

A profile of an offer to buy from the producers A = (A 1 , . . . , A J ), through the CO 2 market clearing, corresponds to a unit price of p CO 2 (A) of the allowance and quantities δ j (A) of allowances bought by each producer (defined by the market rules (9),(10)).

The following minimal assumption on the CO 2 emission related to the electricity production will be restricted in Assumption 4. Assumption 3. We assume that for all producers { j = 1, . . . , J}, the emission rate (originally in CO 2 t/Mwh) q → e j (q) is positive.

For each producer, we fix the maximal amount E j of allowances to buy to κ j 0 e j (z)dz. Then, the marginal production cost function c A j (•), parametrized by the emission regulations, comes out as

q → c A j (q) = c j (q) + e j (q)p CO 2 (A), for q ∈ [0, κ CO 2 j ∧ κ j ] c j (q) + e j (q)p, for q ∈ [κ CO 2 j ∧ κ j , κ j ] (11)
where κ

CO 2 j is such that κ CO 2 j 0 e j (z)dz = δ j (A)
and where c j (•) stands for the marginal production cost without any emission regulation.

Remark 2. The results stated in Section 3.3 remain valid when the CO 2 regulation forbids uncovered electricity production. This strengthened regulation rule leads to the following marginal production cost function

q → c A j (q) = c j (q) + e j p CO 2 (A), for q ∈ [0, κ CO 2 j ∧ κ j ].
In this coupled market setting, the strategy of producer j thus makes a pair (A j , s j ). The set of admissible strategy profile is defined as

Σ Σ = {(A, s); A ∈ ❆, s ∈ S} , (12)
where in the definition of S in (2), we use

C j = c A j ; A ∈ ❆ . ( 13
)
Prices for allowances and electricity, p CO 2 ((A, s)) and p elec ((A, s)), quantities of allowances bought by each producer, δ j ((A, s)) and market shares on electricity market ϕ j ((A, s)) of each producer corresponds to any strategy profile (A, s) ∈ Σ Σ, through the market mechanisms described.

3 Nash Equilibrium analysis

Definition

We suppose that the J producers behave non cooperatively, aiming at maximizing their individual market share on the electricity market. For a strategy profile (A, s) ∈ Σ Σ, the market share of a producer j depends upon its strategy (A j , s j (•)) but also on the strategies (A -j , s -j ) of the other producers 3 . In this set-up the natural solution is the Nash equilibrium (see e.g. [START_REF] Başar | Dynamic Noncooperative Game Theory, 2nd Edition[END_REF]). More precisely we are looking for a strategy profile

(A * , s * ) = ((A * 1 , s * 1 ), • • • , (A * J , s * J )) ∈ Σ Σ
that satisfies Nash equilibrium conditions: none of the producers would strictly benefit, that is, would strictly increase its market share from a unilateral deviation. Namely, for any producer j strategy (A j , s j ) such that

((A * -j , s * -j ); (A j , s j )) ∈ Σ Σ, we have 4 ϕ j ((A * , s * )) ≥ ϕ j ((A * -j , s * -j ); (A j , s j )), (14)
where ϕ j is the quantity of electricity sold. Note that the dependency in terms of A through the marginal cost c A j is now explicit in ϕ j . Condition (14) has to be satisfied for any unilateral deviation of any producer j. In particular (14) has to be satisfied for a producer j admissible deviation (A * j , s j ) such that ((A * j , s * j ); (A * j , s j )) ∈ Σ Σ where producer j would only change its behavior on the electricity market. Consequently, Remark 3. The electricity strategy component s * of the Nash equilibrium (A * , s * ) is also a Nash equilibrium for the restricted electricity game, where producers only behave on the electricity market with marginal electricity production costs c

A * j (•), j = 1, • • • J.
The next section focuses on determining a Nash equilibrium on the game restricted to the electricity market.

Equilibrium on the power market

In this restricted set-up, we consider that the marginal costs {c j , j = 1 . . . , J} are known data, possibly fixed through the position A on the CO 2 market. In this section, we refer to S as the set of admissible strategy profiles, in the particular case where C j = {c j } for each j = 1, . . . , J.

The Nash equilibrium problem is as follows: find a strategy profile

s * = (s * 1 , . . . , s * J ) ∈ S such that ∀ j, ∀ s j = s * j , ϕ j (s * ) ≥ ϕ j (s * -j ; s j ). (15)
The following proposition exhibits a Nash equilibrium, whereby each producer must choose the strategy denoted by C j , and referred to as marginal production cost strategy. It is defined by

C j (q) = c j (q), for q ∈ Dom(c j ) p lolc , for q ∈ Dom(c j ). (16)
Proposition 1.

(i) For any strategy profile s = (s 1 , . . . , s J ), no producer j ∈ {1, . . . , J} can be penalized by deviating from strategy s j to its marginal production cost strategy C j , namely,

ϕ j (s) ≤ ϕ(s -j ;C j ). ( 17
)
In other words, for any producer j, C j is a dominant strategy.

(ii) The strategy profile C = (C 1 , . . .C J ) is a Nash equilibrium.

(iii) If the strategy profile s ∈ S is a Nash equilibrium, then we have p elec (s) = p elec (C) and for any producer j, ϕ j (s) = ϕ j (C).

Point (ii) of the previous proposition is a direct consequence of the dominance property (i). The proof of both (i) and (iii) can be found in [START_REF] Bossy | Design analysis of carbon auction market, through electricity market coupling[END_REF]. Point (ii) of the proposition exhibits a Nash equilibrium strategy profile. Clearly this equilibrium is not unique since we can easily show that a producer's given supply can follow from countless different strategies. Nevertheless point (iii) shows that for any Nash equilibrium, the associated electricity prices are the same and the quantity of electricity bought by any producer j is the same for all equilibrium profiles.

Proof. We prove the dominance property (i). Suppose that one producer, let us say producer 1, deviates and chooses C 1 instead of s 1 . We have to show that its market share cannot be reduced by this deviation. By definition of the admissibility (see Equation (2) we have

s 1 (q) ≥ C 1 (q), ∀q ∈ [0, κ 1 ].
Hence the offer functions defined by (3) satisfy

O(s 1 ; •) ≤ O(C 1 ; •).
And by adding the unchanged offers of the other producers

O O((s -1 , s 1 ); •) ≤ O O((s -1 ,C 1 ); •), (18)
where (s -1 ;C 1 ) denotes the strategy profile that includes producer 1 deviation. The minimum market clearing price (5) for strategy profile s is :

p(s) = inf{p, O O(s; p) > D(p)}.
The minimum market clearing price [START_REF] Carmona | The valuation of clean spread options: Linking electricity, emissions and fuels[END_REF] for strategy profile (s -1 ,C 1 ) is :

p((s -1 ;C 1 )) = inf{p, O O((s -1 ;C 1 ); p) > D(p)}
The inequality (18) together with the fact that the demand D(•) is a decreasing function imply that

p((s -1 ;C 1 )) ≤ p(s).
From which, together with (6) we deduce that

p elec ((s -1 ;C 1 )) ≤ p elec (s).
Now let us show that producer 1 does not reduce its market share by deviating from

s 1 (•) to C 1 (•) , that is that ϕ 1 (s -1 ,C 1 ) ≥ ϕ 1 (s).
For the sake of clarity we adopt, in this paragraph, the following notation: We first consider the case where p elec sC < p elec s .

By definition of the minimum clearing price p sC , the fact that D(p s ) ≤ D(p sC ) and the fact that O O((s -1 ;C 1 ); •) is non-decreasing, we have

D(p s ) ≤ D(p sC ) ≤ O O((s -1 ;C 1 ); p sC ) ≤ O O((s -1 ;C 1 ); p elec sC ) ≤ O O((s -1 ;C 1 ); psC )
Hence, for any π s ∈ {p s , p elec s , ps } and any π sC ∈ {p sC , p elec sC , psC } we have

D(π s ) ≤ O O((s -1 ;C 1 ); π sC ).
Since D(π s ) ≤ D(π sC ), we have

D(π s ) ≤ O O((s -1 ;C 1 ); π sC ) ∧ D(π sC ),
and finally

O O((s -1 ; s 1 ), π s ) ∧ D(π s ) ≤ O O((s -1 ;C 1 ); π sC ) ∧ D(π sC ).
From the market clearing we get

ϕ 1 (s -1 , s 1 ) -ϕ 1 (s -1 ,C 1 ) =O O((s -1 ; s 1 ), p elec s ) ∧ D(p elec s ) -O O((s -1 ;C 1 ); p elec sC ) ∧ D(p elec sC ) + ∑ j>1 (ϕ j (s -1 ,C 1 ) -ϕ j (s -1 , s 1 )) .
Let us denote

E (p elec s ) = j ∈ {2, . . . , J} s.t. ∆ -O(s j ; p elec s ) > 0 .
We have

ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) =O O((s -1 ; s 1 ); p elec s ) ∧ D(p elec s ) -O O((s -1 ;C 1 ); p elec sC ) ∧ D(p elec sC ) + ∑ j>1, j / ∈E (p elec s ) ϕ j (s -1 ;C 1 ) -O(s j ; p elec s ) + ∑ j>1, j∈E (p elec s ) (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 )) ≤O O((s -1 ; s 1 ); p elec s ) ∧ D(p elec s ) -O O((s -1 ;C 1 ); p elec sC ) ∧ D(p elec sC ) + ∑ j>1, j / ∈E (p elec s ) O(s j ; p elec sC ) -O(s j ; p elec s ) + ∑ j>1, j∈E (p elec s ) (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 )) Since p elec sC ≤ p elec s we get ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) ≤ ∑ j>1 j∈E (p elec s ) (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 ))
But for any j ∈ E (p elec s ), the quantity O(s j ; p - s ) ≤ ϕ j (s -1 ; s 1 ). Hence since p sC < p s and O(s j ; •) is non decreasing, we get

O(s j ; p - sC ) ≤ O(s j ; p - s ) ≤ ϕ j (s -1 ; s 1 ).
For such j > 1 we thus have

ϕ j (s -1 ;C 1 )-ϕ j (s -1 ; s 1 ) ≤ ϕ j (s -1 ;C 1 )-O(s j ; p elec s -) ≤ ϕ j (s -1 ;C 1 )-O(s j ; p elec sC ) ≤ 0.
from which it follows that

ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) ≤ 0.
Now consider the case where p elec s = p elec sC := p elec .

Due to the rule (6), we necessarily have p s = p sC := p.

• If O O((s -1 , s 1 ); p elec ) ≤ O O((s -1 ,C 1 ); p elec ) ≤ D(p elec ),
hence by the market clearing

ϕ 1 (s -1 ; s 1 ) = O(s 1 ; p elec ) ≤ O(C 1 ; p elec ) = ϕ 1 (s -1 ;C 1 ). • If O O((s -1 ; s 1 ); p elec ) ≤ D(p elec ) ≤ O O((s -1 ;C 1 ); p elec ), ϕ 1 (s -1 ; s 1 ) = O(s 1 ; p elec ) ≤D(p elec ) -∑ j>1 ϕ j (s -1 ; s 1 ) = D(p elec ) -∑ j>1 O(s j ; p elec ) ≤D(p elec ) -∑ j>1 ϕ j (s -1 ;C 1 ) = ϕ 1 (s -1 ;C 1 ). • If D(p elec ) ≤ O O((s -1 , s 1 ); p elec ) ≤ O O((s -1 ,C 1 ); p elec ),
by the market clearing we get

ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) =O O((s -1 ; s 1 ), p elec ) ∧ D(p elec ) -O O((s -1 ;C 1 ); p elec ) ∧ D(p elec ) + ∑ j>1 (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 )) ≤ ∑ j>1 (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 )) ≤ ∑ j>1, j∈E (p elec ) (ϕ j (s -1 ;C 1 ) -ϕ j (s -1 ; s 1 )).
From (7), we have for j ∈ E ( p)

ϕ j (s -1 ; s 1 ) = O(s j , p -) + ∆ -O(s j ; p) (D( p) -O O((s -1 ; s 1 ), p -)) ∆ -O O((s -1 ; s 1 ), p)
and

ϕ j (s -1 ;C 1 ) = O(s j ; p -) + ∆ -O(s j ; p) (D( p) -O O((s -1 ;C 1 ); p -)) ∆ -O O((s -1 ;C 1 ); p) .
Hence, if E ( p) is non empty then at least one producer exists, j = 1 such that ∆ -O(s j ; p) > 0. and from the desegregation of O O and definition of ∆ -it results that

ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) = ∑ j>1, j∈E ( p) ∆ -O(s j , p) (D( p) -O O(s -1 ; p -) -O(C 1 ; p -)) O O((s -1 ;C 1 ); p) -O O(s -1 ; p -) -O(C 1 ; p -) - (D( p) -O O(s -1 , p -) -O(s 1 , p -)) O O((s -1 , s 1 ), p) -O O(s -1 ; p -) -O(s 1 ; p -) We note that 0 < O O((s -1 ; s 1 ); p) -O O(s -1 ; p -) -O(C 1 ; p -) ≤ O O((s -1 ;C 1 ); p) -O O(s -1 ; p -) -O(C 1 ; p -) and that D( p) -O O((s -1 ;C 1 ); p -) > 0 by assumption. Then ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) ≤ ∑ j>1, j∈E ( p) ∆ -O(s j ; p) × (D( p) -O O(s -1 ; p-) -O(C 1 ; p-)) O O((s -1 ; s 1 ); p) -O O(s -1 ; p -) -O(C 1 ; p -) - (D( p) -O O(s -1 ; p-) -O(s 1 ; p-)) O O((s -1 ; s 1 ); p) -O O(s -1 ; p -) -O(s 1 ; p -) Since D( p) ≤ O O((s -1 ; s 1 ); p) and O(C 1 ; p -) ≥ O(s 1 ; p -), we can deduce that ϕ 1 (s -1 ; s 1 ) -ϕ 1 (s -1 ;C 1 ) ≤ 0.
This follows from the fact that x → Ax Bx with A ≤ B, is decreasing with respect to x.

⊓ ⊔

Coupled market design using the Nash equilibrium

From this point we restrict our attention to a particular market design. In the following, the scope of the analysis applies to a special class of producers, a specific electricity market price clearing (satisfying Definition 1) and a range of quantities Ω of allowances available on the CO 2 market. Although not necessary, the following restriction simplifies the development.

Assumption 4. On the producers. Each producer j operates a single production unit, for which (i) The marginal cost contribution that does not depend on the producer positions A in the CO 2 market is constant, q → c j (q) = c j . The related emission rate q → e j (q) = e j is also assumed to be a positive constant. (ii) The producers are different pairwise: ∀i, j ∈ {1, • • • J}, (c i , e i ) = (c j , e j ).

For each producer, the maximal amount E j of allowances to buy is now e j κ j .

As a consequence of Assumption 4, the marginal production cost in (11) simply writes as

q → c A j (q) =        c j + e j p CO 2 (A), for q ∈ [0, δ j (A) e j ∧ κ j ] c j + e j p, for q ∈ [ δ j (A) e j ∧ κ j , κ j ] (19)
For a given strategy profile on the electricity market, Definition 1 gives a range of possible determinations for the electricity price. Previously, the analysis of the Nash Equilibrium restricted to the electricity market did not require a precise clearing price determination. Nevertheless to extend our analysis of the coupling we need to explicit this determination and assume the following: Assumption 5. On the electricity market. For a given strategy profile s of the producers, the clearing price of electricity is p elec (s). The market rule fixes p elec (•) = p(•) or p elec (•) = p(•) as defined in [START_REF] Carmona | The valuation of clean spread options: Linking electricity, emissions and fuels[END_REF].

We will illustrate below that this choice of clearing price ensures the increasing behavior of p elec (•) in terms of the carbon price (see Lemma 1).

The quantity Ω of CO 2 allowances available on the market plays a crucial role in the market design. As a matter of fact, if this quantity is too high, its market price will drop to zero, leaving the market incapable of fulfilling its role of decreasing CO 2 emissions. Therefore we clearly need to make an assumption that restricts the number of allowances available. Capping the maximum quantity of allowances available requires information on which producers are willing to obtain allowances. This is the objective of the following paragraph where we define a willing to buy function that plays a central role in the analysis of Nash equilibria.

Willing to buy functions

In this paragraph, we aim at guessing a Nash equilibrium candidate. We base our reasoning on the dominant strategies on the electricity market alone (see Proposition 1). Remark 3 allows us to fix the electricity market strategy as a marginal production cost strategy, given the marginal cost functions C A = {c A j , j = 1, . . . J} imposed by the output of the CO 2 clearing, as in (19).

In particular, when A ∈ ❆, we observe that the strategies (A, {c A j , j = 1, . . . J}) are in the set of admissible strategies defined in (12).

From now on, all the strategy profiles that we consider on the carbon market are assumed to be admissible.

In the following, as the discussion will mainly focus on the impact of strategies A through the carbon market, we denote the electricity market output as:

p elec (A) instead of p elec (C A ) (ϕ 1 (A), . . . , ϕ J (A)) instead of (ϕ 1 (C A ), . . . , ϕ J (C A )). (20)
To begin with, we consider an exogenous CO 2 cost τ similar to a CO 2 tax: the producers' marginal cost becomes for any τ ∈ [0, p], c τ j (•), c τ j (q) = c j + τe j , for q ∈ [0, κ j ], j = 1, . . . , J.

In this tax framework, the dominant strategy on the electricity market is also parametrized by τ as C τ = {c τ j , j = 1, . . . J} defined in (16). The clearing electricity price and quantities derive as

p elec (τ) = p elec (C τ ) (ϕ 1 (τ), . . . , ϕ J (τ)) = (ϕ 1 (C τ ), . . . , ϕ J (C τ )). (21)
Price p elec (τ) will be referred to as the taxed electricity price, by contrast with price p elec (A) issued from the marginal production cost strategy that results from the position A on the carbon market.

Remark 4. Considering a carbon tax τ and a carbon market strategy A such that τ = p CO 2 (A), we emphasize the fact that the corresponding electricity prices are not equivalent, but we always have the following inequality p elec (τ) ≤ p elec (A).

This comes from the fact that C τ and C A differ only on the width of their steps, and that O(c A i ; •) ≤ O(c τ i ; •). We start with the following, the proof of which is set out at the end of this subsection: Lemma 1. Under Assumption 5, the map τ → p elec (τ) is increasing and right continuous.

We determine the willing-to-buy-allowances functions W j (•) and W (•), as follows:

W j (τ) = e j ϕ j (τ) and W (τ) = J ∑ j=1 W j (τ) (22)
For producer j, W j is the quantity of emissions it would produce under the penalization τ, and consequently the quantity of allowances it would be ready to buy at price τ. Given the CO 2 value τ, the total amount W (τ) represents the allowances needed to cover the global emissions generated by the players who have won electricity market shares. We also define the functions W j (τ) = e j κ j ✶ {ϕ j (τ)>0} , and

W (τ) = J ∑ j=1 W j (τ) (23)
Given that the CO 2 value τ, W (τ) is the amount of allowances needed by the producers who have won electricity market shares and want to cover their overall production capacity κ j . Obviously we have

W (τ) ≤ W (τ), ∀τ ∈ [0, p].
We now can state our last design assumption Assumption 6. On the carbon market design. The available allowances Ω satisfy

W (p) < Ω < W (0).
Assumption 6 allows us to define two prices of particular interest for the construction of the equilibrium strategy:

τ lower = sup{τ ∈ [0, p] s.t. W (τ) > Ω }, (24)
and

τ higher = sup{τ ∈ [0, p] s.t. W (τ) > Ω }. (25)
Observe that we always have τ lower ≤ τ higher .

Lemma 2. The function τ → W (τ) is non increasing:

W (t ′ ) ≤ W (t), ∀ 0 ≤ t < t ′ ≤ p.
We end this subsection by successively giving the proofs of Lemmas 1 and 2.

Proof of Lemma 1. Although the result of this lemma is intuitive, the proof is rather technical. This is due to our assumptions, in particular regarding demand, that allow the demand function to have discontinuity points and some non-elasticity areas (see Assumption 1). More precisely, if we define the map τ → O O(τ; p) by

O O(τ; p) = J ∑ i=1 O(C τ j (•); p) = J ∑ i=1 κ i ✶ {p≥c i +τe i } = J ∑ i=1 κ i ✶ {τ≤ p-c i e i } ,
then we can observe that, for any p > 0 far enough from the c i , for any ε ≥ 0 We call S κ = {p c ; D(p c ) = ∑ κ i }, the set of prices that make demand coincide with some accumulation of production capacities.

O O(τ + ε; p) ≤ O O(τ; p)
We observe that p elec (τ) ∈ {c i + τe i , i = 1, . . . , j} ∪ S D ∪ S κ . In particular, from Definition 1,

p(τ) = inf{p > 0; O O(τ; p) > D(p)},
and we obtain that

D(p(τ + ε)) ≤ O O(τ + ε; p(τ + ε)) ≤ O O(τ; p(τ + ε))
from which we conclude that p(τ + ε) ≥ p(τ). Now we prove the right continuity of τ → p(τ). Let us fix a τ.

(i) We first consider the case D(p(τ)) < O O(τ; p(τ)).

This means that p(τ) is of the form c ℓ + τℓ, for a given ℓ. Then when ε > 0 is small enough, we also have p(τ + ε) = c ℓ + (τ + ε)e ℓ . Indeed, D(c ℓ + (τ + ε)e ℓ ) ≤ D(c ℓ + τe ℓ ) and for a small enough ε,

O O(τ; c ℓ + τe ℓ ) = κ ℓ + ∑ i =ℓ κ i ✶ {τ≤ c ℓ -c i 1-e i /e ℓ } = O O(τ + ε; c ℓ + (τ + ε)e ℓ ) Thus, D(c ℓ + (τ + ε)e ℓ ) < O O(τ + ε; c ℓ + (τ + ε)e ℓ ) which implies that p(τ) + e ℓ ε = c ℓ + (τ + ε)e ℓ ≥ p(τ + ε) and hence e ℓ ε ≥ p(τ + ε) -p(τ).
(ii) We consider next the case

D(p(τ)) > O O(τ; p(τ)).
This means that p(τ) ∈ S D is at a discontinuity point, say p d of the demand, p(τ) = p d . Then, for any δ > 0,

D(p(τ) + δ ) < O O(τ; p(τ) + δ ). But, O O(τ; p d + δ ) = J ∑ i=1 κ i ✶ {τ≤ p d +δ -c i e i }
and we can choose δ to be small enough so that τ = p d +δ -c i e i

. Then, for a small enough ε,

D(p(τ) + δ ) < O O(τ; p(τ) + δ ) = O O(τ + ε; p(τ) + δ )
which implies that p(τ) + δ ≥ p(τ + ε), so we obtain δ ≥ p(τ + ε)p(τ) ≥ 0.

(iii) We consider now the case D(p(τ)) = O O(τ; p(τ)).

This means that p(τ) ∈ S κ , say p(τ) = p c Then, for any δ > 0,

D(p(τ) + δ ) < O O(τ; p(τ) + δ ). But, O O(τ; p c + δ ) = J ∑ i=1 κ i ✶ {τ≤ pc+δ -c i e i }
and we can choose δ small enough such that τ = p c +δ -c i e i

. Then, for ε small enough,

D(p(τ) + δ ) < O O(τ; p(τ) + δ ) = O O(τ + ε; p(τ) + δ )
which implies that p(τ) + δ ≥ p(τ + ε), so we get

δ ≥ p(τ + ε) -p(τ) ≥ 0.
The right-continuity of τ → p(τ) follows, by definition as p(τ) is a continuous transformation of p(τ).

⊓ ⊔

Proof of Lemma 2. The proof consists in a complete analysis of the entire combination of situations, but each situation is elementary. We present some cases here, but we relegate the rest in the appendix.

Let us suppose the opposite, that is there exists 0 ≤ t < t ′ ≤ p such that the emission levels are W (t ′ ) > W (t).

We define the function τ → I(τ) valued in the subsets of {1, . . . , J} that lists the producers in the electricity market producing at tax level τ:

i ∈ I(τ) if ϕ i (τ) > 0.
In particular we have for all τ ∈ [0, p],

W (τ) = ∑ i∈I(τ) e i ϕ i (τ).

(i) We first examine the situation I(t ′ ) = I(t).

To shorten the expressions, we adopt the following shorten notation

I(t) = I and I(t ′ ) = I ′ .
(i-a) If ∑ i∈I ϕ i (t) = D(t) then, from the demand constraint (DC) and the emission levels hypothesis (EH), we have

∑ i∈I ϕ i (t) = D(t) ≥ D(t ′ ) ≥ ∑ i∈I ′ ϕ i (t ′ ) (DC) ∑ i∈I ϕ i (t)e i < ∑ i∈I ′ ϕ i (t ′ )e i . (EH)
We denote by I the subset of I of index such that c i + te i = p(t). In particular, when j ∈ I \ I, then ϕ j (t) = κ j .

Note that there exists at most one index (say ℓ) in the set

I ∩ I ′ . If j ∈ I \ I ′ , if k ∈ I ′ \ I,
then, by the definition of the sets c j + e j t =c ℓ + e ℓ t c j + e j t ′ <c ℓ + e ℓ t ′ c k + e k t <c j + e j t c k + e k t ′ =c ℓ + e ℓ t ′ c k + e k t <c ℓ + e ℓ t c j + e j t ′ <c k + e k t

From which, we easily deduce that

max{e j , j ∈ I \ I ′ } < e ℓ < min{e k , k ∈ I ′ \ I}. ( 26
)
Now, we decompose the sets I and I ′ in the demand constraint (DC) and the emission levels hypothesis (EH) as follows:

∑ n∈I\ I∪ I ′ κ n + ϕ ℓ (t) + ∑ i∈ I\ I ′ ϕ i (t) + ∑ k∈ I ′ \ I κ k ≥ ∑ n∈I\ I∪ I ′ κ n + ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ κ i + ∑ k∈ I ′ \ I ϕ k (t ′ ) (DC) ∑ n∈I\ I∪ I ′ e n κ n + e ℓ ϕ ℓ (t) + ∑ i∈ I\ I ′ e i ϕ i (t) + ∑ k∈ I ′ \ I e k κ k < ∑ n∈I\ I∪ I ′ e n κ n + e ℓ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ e i κ i + ∑ k∈ I ′ \ I e k ϕ k (t ′ ) (EH) .
After simplification, we obtain

ϕ ℓ (t) + ∑ i∈ I\ I ′ ϕ i (t) + ∑ k∈ I ′ \ I κ k ≥ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ κ i + ∑ k∈ I ′ \ I ϕ k (t ′ ) (DC) e ℓ ϕ ℓ (t) + ∑ i∈ I\ I ′ e i ϕ i (t) + ∑ k∈ I ′ \ I e k κ k < e ℓ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ e i κ i + ∑ k∈ I ′ \ I e k ϕ k (t ′ ) (EH) . Assume first that ϕ ℓ (t) + ∑ i∈ I\ I ′ ϕ i (t) ≥ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ κ i . Equivalently, we have ϕ ℓ (t) -ϕ ℓ (t ′ ) ≥ ∑ i∈ I\ I ′ (κ i -ϕ i (t))
and from ( 26),

e ℓ ϕ ℓ (t) -ϕ ℓ (t ′ ) ≥ ∑ i∈ I\ I ′ e i (κ i -ϕ i (t)).
By combining the above with the emission levels hypothesis (EH), we obtain the following contradiction:

∑ k∈ I ′ \ I e k κ k < ∑ k∈ I ′ \ I e k ϕ k (t ′ ). Assume now that ϕ ℓ (t) + ∑ i∈ I\ I ′ ϕ i (t) < ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ κ i . Multiplying the de- mand constraint (DC) by ê := min{e k , k ∈ I ′ \ I}, we get ∑ k∈ I ′ \ I e k (κ k -ϕ k (t ′ )) ≥ ê ϕ ℓ (t) -ϕ ℓ (t ′ ) + ê ∑ i∈ I\ I ′ (κ i -ϕ i (t))
But from (EH) and (26), we also have

∑ k∈ I ′ \ I e k (κ k -ϕ k (t ′ )) < e ℓ ϕ ℓ (t) -ϕ ℓ (t ′ ) + e ℓ ∑ i∈ I\ I ′ (κ i -ϕ i (t)) Then 0 ≥ ( ê -e ℓ ) ϕ ℓ (t) -ϕ ℓ (t ′ ) + ( ê -e ℓ ) ∑ i∈ I\ I ′ (κ i -ϕ i (t))
which contradicts our assumption.

(i-b) If ∑ i∈I ϕ i (t) < D(t) then, for all i ∈ I, ϕ i (t) = κ i and (EH) is necessarily false.
We complete the proof of the lemma with the case I ′ = I in Appendix 5.1. ⊓ ⊔

Towards an equilibrium strategy

In the following we do not explicit a Nash equilibrium. Instead we establish the existence of an interval in which the coupled carbon market Nash equilibria evolve. We demonstrate that there is no possible deviation enabling a Nash equilibrium outside the interval establishing the carbon price. This derives from a series of lemmas in which we explicit the market.

To begin with, we propose an analysis of the three following statements:

Lower price strategy

Consider any strategy

A W = (A W 1 , . . . , A W J ) such that A W j (τ) = W j (τ lower ), for 0 ≤ τ ≤ τ lower anything admissible, for τ > τ lower . ( 27
) Lemma 3. (i) p CO 2 (A W ) ≥ τ lower .
(ii) In the case where p CO 2 (A W ) = τ lower , there is no unilateral favorable deviation that clears the market at a CO 2 price lower than τ lower .

Proof.

Point (i) is a consequence of the definition of τ lower = sup{τ ∈ [0, p], s.t. W (τ) > Ω }. Since A W j (τ) = W j (τ) for τ ≤ τ lower , it follows that p CO 2 (A W ) = sup{τ ∈ [0, p], s.t. ∑ j A W j (τ) > Ω } ≥ τ lower .
To prove (ii), first note that, since we assume p CO 2 (A W ) = τ lower , we have ϕ j (A W ) = ϕ j (τ lower ) = 1 e j W j (τ lower ).

Suppose one producer, say Producer 1, deviates and chooses

A 1 (•) instead of A W 1 (•). Suppose the new carbon price τ := p CO 2 (A W -1 ; A 1 ) < τ lower . Since A W j ( τ + ) = A W j ( τ) for j = 1, necessarily we have A 1 ( τ + ) < A 1 ( τ), by definition of τ. Then ∆ (A 1 ) > 0 and it follows that δ 1 (A W -1 ; A 1 ) = δ 1 (A W ).
From the fact that the marginal production costs of all Producers have decreased (the emission cost is τ instead of τ lower ), it comes that p elec (A W -1 ; A 1 ) ≤ p elec (A W ). This means that the part of electricity production capacity that is not covered by allowances (and hence penalized with p) has a marginal production cost greater than p elec (A W -1 ; A 1 )). We then deduce that, at best,

ϕ 1 (A W -1 ; A 1 ) ≤ ϕ 1 (A W ). ⊓ ⊔ Lemma 4. Suppose A is such that p CO 2 (A) < τ lower . Then A is not a Nash equilib- rium.
Proof. To prove this lemma we exhibit an unilateral deviation from A of a producer, improving its market share on the electricity market.

• Assume first that at least one producer exists, say Producer 1, such that ϕ 1 (A) < κ 1 and there exists a tax value τ1 such that p CO 2 (A) < τ1 ≤ τ lower and,

W 1 (τ) = e 1 κ 1 for any τ ∈ [p CO 2 (A), τ1 ].
This means that Producer 1 may sell κ 1 , for any tax level τ in [p CO 2 (A), τ1 ], and consequently we have

c 1 + τe 1 < p elec (τ) for τ in [p CO 2 (A), τ1 ].
Consider a deviation A 1 of player 1, such that the resulting clearing price on CO 2 market, p CO 2 (A -1 ; A 1 ) ∈ [p CO 2 (A), τ1 ]. From Remark 4, we have

p elec (p CO 2 (A -1 ; A 1 )) ≤ p elec (A -1 ; A 1 ).
This means that Producer 1 may sell its overall covered capacity:

ϕ 1 (A -1 ; A 1 ) = 1 e 1 δ 1 (A -1 ; A 1 ).
Now we define τ → A 1 (τ) as follows, for ε > 0 arbitrarily small, for p CO 2 (A) < τ,

A 1 (τ) =e 1 κ 1 ✶ { ∑ j =1 A j (τ) + δ 1 (A) < Ω } ✶ {p CO 2 (A) < τ ≤ τ1 } + Ω -∑ j>1 A j (τ) -ε ✶ { ∑ j =1 A j (τ) + δ 1 (A) ≥ Ω } ✶ {p CO 2 (A) ≤ τ ≤ τ1 } + A 1 (τ)✶ {τ > τ1 } and A 1 (p CO 2 (A)) =e 1 κ 1 . Note that A 1 (τ) ≥ A 1 (τ) for p CO 2 (A) ≤ τ ≤ τ1 , and consequently p CO 2 (A -1 ; A 1 ) ≥ p CO 2 (A). If p CO 2 (A -1 ; A 1 ) > p CO 2 (A), then e 1 ϕ 1 (A -1 ; A 1 )) = δ (A -1 ; A 1 ) > δ (A) ≥ e 1 ϕ 1 (A)
, and we get our favorable deviation.

If p CO 2 (A -1 ; A 1 ) = p CO 2 (A), we observe that when ∆ (A 1 ) ≥ 0, we also have ∆ ( A 1 ) = 0. Then by CO 2 market clearing mechanism, Producer 1 get e 1 κ 1 allowances instead of δ (A) and strictly improves its electricity market share. when ∆ (A 1 ) < 0, we have A 1 (p CO 2 (A) + ) > A 1 (p CO 2 (A) + ), that also insure that Producer 1 increase δ (A -1 ; A 1 ) > δ (A) (see (10)).

• Assume now that all producers are either such that ϕ j (A) = κ j or such that ϕ j (A) < κ j and W j (p CO 2 (A) + ) < e j κ j . Among the second category, there exists at least one producer (say Producer 1) such that ϕ 1 (A) < ϕ 1 (p CO 2 (A)) with ϕ 1 (p CO 2 (A)) > 0 (unless to contradict p CO 2 (A) < τ lower ). Here we have used the notation (20) and (21). W 1 (p CO 2 (A) + ) < e 1 κ 1 means that c 1 +e 1 p CO 2 (A) = p elec (p CO 2 (A)) (as p elec (•) is right-continuous).

A strictly favorable deviation A 1 of Producer 1, thus consists in increase its ask at the price p CO 2 (A) + , in order to increase its δ ((A -1 , A 1 )) (see (10)):

A 1 (τ) = Ω -∑ j>1 A j (τ) -ε ✶ {p CO 2 (A) < τ} + e 1 κ 1 ✶ {p CO 2 (A) = τ} . Then p CO 2 ((A -1 ; A 1 )) = p CO 2 (A), A 1 (p CO 2 (A)) ≥ A 1 (p CO 2 (A)), but A 1 (p CO 2 (A) + ) > A 1 (p CO 2 (A) +
), for ε sufficiently small. This last inequality guaranties that δ 1 ((A -1 ; A 1 )) > δ 1 (A) and finally ϕ 1 (p CO 2 (A)) ≥ ϕ 1 ((A -1 ; A 1 )) > ϕ 1 (A).

⊓ ⊔

High price strategy

Consider any strategy

A W = (A W 1 , • • • , A W J ) such that A W j (τ) =
anything admissible, for τ ≤ τ higher W j (τ), for τ > τ higher . (28) Lemma 5.

(i) p CO 2 (A W ) ≤ τ higher .

(ii) In the case where p CO 2 (A W ) = τ higher , there is no unilateral favorable deviation that clears the market at a CO 2 price higher than τ higher .

Proof. Point (i) follows directly from the definition of τ higher .

To prove (ii), suppose one producer, say Producer 1, chooses its strategy A 1 (•) instead of A W 1 (•), and that the resulting CO 2 price is τ := p CO 2 (A W -1 ; A 1 ) > τ higher .

Necessarily, due to the definition of A W , this means that W 1 ( τ) = 0, which in turn means that c 1 + τe 1 > p elec ( τ). To conclude, it is sufficient to notice that any Producer j = 1 obtains what he asks for, i.e. δ j (A W -1 ; A 1 ) = W j ( τ + ), from which it follows that the coupled electricity price equals the taxed electricity price: p elec (A W -1 ; A 1 ) = p elec ( τ), and then ϕ 1 (A W -1 ; A 1 ) = W i ( τ) = 0 and the deviation of 1 is not favorable.

⊓ ⊔ Lemma 6. Suppose A is such that p CO 2 (A) > τ higher . Then A is not a strong Nash equilibrium.

Proof. Given A, such that p CO 2 (A) > τ higher , we consider the coalition of producers K such that for j ∈ K , δ j (A) > 0 whereas W j (p CO 2 (A)) = 0. Consider the following cooperating deviation of K :

A j (•) = A W j (•), for j ∈ K .
Then p CO 2 (A -K ; A K ) < p CO 2 (A) and at least for one member of K , δ j (A) > 0 when W j (p CO 2 (A)) > 0. This means that ϕ j (A -K ; A K ) > 0 which is a strictly favorable deviation of j, whereas the situation is unchanged for the others in K that still produce nothing. We exhibit a coalition that allows a deviation from A that benefits to all of its members. Then A is not a strong Nash equilibrium. ⊓ ⊔

Intermediate strategy

Consider any strategy profile B = (B 1 , • • • , B J ) satisfying the following:

B j (τ) =    W j (τ),
for τ > τ higher anything admissible, for τ lower < τ ≤ τ higher W j (τ lower ), for τ ≤ τ lower .

(29) This is not in general an equilibrium, nevertheless we have the following properties :

Lemma 7. (i) p CO 2 (B) ∈ [τ lower , τ higher ].
(ii) If there exists a favorable deviation from a producer, say Producer 1, that chooses B 1 instead of B 1 , such that p CO 2 (B -1 ; B 1 ) < τ lower , then there exists another favorable deviation B 1 such that p CO 2 (B -1 ; B 1 ) = τ lower , and such that ϕ 1 (B -1 ; B 1 ) ≥ ϕ 1 (B -1 ; B 1 ).

Proof. Point (i) follows directly from Lemma 3-(i) and Lemma 5-(i).

To prove point (ii), we denote τ B := p CO 2 (B -1 ; B 1 ) < τ lower . We first observe that, as producers j = 1 are served first on the carbon market,

δ 1 ( B) = Ω -∑ j =1 W j (τ lower ).
Define B 1 , the new deviation of 1 as

B 1 = B 1 (τ), for τ > τ lower , W 1 (τ lower ), for τ ≤ τ lower .
The CO 2 price is now fixed to p CO 2 ((B -1 ; B 1 )) = τ lower , and from the CO 2 market mechanism it follows that

δ 1 ((B -1 ; B 1 )) ≥ δ 1 ((B -1 ; B 1 )).
Since B j (τ B ) = B j (τ + B ) = W j (τ lower ) for any j = 1, it comes that δ 1 ((B -1 ; B 1 )) = Ω -∑ j =1 W j (τ lower ). Indeed, for strategy (B -1 ; B 1 ), the producers j = 1 such that B j (τ lower + ) < W j (τ lower ) receive a quantity of quotas δ j ((B -1 ; B 1 ) ≤ W j (τ lower ), from which it follows that δ 1 ((B -1

; B 1 )) = Ω -∑ j δ j ((B -1 ; B 1 )) ≥ δ 1 ((B -1 ; B 1 )).
We also deduce that ϕ 1 ((B -1 ; B 1 )) = 1 e 1 δ 1 ((B -1 ; B 1 )). To conclude, it is sufficient to notice that ϕ 1 ((B -1

; B 1 )) = 1 e 1 δ 1 ((B -1 ; B 1 )) ≥ 1 e 1 δ 1 ((B -1 ; B 1 )) ≥ ϕ 1 ((B -1 ; B 1 )). ⊓ ⊔
The following corollary is a direct consequence of Lemmas 4, 5, 6 and 7.

Corollary 1. Let E be a (strong) Nash equilibrium. Then the following E ′ is also a (strong) Nash equilibrium:

E ′ j (τ) =    W j (τ), for τ > τ higher E j (τ),
for τ lower < τ ≤ τ higher W j (τ lower ), for τ ≤ τ lower (30) It is worthy of mentioning that same results of the section 3.3 apply when producers have an electricity production power plants portfolio, or when one modify the maximal amount E j of allowances to buy. The interval remain relevant, with straightforward modification on the functions W j (•) and W j (•) and the related price bounds τ lower and τ higher .

Conclusion

Once CO 2 is emitted into the atmosphere, it remains there for more than a century. Estimating its value is an essential indicator for efficiently defining policy. Carbon valuation is crucial for designing markets that foster emission reductions. In this paper, we established the links between an electricity market and a carbon auction market through an analysis of electricity producers' strategies. We proved that they lead to the interval where relevant Nash equilibria evolve, enabling the computation of equilibrium prices on both markets. It has been established that Nash equilibria driver on the carbon market rely more on the producers' emission rate than on their marginal costs. For each producer, each equilibrium derives the level of electricity produced and the CO 2 emissions covered.

For a given design and set of players, the information provided by the interval may be interpreted as a diagnosis of market behavior in terms of prices and volume.

In addition to this analysis of the Nash equilibrium we plan to analyze the electricity production mix, with a particular focus on renewable shares that do not participate in emissions.

Acknowledgements This work was partly supported by Grant 0805C0098 from ADEME.

Appendix

End of the proof of Lemma 2 (ii) We examine the situation I(t ′ ) = I(t)

To shorten the expressions, we still adopt the following shorten notation

I(t) = I and I(t ′ ) = I ′ . I(t) ∩ I ′ (t) = I I ′
We break down I and I ′ into the sets I I ′ , I\I ′ and I ′ \I. We denote by I the set of index i ∈ I such that c i + te i = p(t). In particular, when j ∈ I\ I, then ϕ j (t) = κ j .

We first derive some generic relations between the emission rates for these sets. Among the indexes in the set I I ′ , we observe that at most one index exists (say ℓ) in the set

I ∩ I ′ . If j ∈ I\ I ′ , if k ∈ I ′ \ I,
j , j ∈ I\I ′ } > (e ℓ , ê) ∨ max{e k , k ∈ I ′ \I} max{e k , k ∈ I ′ \I} < (e ℓ , ê′ ) ∧ min{e j , j ∈ I\I ′ }. (33)
We divide your analysis in cases. In the first one the demand is fully satisfied for the price p elec (t).

(ii-a) If ∑ i∈I ϕ I (t) = D(p elec (t)) ∑ i∈I\I ′ ϕ I + ∑ i∈ I I ′ ϕ i (t) = D(p elec (t)) ≥ D(p elec (t ′ )) ≥ ∑ i∈ I I ′ ϕ i (t ′ ) + ∑ i∈I ′ \I ϕ i (t ′ ) (DC) ∑ i∈I\I ′ ϕ i (t)e i + ∑ i∈ I I ′ ϕ i (t)e i < ∑ i∈ I I ′ ϕ i (t ′ )e i + ∑ i∈I ′ \I ϕ i (t)e i (EH)
We must then examine the following two subcases, relative to the situations where the demand is satisfied or not at the price p elec (t ′ ).

(ii-a-1)

If ∑ i∈I ′ ϕ i (t ′ ) < D(p elec (t ′ )), then ϕ i (t ′ ) = κ i for all i ∈ I ′ and ∑ j∈I\I ′ ϕ j (t) + ∑ i∈ I I ′ ϕ i (t) > ∑ i∈ I I ′ κ i + ∑ k∈I ′ \I κ k (DC) ∑ j∈I\I ′ ϕ j (t)e j + ∑ i∈ I I ′ ϕ i (t)e i < ∑ i∈ I I ′ κ i e i + ∑ k∈I ′ \I κ k e k (EH)
As ϕ i (t) = κ i when i ∈ (I\ I) ∩ I I ′ , we can simplify the two sides of (DC) and (EH) by the sum over (I\ I) ∩ I I ′ . The remaining part of

I I ′ is {ℓ} ∪ I\ I ′ ∩ I I ′ : ∑ j∈I\I ′ ϕ j (t) + ϕ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ ϕ i (t) > κ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ κ i + ∑ k∈I ′ \I κ k (DC) ∑ j∈I\I ′ e j ϕ j (t) + e ℓ ϕ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ e i ϕ i (t) < e ℓ κ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ e i κ i + ∑ k∈I ′ \I e k κ k (EH)
Then, we multiply (DC) by ē := (e ℓ , ê) ∨ max{e k , k ∈ I ′ \I}, and we obtain by (33)

∑ j∈I\I ′ e j ϕ j (t) + ēϕ ℓ + ē ∑ i∈ I\ I ′ ∩ I I ′ ϕ i (t) > ēκ ℓ + ē ∑ i∈ I\ I ′ ∩ I I ′ κ i + ∑ k∈I ′ \I e k κ k .
We subtract with (EH) : (ii-b-2) If ∑ i∈I ′ ϕ i (t ′ ) = D(p elec (t ′ )), we go back to the analysis of the case (ii-a-2), with the main difference that all quantities ϕ i (t) are now equal to κ i . We go to inequalities (34) and (35) which are simplified as the right-had sides are now zero. The contradiction follows with the same arguments.

Fig. 2

 2 Fig. 2 Electricity clearing price p(s) and p(s).

  p s := p(s) p elec s := p elec (s) ps := p(s) and p sC := p((s -1 ;C 1 )) p elec sC := p elec ((s -1 ;C 1 )) psC := p((s -1 ;C 1 ))

  τ + ε; p) = O O(τ; p). We call S D = {p d ; lim ε→0;ε>0 D(p d +ε) < D(p d )}, the set of discontinuity points of the Demand function.

(

  ēe ℓ )ϕ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )ϕ i (t) > ( ē -e ℓ )κ ℓ + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )κ i .But ē ≥ e ℓ when ℓ exists, and ē ≥ ê ≥ e i for i ∈ I\ I ′ ∩ I I ′ . So we obtain our contradiction.(ii-a-2) If ∑ i∈I ′ ϕ i (t ′ ) = D(p elec (t ′ )), then ∑ j∈I\I ′ ϕ j (t) + ∑ i∈ I I ′ ϕ i (t) > ∑ i∈ I I ′ ϕ i (t ′ ) + ∑ k∈I ′ \I ϕ k (t ′ ) (DC) ∑ j∈I\I ′ ϕ j (t)e j + ∑ i∈ I I ′ ϕ i (t)e i < ∑ i∈ I I ′ ϕ i (t ′ )e i + ∑ k∈I ′ \I ϕ k (t ′ )e k . (EH)We decomposeI\I ′ = I\(I ′ ∪ I) ∪ I\I ′ and I ′ \I = I ′ \(I ∪ I ′ ) ∪ I ′ \I: ∑ j∈I\(I ′ ∪ I) κ j + ∑ j∈ I\I ′ ϕ j (t) + ∑ i (t ′ ) + ∑ k∈ I ′ \I ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ ) κ k (DC)∑ j∈I\(I ′ ∪ I) e j κ j + ∑ j∈ I\I ′

  e i ϕ i (t ′ ) + ∑ k∈ I ′ \I e k ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ ) e k κ k . (EH)We also break down the set I I ′ = (I ∩ I ′ ):I I ′ = I I ′ ∩ {ℓ} ∪ I I ′ ∩ I\ I ′ ∪ I I ′ ∩ I ′ \ I ∪ I\ I ∩ I ′ \ I ′ ) . ∑ j∈I\(I ′ ∪ I)) κ j + ∑ j∈ I\I ′ ϕ j (t) + ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′ ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ ϕ i (t) > ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′ ϕ i (t ′ ) + ∑ i∈ I ′ \ I∩ I I ′ ϕ i (t ′ ) + ∑ k∈ I ′ \I ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ ) κ k (DC)∑ j∈I\(I ′ ∪ I) e j κ j + ∑ j∈ I\I ′

  e j ϕ j (t) + e ℓ ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′

  e i ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ e i ϕ i (t) < e ℓ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′

  e i ϕ i (t ′ ) + ∑ i∈ I ′ \ I∩ I I ′

  e i ϕ i (t ′ ) + ∑ k∈ I ′ \I e k ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ ) e k κ k (EH)For index i in the last subset (I\ I ∩ I ′ \ I ′ ), we have ϕ i (t) = κ i and ϕ i (t ′ ) = κ i , so we simplify (DC) and (EH) from this last subset. Thus, ∑ j∈I\(I ′ ∪ I)κ j + ∑ j∈ I\I ′ ϕ j (t) + ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′ ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ κ i > ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′ κ i + ∑ i∈ I ′ \ I∩ I I ′ ϕ i (t ′ ) + ∑ k∈ I ′ \I ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ ) κ k (DC)∑ j∈I\(I ′ ∪ I) e j κ j + ∑ j∈ I\I ′

  e j ϕ j (t) + e ℓ ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′

  e i ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ e i κ i < e ℓ ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′ e i κ i + ∑ i∈ I ′ \ I∩ I I ′

  e i ϕ i (t ′ ) + ∑ k∈ I ′ \I e k ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ e k κ k (EH)We multiply (DC) by ē := (e ℓ , ê) ∨ max{e k , k ∈ I ′ \I} , we get by (33) ∑ j∈I\(I ′ ∪ I) e j κ j + ∑ j∈ I\I ′

  e j ϕ j (t) + ēϕ ℓ (t) + ē ∑i∈ I\ I ′ ∩ I I ′ ϕ i (t) + ē ∑ i∈ I ′ \ I∩ I I ′ κ i > ēϕ ℓ (t ′ ) + ē ∑ i∈ I\ I ′ ∩ I I ′ κ i + ē ∑ i∈ I ′ \ I∩ I I ′ ϕ i (t ′ ) + ∑ k∈ I ′ \I e k ϕ k (t ′ ) + ∑ k∈I ′ \(I∪ I ′ e k κ kWe subtract (EH)( ēe ℓ )ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ ( ēe i )κ i > ( ēe ℓ )ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )κ i + ∑ i∈ I ′ \ I∩ I I ′ ( ēe i )ϕ i (t ′ )We arrange the terms( ēe ℓ )ϕ ℓ (t) + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )ϕ i (t) + ∑ i∈ I ′ \ I∩ I I ′ ( ēe i )κ i > ( ēe ℓ )ϕ ℓ (t ′ ) + ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i )κ i + ∑ i∈ I ′ \ I∩ I I ′ ( ēe i )ϕ i (t ′ ) If ℓ exists, then ē = e ℓ and ∑ i∈ I ′ \ I∩ I I ′ (e ℓ -e i ) κ i -ϕ i (t ′ ) > ∑ i∈ I\ I ′ ∩ I I ′ (e ℓe i ) (κ iϕ i (t)) ∑ i∈ I ′ \ I∩ I I ′ (e ℓ -ê′ ) κ i -ϕ i (t ′ ) > ∑ i∈ I\ I ′ ∩ I I ′ (e ℓê) (κ iϕ i (t)) .(34)But ê < e ℓ < ê′ , and the contradiction follows.If ℓ does not exist, then ē = ê ∨ max{e k , k ∈ I ′ \I} ∑ i∈ I ′ \ I∩ I I ′ ( ēe i ) κ i -ϕ i (t ′ ) > ∑ i∈ I\ I ′ ∩ I I ′ ( ēe i ) (κ iϕ i (t)) ∑ i∈ I ′ \ I∩ I I ′ ( ē -ê′ ) κ i -ϕ i (t ′ ) > ∑ i∈ I\ I ′ ∩ I I ′ ( ēê) (κ iϕ i (t))(35)But max{e k , k ∈ I ′ \I} < ê′ , and the contradiction follows.(ii-b) If ∑ i∈I ϕ i (t) < D(p elec (t)) then for all i ∈ I, ϕ i (t) = κ i . (ii-b 1) If ∑ i∈I ′ ϕ i (t ′ ) < D(p elec (t ′ )), then ϕ i (t ′ ) = κ i for all i ∈ I ′ . Moreover, we have that O O(t, p(t)) ≥ D(p(t)) + ε) ≥ D(p(t ′ )) > O O(t ′ , p(t ′ )) and (DC)-(EH) becomes ∑ j∈I\I ′ κ j > ∑ k∈I ′ \I κ k (DC) ∑ j∈I\I ′ e j κ j < ∑ k∈I ′ \I e k κ k (EH)Then, we multiply (DC) by min{e j ; j ∈ I\I ′ } ≥ max{e k ; k ∈ I ′ \I}, and we obtain a contradiction with (EH).

  then, by the definition of the sets c j + e j t =c ℓ + e ℓ t c j + e j t ′ <c ℓ + e ℓ t ′ c k + e k t <c j + e j t c k + e k t ′ =c ℓ + e ℓ t ′ c k + e k t <c ℓ + e ℓ t c j + e j t ′ <c k + e k t I\I ′ and k ∈ I ′ \I, we havec j + e j t < c k + e k t c j + e j t ′ > c k + e k t ′from which, we also easily deduce that max{e k , k ∈ I ′ \I} < min{e j , j ∈ I\I ′ }.(32)For the same j and k, for ( ĉ, ê) representative of index in I I ′ ∩ I \ I ′ , and ( ĉ′ , ê′ ) representative of index in I I ′ ∩ I ′ \ I, we also have c j + e j t ≤ ĉ + êt c j + e j t ′ > ĉ + êt ′ and c k + e k t > ĉ′ + ê′ t c k + e k t ′ ≤ ĉ′ + ê′ t ′

	from which, we easily deduce that ê := max e j , j ∈ I I ′ ∩ I\ I ′ For j ∈ from which, we deduce that min{e	< e ℓ < min e k , k ∈ I I ′ ∩ I ′ \ I	:= ê′ . (31)

European Emission Trading System

Also called Dutch auction market with several units to sell, in a second item auction market, the seller begins with a very high price and reduces it. The price is lowered until a bidder accepts the current price.

Here v -j stands for the profile(v i , • • • , v j-1 , v j+1 , • • • , v J ).

(v -j ; v) stands for (v 1 , • • • v j-1 , v, v j+1 , • • • v J )


Lakhdar Hachani

Redouane Boussaa

Bachir Saadi

X D Wang

Kader Zaidat

Aissaa Belgacem Bouzida

Dominique Henry

Valéry Botton

H Ben Hadid

Yves Fautrelle

Experimental investigation of the natural and forced convection on solidification of Sn-3wt. %Pb alloy using a benchmark experiment

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Experimental investigation of the natural and forced convection

on solidification of Sn-3wt. %Pb alloy using a benchmark experiment L. HACHANI 1,2,4 , R. BOUSSAA 1,3 , B. SAADI 1 , X-D. WANG 1 , K. ZAIDAT 1 , A. BELGACEM BOUZIDA 2 , D. HENRY 3 , V. BOTTON 3 , H. BEN HADID 3 ABSTRACT: We deal with the development of a solidification benchmark experiment in order to investigate the structure formation as well as solute macro-mesosegregation, by means of a wellcontrolled solidification experiment. The experiment consists in solidifying a rectangular ingot of Sn-3wt.%Pb alloy, by using two lateral heat exchangers which allow extraction of the heat flux from one or two vertical sides of the sample. The domain is a quasi two dimensional parallepipedic ingot (100x60x10) mm. The temperature difference ΔT between the two lateral sides is 40 K and the cooling rate CR = 0.03 K/s. The instrumentation consists in recording the instantaneous temperature maps by means of an array of 50 thermocouples in order to provide the time evolution of the isotherms. After each experiment the patterns of the segregations have been obtained by X-ray radiograph and confirmed by eutectic fraction measurements. The local solute distribution determined by means of induction coupled plasma analysis is provided. The originality of the present study is to examine the effect of the forced convection driven by a travelling magnetic field (TMF) induced by a linear inductor located on the bottom part of the sample. A periodically reversed stirring with a modulation frequency equal to 0.5 Hz stirring have been investigated. This study allows us to evaluate the evolution due to the forced convection induced by a TMF field, as well as its influence on the initial conditions, the solidification macrostructure and the segregation behavior. Measurements of the velocity field by ultrasonic Doppler velocimetry (UDV) method in a Ga-In-Sn pool were performed and transposed to the tin-lead alloy case before solidification. Post-mortem patterns of the macromesosegregations have been obtained by X-ray radiography. The results show the transport effects of the flow on both the macrosegregations and the channel formation. The reversal of the TMF produces a decrease of the level of mesosegregations, namely channel formation.

Experimental setup

The properties of most alloys depend largely upon the degree of control, which can be applied during the solidification process. One of the major problems lies in the non-uniform distribution of solute concentration that is inherent to solidification. These concentration variations, also known as segregations, appear at the mesoscopic scale (freckles) and/or at the ingot scale (macrosegregation) [START_REF] Sarazin | A Channel Formation in Pb-Sn, Pb-Sb, and Pb-Sn,Sb Alloy Ingots and comparison with the system NH4 Cl-H2O. Metall Met Trans A[END_REF][START_REF] Hebditch | Observations of ingot macrosegregation on model systems[END_REF] in a solidified sample. Naturally these concentration variations have important repercussions on materials properties; therefore, minimizing segregation is essential for the proper alloys performance. It appears that the role of the gravity may be prominent for the evolution of segregation and can create unexpected concentration patterns. One of the possibilities to counter this effect is to impose a forced convection.

The introduction of magnetic field in the solidification process is one of the effective methods to improve the microstructure and mechanical performance of alloys [START_REF] Campanella | Grain refinement induced by electromagnetic stirring: A dendrite fragmentation criterion[END_REF][START_REF] Zaïdat | Control of melt convection by a traveling magnetic field during the directional solidification of Al-Ni alloys[END_REF][START_REF] Willers | The columnar to equiaxed transition in Pub-Sn alloys affected by electromagnetically driven convection[END_REF][START_REF]Experimental analysis of the solidification of Sn-3 wt.%Pb alloy under natural convection[END_REF] In this study, the model alloy (Sn-3-wt.%Pb ) was chosen as the experimental study subject on account of its low melting point. Based on the experiment result, we examine the effect of the forced convection driven by a travelling magnetic field on the solidification process in terms of temperature history, velocity field, structure maps and solute distribution in the solid sample during the horizontal directional solidification.

The experiment principle is similar, to the well-known Hebditch and Hunt experiment [START_REF] Hebditch | Observations of ingot macrosegregation on model systems[END_REF], with a special emphasis on controlling initial and thermal boundary conditions, and obtaining reproducible quantitative measurements [START_REF] Wang | An investigation of the influence of natural convection on tin solidification using a quasi two-dimensional experimental benchmark[END_REF].

In this study a periodically reversed stirring with a modulation frequency equal to 0.5 Hz has been investigated. The present work is aimed at evaluating: 1. How the fluid flow in the mushy zone transports the solute and generates segregations. 2. Whether it is possible to control the fluid flow by mean of an electromagnetic stirring device and accordingly to control segregation.

In the next section, the experimental process is detailled. The heat transfer and the macrosegregation are discussed. The description of the experimental device represented on Fig. 1, and process are presented elsewhere with more details in the work [START_REF]Experimental analysis of the solidification of Sn-3 wt.%Pb alloy under natural convection[END_REF]. However in this experimental study we are interested primarily on the part of forced convection driven by travelling magnetic field stirring and its influence on the solidification process. The travelling magnetic field is generated by linear motor placed underneath the bottom wall of stainless steel crucible. The corresponding electromagnetic force is imposed from the melting to solidifying steps. Two experiments have been investigated: Case I, the melt is solidifying under the natural convection. The temperature difference between the two lateral sides of the sample produces clockwise fluid flow recirculations. Case II the experimental conditions are the same as case I, but the electromagnetic force direction is periodically reversed. The modulation frequency is equal to 0.5 Hz.

Experimental result

Magnetic fields evolution

The linear inductor has eight poles and is located underneath the bottom wall of the crucible about 5 mm. Its length is L ind = 244 mm. This inductor is fed by a three-phase AC current of normal frequency (

Hz f 50 0 
). The wave-length l and the pole pitch are l = 2 = 48 mm, so that the wave number is k = 2π/l = 130.899 m -1 . The synchronism velocity of the moving magnetic field is

s m k l f U e S / 4 . 2 0     .
A measure of the modulus of the magnetic field

2 2 y x B B B   
in different x-positions in the absence of liquid metal for various applied current (7.5, 8.5 and 10 A) at a frequency of 50 Hz as show that for an intensity rating of 8.5A the magnetic field strength is about 50 mT.

Velocity measurement by ultrasonic Doppler velocimetry (UDV)

In order to assess the flow characteristics before solidification, velocity measurements were performed with the eutectic (GaInSn: 67wt.%Ga-20.5wt.%Sn-12.5wt.%In) alloy. The liquid metal, whose density is 6400 kg/m 3 , and dynamic viscosity 2.16×10 -3 Pa.s [START_REF] Wang | A periodically reversed flow driven by a modulatedtraveling magnetic field. Part I: Experiments with GaInSn[END_REF], is contained in a transparent Plexiglas rectangular tank of width W=1cm, length L=10cm, and height H=6cm. An Ultrasonic Doppler Velocimetry (UDV) sensor is fixed on a vertical sidewall of the tank to probe the horizontal velocity components. The data provided by the ultrasonic velocimetry UDV consist in instantaneous velocities in the x direction V x (x, t i ). The results show that for applied current I = 7.5A the mean (time averaged) velocity varies over a large range from 17.96 to 34 mm/s and exhibits significant fluctuations around 25.19 mm/s, and the corresponding Reynolds number is ( ). It is very important to note that all experimental measurements performed on (GaInSn), are achieved in order to estimate the configuration dynamics in the melt of Sn-03wt.%Pb with an approximation by taking into account the similarity coefficient

4294 . 0 ) ( ) ( ) ( ) ( ) ( ) (            Sn In Ga Sn In Ga Pb Sn Pb Sn Sn In Ga velocity Pb Sn velocity    

Temperature fields and thermal gradient evolution

In the following subsection, we show in Fig. 2 temperature contours coupled with thermal gradient for two cases of solidification discussed previously at selected moment (t = 14740 s) during the cooling. The experimental conditions are: ΔT = 40 K, CR = 0.03 K/s. At the initial stage the temperature difference between the two lateral sides of the heat exchangers the left wall temperature is T L = 280 °C and the right one T R = 240 °C. The magnetic force with the applied current I = 9.56 A, is imposed on liquid Sn-Pb alloy (electrical conductivity = 3.86x10 6 Ω -1 .m -1 ). The temperature gradients were calculated at each thermocouple location from the temperature difference between two adjacent points by means of a numerical centered difference scheme. The temperature data are analyzed to obtain some clues as to the magnitude of the velocity field during solidification. The method consists in calculating the projection of the velocity field on the temperature gradient vector at each node corresponding to the location of a thermocouple. The principle of the method is presented with more details in L. Hachani et al [START_REF]Experimental analysis of the solidification of Sn-3 wt.%Pb alloy under natural convection[END_REF].The algebraic velocity estimate U is defined as

) /( . T T U U    
where T  is the norm of the temperature gradient. Figure [START_REF] Hebditch | Observations of ingot macrosegregation on model systems[END_REF] shows the time evolution of the velocity projections on the thermal gradient vector for various internal nodes L14, L15, L16, L17 and L18 (shown in Fig. 1) located in the upper part of the ingot where the scalar products of the velocity vector u  and the temperature gradient T  are supposed not to be too small. Various observations may be drawn from such analysis. Firstly, we retrieve the various steps of the solidification, namely the global stabilization of the velocity during the thermal natural convection regime before cooling. Secondly the velocity decays when the columnar front becomes close to the node.

Post-mortem analysis

Concentration map for Sn-3wt.%Pb alloy

A chemical method coupled to ICP is used to analyze quantitatively the lead concentration distribution. We drilled fifty holes into ingot to extract about 200 mg for each analyzed sample. The holes positions which correspond to the thermocouple positions are presented in Fig. 1. The results illustrated by Fig. 4, show the segregation distribution for the two cases of solidification for identical experimental conditions (Sn-3wt.%Pb alloy. DT = 400K/m, CR = 0.03K/s). We have checked that the measured average concentration was close to the nominal one.

X-Ray analysis

X-Ray radiographs of the samples were performed by CEA-Saclay laboratory (France) (Fig. 5) in order to analyze qualitatively the lead concentration distribution. The main objective is to observe all types of macro-meso segregations, specially the channel distributions. The lead easily absorbs the X-rays, and the color contrast gives indications of the solute distribution within the ingot. Lead-rich solute zone is easily visible and appears in the photography. The modulated electromagnetic stirring has shown its effectiveness, particularly in the significant reduction of the number of segregated channel but doesn't suppress them completely. The forced convection changes also the channel locations

Conclusion

This work constitutes an experimental study of the effect of natural and forced convections in a benchmark solidification experiment using Sn-wt.3%Pb alloys. The thermal phenomena, grain structure and macro-mesosegregation (freckles) were examined. The role of melt convection has been studied during solidification. The melt was electromagnetically agitated by means of a travelling magnetic field (TMF) of moderate intensity. A periodically reversed stirring with a modulation frequency equal to 0.5 Hz stirring has been investigated. The following major conclusions can be drawn: The forced convection influences significantly the macrostructure and segregation distribution.The effect of the electromagnetic stirring on the flow pattern in the molten metal is very important. The thermal process quantities, namely temperature gradients, cooling rate at the solidifying front and liquidus interface velocity, can be used to characterize the morphology of the grain structures.

Fig. 1 .

 1 Fig.1. Sketch of the sample surrounded by the two lateral heat exchangers. The location of the lateral thermocouple lattice (L1 to L50) welded on one of the largest surface of the crucible is also shown.

Fig. 2 .

 2 Fig.2. Temperature field maps selected at time (t = 14740s), for the two cases of solidification (I and IV). The experimental conditions: Sn-3wt.%Pb, ΔT = 40 K, CR = 0.03 K/s, applied current I = 9.56 A, f = 50 Hz, σ = 1.86x10 6 Ω -1 .m -1 , B 0 = 30 mT, liquidus temperature T l = 227.7°C, eutectic temperature T s = 183°C. The arrows correspond to the T   vector which is proportional to the local heat flux density

Fig. 3 .

 3 Fig.3. Velocity estimate U versus time for various internal nodes (L14, L15, L16, L17 and L18) located in the upper part of the ingot. Fig. 5 also shows the extrapolation to zero used to estimate the instant when the columnar front crosses the node position

Fig. 4 .

 4 Fig.4. Lead concentration for Sn-3wt.%Pb. Applied temperature difference ΔT = 40 K. Cooling rates 0.03 K/s. Two cases are presented: case I natural convection and case II natural and modulated forces (modulation frequency f = 0.5 Hz), I = 9.56 A.

Fig. 5 .

 5 Fig.5. X-ray radiograph of the solidified ingot showing the freckles, Sn-3wt.%Pb alloy. DT = 400K/m, CR = 0.03K/s. left natural convection, right modulated forces (modulation frequency f=0.5 Hz), I = 9.56 A.

  and Y. FAUTRELLE 1 1. Grenoble Institute of Technology, SIMAP-EPM Laboratory, PHELMA BP75 38402 Saint Martin d'Hères Cedex, France 2. Laboratory of Mechanics, University Amar Telidji of Laghouat, Algeria 3. LMFA, Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France 4. University Hadj Lakhdar of Batna, Algeria

Acknowledgments

The authors acknowledge the European Space Agency through the CETSOL project (ESA-MAP AO-99-117) as well as the SMACS ANR project for their financial support. The authors are indebted to B. Rattoni (CEA-Saclay, France) who performed the X-ray characterizations.