Ramzi Ben Mhenni

Sébastien Bourguignon

Jordan Ninin

Frédéric Schmidt

SPECTRAL UNMIXING WITH SPARSITY AND STRUCTURING CONSTRAINTS

Keywords: sparse spectral unmixing, 0 -norm optimization, structured sparsity, mixed-integer programming

This paper addresses the linear spectral unmixing problem, by incorporating different constraints that may be of interest in order to cope with spectral variability: sparsity (few nonzero abundances), group exclusivity (at most one nonzero abundance within subgroups of endmembers) and significance (non-zero abundances must exceed a threshold). We show how such problems can be solved exactly with mixed-integer programming techniques. Numerical simulations show that solutions can be computed for problems of limited, yet realistic, complexity, with improved estimation performance over existing methods, but with higher computing time.

INTRODUCTION

Spectral unmixing consists in decomposing an observed reflectance spectrum into a linear mixture of elementary spectra (endmembers) and estimating their proportions (abundances) [START_REF] Singer | Mars -Large scale mixing of bright and dark surface materials and implications for analysis of spectral reflectance[END_REF][START_REF] Keshava | Spectral unmixing[END_REF]. We focus on abundance estimation, and endmembers are supposed to be known, e.g., obtained by laboratory spectroscopic measurements or estimated from the hyperspectral data set, forming a dictionary used for decomposition. In order to cope with spectral variability, spectral libraries may contain an increasingly high number of spectra-the minerals data base provided by the United States Geological Survey (USGS), for example, increased from 504 spectra in 2003 [START_REF] Clark | USGS digital spectral library splib05a[END_REF] to 1276 spectra in 2017 [START_REF] Kokaly | USGS Spectral Library Version 7 Data[END_REF]. Dealing with a large number of endmembers is acknowledged as a critical issue, especially because many spectra are highly correlated [START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches[END_REF][START_REF] Iordache | Collaborative sparse regression for hyperspectral unmixing[END_REF]. In such cases, the standard approach based on least-squares fitting under positivity and possibly sum-to-one constraints [START_REF] Heinz | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery[END_REF], may perform badly, and stronger constraints are required, some of which are considered in this paper.

The first one is sparsity: usually, each mixture contains only a few components. Second, imposing structural constraints on the dictionary is also of interest in order to cope with variability. In particular, experimental data concerning a mineral are usually acquired in various conditions, and the dictionary may include such different spectra, although at most one is actually used in the decomposition (exclusivity constraint). Last, imposing a minimum value to each non-zero abundance (significance constraint) also promotes sparsity: conjugated with positivity and sum-to-one constraints, it concentrates the solution into few components, without imposing a maximum number of non-zero abundances.

To our knowledge, such constraints have never been addressed exactly, probably because of their numerical difficulty. Cardinality constraints resort to combinatorial optimization. Sparse spectral unmixing has been proposed e.g., with 1 -norm [START_REF] Guo | L1 unmixing and its application to hyperspectral image enhancement[END_REF][START_REF] Iordache | Sparse unmixing of hyperspectral data[END_REF] or p, p<1 -norm [START_REF] Tuia | Nonconvex regularization in remote sensing[END_REF] regularization, or with a greedy Orthogonal Matching Pursuit procedure [START_REF] Iordache | Sparse unmixing of hyperspectral data[END_REF]. While computationally attractive, such approaches usually do not solve the original sparse problem, and they face difficulties in considering the sum-to-one constraint and nonnormalized dictionaries. Both exclusivity and significance constraints are basically logical constraints, that cannot be addressed by standard optimization (exclusivity is mentioned e.g. in [START_REF] Iordache | Collaborative sparse regression for hyperspectral unmixing[END_REF], but no algorithmic solution is proposed). This paper considers the exact resolution of such problems through mixed-integer programming (MIP) techniques, which mix continuous and discrete variables and offers a natural framework for such constraints. Although MIP optimization is usually NP-hard, it has been investigated for long in operations research and very efficient solvers were developed [START_REF] Bixby | A brief history of linear and mixed-integer programming computation[END_REF].

In this paper, we first show how the above constraints can be exactly translated into MIPs; their exact resolution is then performed through dedicated software. Second, we evaluate the obtained solutions through numerical simulations in their ability to correctly solve unmixing problems, as a function of the problem complexity (cardinality and noise level).

CONSTRAINED LINEAR SPECTRAL UNMIXING

We consider the standard linear mixture model [START_REF] Singer | Mars -Large scale mixing of bright and dark surface materials and implications for analysis of spectral reflectance[END_REF][START_REF] Keshava | Spectral unmixing[END_REF]:

y = N n=1 a n s n + = Sa + ,
where y ∈ R L is a reflectance spectrum acquired in L spectral bands, s n ∈ R L represents the n-th endmember, a n ∈ R is the associated abundance, a = [a 1 , . . . , a n ] T , represents noise and model errors and S = [s 1 , . . . , s N ] is the dictionary of N endmembers, with usually N L. In order to account for physical considerations (abundances are percentages), positivity and sum-to-one constraints are usually added. The resulting Fully Constrained Least-Squares (FCLS) estimate [START_REF] Heinz | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery[END_REF] reads:

min a∈ R N y -Sa 2 such that (s.t.) a ≥ 0, N n=1 a n = 1, (1)
where vector inequality is taken componentwise. In the presence of noise, and if the dictionary S contains highly correlated spectra, FCLS may not be efficient, as shows the example in Figure 1, Section 4: estimated abundances are spread over many non-zero, but small, values, which does not correspond to physical reality and lead to erroneous interpretation. In such cases, imposing more constraints may be necessary.

Cardinality constraint ( 0 -norm sparsity)

In most cases, only a small number of elementary spectra are used for representing a mixture: a should be sparse, that is, only a few a n are non-zero. Let us remark that formulation (1), through the positivity constraint, produces solutions for which some abundances are zero. However, on difficult problems, FCLS generally produces a large number of smallvalued abundances and may fail in locating the true components (see simulation results in Section 4). Enforcing more sparsity may contribute to improve abundance estimation [START_REF] Iordache | Sparse unmixing of hyperspectral data[END_REF][START_REF] Greer | Sparse demixing of hyperspectral images[END_REF][START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches[END_REF]. Introducing the 0 "norm" a 0 = Card{n|a n = 0}, the sparse linear unmixing problem can be written as the cardinality-constrained optimization problem:

min a≥0 y -Sa 2 s. t. a 0 ≤ K, N n=1 a n = 1. (2)
Since solving (2) is essentially a combinatorial problem, suboptimal estimation strategies are usually preferred for computational efficiency. Substituting the 0 norm with the convex 1 norm has been a standard in sparsity-enhancing methods for a long time. However, it is not fully appropriate here: since abundances are positive and sum to one, their 1 norm is fixed to 1! Therefore, the 1 norm cannot be used for enforcing sparsity in problem [START_REF] Singer | Mars -Large scale mixing of bright and dark surface materials and implications for analysis of spectral reflectance[END_REF]. Indeed, we note that the works in [START_REF] Guo | L1 unmixing and its application to hyperspectral image enhancement[END_REF][START_REF] Iordache | Sparse unmixing of hyperspectral data[END_REF] exploiting 1 -norm strategies discard the sum-to-one constraint. While we acknowledge that imposing a lower value on a n may act as a sparse regularizer, in our experiments, however, the best 1 -norm-based solution was always found with a n 1, that is, the FCLS solution [START_REF] Singer | Mars -Large scale mixing of bright and dark surface materials and implications for analysis of spectral reflectance[END_REF]. The non-convex p,p<1 -norm regularization recently proposed in [START_REF] Tuia | Nonconvex regularization in remote sensing[END_REF] may be more suited for enforcing sparsity, although optimization strategies in [START_REF] Tuia | Nonconvex regularization in remote sensing[END_REF] also discard the sumto-one constraint. Standard greedy iterative algorithms [START_REF] Tropp | Greed is good: Algorithmic results for sparse approximation[END_REF] are also unadapted to cope with the sum-to-one constraint and with non-normalized dictionaries: since endmembers are reflectance spectra, the dictionary cannot be normalized without affecting the abundance estimation [START_REF] Greer | Sparse demixing of hyperspectral images[END_REF]. Finally, the backward procedure in [START_REF] Greer | Sparse demixing of hyperspectral images[END_REF] iteratively removes one by one the small-but-non-zero components in the FCLS solution.

Although this method has no optimality guarantee with respect to problem [START_REF] Keshava | Spectral unmixing[END_REF], it provides a sparse solution by running iterations until a few abundances are non-zero. Therefore, it will be used for comparison in our experiments.

Group exclusivity constraint

In order to improve the model accuracy in dictionary-based spectral unmixing, a given mineral can be represented by several possible spectra, depending on grain size, acquisition geometry, impurities, etc. Also, mineral samples used for endmember laboratory measurements may come from different places on Earth, resulting in slight differences in impurities. In addition, some minerals (e.g., olivine or pyroxene) are actually solid-state solutions with a continuum of possible compositions, creating a variability in the spectral shape. In such cases, the spectral library may include different possible variations of a given mineral, although at most one may be activated in a given mixture [START_REF] Iordache | Collaborative sparse regression for hyperspectral unmixing[END_REF]. Let the component indices be split 1 into J groups G 1 , . . . , G J , and let a (j) denote the corresponding abundance subset. We address the following optimization problem, which imposes that the mixture contains at most one spectral element in each group G j :

min a≥0 y -Sa 2 s. t. a (j) 0 ≤ 1, j = 1, . . . , J N n=1 a n = 1, (3)
Note that such group exclusivity (GE) constraints also enhance more sparsity in the FCLS solution (1).

Significant abundances only

If the spectral dictionary is highly correlated and in the presence of noise, the FCLS solution usually shows (possibly many) small-amplitude components. Such artifacts can be easily eliminated after some post-processing step; however, they may also perturbate detection of the true components (see Section 4, Figure 1, left column). Therefore, one may want to impose a minimum value to non-zero abundances, in order to detect only significant contributions. We call these constraints significant abundances (SA). The last optimization problem we are interested in therefore reads:

min a≥0 y-Sa 2 s. t. (a n = 0 ⇒ a n ≥ τ ) , n = 1, . . . , N N n=1 a n = 1. (4)
It represents another sparsity-promoting formulation (less than 1/τ abundances are non-zero), that may be preferred to the 0 -norm problem (2). In particular, tuning the related threshold τ can be easier in practice than specifying the number of non-zero abundances that are searched in the mixture.

REFORMULATIONS AS MIXED-INTEGER PROGRAMS

We now formulate the three constrained spectral unmixing problems (with sparsity, exclusivity and significance constraints) as mixed-integer programs (MIPs). Since these constraints are discrete or logical ones, we introduce binary variables b n indicating the presence or absence of a given endmember in the mixture: b n = 1 ⇔ a n = 0. Remarking that abundances a n ∈ [0, 1], one can easily show that the former equivalence can be exactly translated as 0 ≤ a n ≤ b n , giving N linear inequality constraints mixing binary and real variables, that can be efficiently tackled by MIP optimization. The 0 norm can now be simply written as the sum of the binary variables: a 0 = n b n . Therefore, the cardinalityconstrained problem (2) is equivalently reformulated as:

min a∈R N ,b∈{0,1} N y -Sa 2 s. t.      0 ≤ a ≤ b n b n ≤ K n a n = 1 .
Similarly, the exclusivity-constrained problem (3) reads:

min a∈R N ,b∈{0,1} N y-Sa 2 s. t.      0 ≤ a ≤ b i∈ Gj b i ≤ 1, j = 1, . . . , J N n=1 a n = 1.
Finally, one can show that the logical constraints in (4) can be equivalently written as τ b n ≤ a n ≤ b n , so that the problem with significance constraints (4) reads:

min a∈R N ,b∈{0,1} N y -Sa 2 s. t. τ b n ≤ a n ≤ b n , ∀n N n=1 a n = 1 .
These three optimization problems share a common structure: a quadratic cost function, linear equality and inequality constraints, and binarity of some variables. Efficient resolution methods have been developed in operations research for such quadratic MIPs, based on branch-and-bound tree exploration strategies. In this paper, resolution is performed with IBM ILOG CPLEX, which is unanimously considered among the best MIP solvers [START_REF] Bixby | A brief history of linear and mixed-integer programming computation[END_REF]. Finally, although considered separately in the former description, the three constraints can obviously be mixed in order to build more complex estimates, and resulting problems still form quadratic MIPs. Note also that alternate formulations that would switch the cost function and the constraints (e.g., minimize the 0 norm under a bounded approximation error constraint) result in quadratically-constrained MIPS, whose resolution by MIP solvers is much less efficient [START_REF] Bourguignon | Exact sparse approximation problems via mixedinteger programming: Formulations and computational performance[END_REF].

SIMULATION RESULTS

We study the performance of these approaches on simulated spectral unmixing problems. The dictionary is composed of reflectance spectra of minerals taken in the USGS database [START_REF] Clark | USGS digital spectral library splib05a[END_REF], possibly present on planetary surfaces [START_REF] Schmidt | Minerals detection for hyperspectral images using adapted linear unmixing: LinMin[END_REF]. It contains N = 481 spectra, among which J = 85 groups of spectra with same mineral name contain from 2 to 18 endmembers, and 146 endmembers are not clustered into groups. Spectral sampling represents L = 113 wavelengths from 1 to 2.5 microns. Abundances are uniformly distributed above τ = 0.1 and sum to one. Spectral components are randomly chosen, with at most one endmember in each group. Gaussian noise is added with SNR dB = 10 log( Sa 2 / 2 ) equal to 55 and 40 dB. Due to the positivity of the spectra (which have therefore nonzero mean), such values seem particularly high, but they actually correspond to low and medium noise levels (see Figure 1 for an example). In both cases, data are sufficiently noisy so that the FCLS estimate almost always fails in retrieving the true abundance set, as will be seen later. Abundances are estimated by FCLS, by the backward method in [START_REF] Greer | Sparse demixing of hyperspectral images[END_REF] and by the following constrained estimates: 0 -norm sparsity ( 0 ), exclusivity (GE), 0 -norm sparsity and exclusivity ( 0 +GE), significance and exclusivity (SA+GE). The best 1 -norm-based solutions were obtained by tuning the 1 norm close to 1 (see § 2.1), and therefore gave similar results to FCLS. The threshold τ for significance constraints is set to 0.1 and cardinality constraints are set to their true value. Resolution is performed on a standard desktop computer. Figure 1 shows an example of results, with SNR = 40 dB and K = 4. FCLS correctly detects only 3 spectra, with underestimated amplitudes, and produces many erroneous detections. In this example, adding the GE constraint improves the detection since the 4 spectra are correctly detected. However, some artifacts can still be found, and abundances of true components are therefore badly estimated. The backward method correctly detects 3 spectra, with slightly misestimated amplitudes. On the other hand, the three 0 , 0 +GE and SA+GE estimates perfectly detect the four components, with fairly estimated amplitudes (we only show the first estimate, since the three are identical).

Figure 2 shows the quadratic error between true and estimated abundances as a function of K, averaged over 30 data realizations, with low (55 dB) and medium (40 dB) noise levels. GE improves over simple FCLS, and the backward performs better FCLS and GE at high SNR and low K. The three 0 , 0 +GE and SA+GE estimates always give the best results, with perfect detections at high SNR (even for K = 7, whereas FCLS and GE fail from K = 2 and the backward method fails from K = 4). Of course, their performance decreases as the problem complexity increases: then, the 0 minimizer, even correctly computed, does not find the true solution. For SNR = 55 dB and K > 6, the problem complexity does not make it possible to compute the solution in the maximum time al- lowed for MIP resolution (1000 s). Table 1 gives average computing times obtained on the former instances, for GE, 0 , 0 +GE and SA+GE constraints. Including GE constraints is never computationally prohibitive, although it requires the resolution of a MIP. The computing times of the three 0 -norm-based estimates are still reasonable for low noise level or small K, but strongly increase with these two factors. In particular, the optimal solution of 0 +GE is obtained in less than 1000 s, on all instances up to K = 6 for SNR = 55, and up to K = 4 for SNR = 40. Problems with significance constraints are much more difficult to solve, as show the rows 'SA+GE' in Table 1. In all our simulations, 0 and 0 +GE always gave the same solution. However, adding GE constraints allows better numerical resolution for the most difficult problems. Finally, the computing time of the backward method in [START_REF] Greer | Sparse demixing of hyperspectral images[END_REF] is rather high (approximately 32 s for all problems), since it requires the resolution of many (up to N -K) FCLS problems.

CONCLUSION

Complex constraints for spectral unmixing (cardinality, group exclusivity, significant abundances) were taken into account 8.1 75 515 (11) 929 (25) 982 (28) SA+GE 282 (6) 426 (8) 746 (19) 936 (26) 997 (29) Table 1. Computing times (s) for optimization of MIP problems, averaged over 30 instances. In parentheses: number of instances for which optimization did not terminate in 1000 s. The computing time of the backward method is almost constant same for all problems, around 32 s. via MIP formulations. When such constraints are valid in practice, they may be advantageously used for improving abundance estimation. Exact optimization of the resulting problems was shown to be possible for problems of limited complexity. In particular, accounting for group exclusivity only reasonably increases the computing time. More involved sparsity-inducing constraints (cardinality or significant abundances) are computationally more demanding. In our simulations with 481 endmembers, exact resolution of 0 -norm problems was still possible up to K = 6 at high SNR and K = 4 at medium SNR, which is still realistic in many cases.

Incorporating binary indicating variables offers a flexible framework, which may be used for considering other highlevel structuring constraints. Adding more constraints to the problems studied in this paper is also a lever for improving their numerical resolution. Last, dedicated MIP optimization strategies are currently under investigation in order to reduce the computing time.

2 λFig. 1 .a -å 2 1Fig. 2 .

 2122 Fig. 1. Example of result, K = 4 spectra, SNR = 40 dB. Top: estimated (black squares) and true (red stars) abundances. Bottom: true (solid red line) and estimated (dashed black line) endmembers weighted by their abundances, and noise (blue solid line).

We consider here a partition of the dictionary, but similar constraints operating on overlapping groups could also be accounted for.


Meriem Amara
email: meriem.amara@doctorant.ec-lyon.fr

Christian Vollaire

Marwan Ali

François Costa

Francois Costa Safran

Black box EMC modeling of a three phase inverter

Keywords: power electronic converters, electromagnetic compatibility, EM modeling, EMC filter I

come

Black box EMC modeling of a three phase inverter

INTRODUCTION

Nowadays, the use of power converters in systems "more electric" (actuation, pumping, conversion, lighting, etc.) requires to control conducted emissions. The electromagnetic compatibility (EMC) of any equipment in its environment avoids any risk of interference, or alteration of the functional performance of sensitive systems.

Thanks to the progress of power electronics, and in order to gain mass on the distribution of electrical energy, more and more aircraft manufacturers are planning to migrate to a high-voltage DC distribution (HVDC). With the voltage rise, dv/dt have a greater impact on conducted emissions compared to what is currently known. In addition, the use of higher switching frequencies in high power converters enabled by wide bandgap semiconductors constitutes a major electromagnetic risk that must be controlled. In order to protect the networks and loads from these emissions, and to meet the standards for electromagnetic compatibility (EMC) including the aeronautic standard "DO160F", EMC filters are absolutely necessary [START_REF] Akagi | Attenuation of Conducted EMI Emissions[END_REF], [START_REF] Raggl | Guideline for a Simplified Differential-Mode EMI Filter Design[END_REF].

An efficient and optimized EMC filter requires a specific design for the two modes of propagation of parasitic currents, i.e., Common Mode (CM) and Differential Mode (DM). The design is based on the use of an accurate EMI model of the converter. Behavioral model has proved its efficiency in representing accurately electromagnetic interferences [START_REF] Baisden | Generalized Terminal Modeling of Electromagnetic Interference[END_REF], [START_REF] Chen | Effective EMI Filter Design Method for Three-Phase Inverter Based Upon Software Noise Separation[END_REF]. This kind of model is constituted by disturbance equivalent sources associated to CM and DM equivalent impedances [START_REF] Ali | Nouvelles Architectures intégrées de filtre CEM Hybride[END_REF].

In the following paragraphs, we will present the application of this method on a three-phase inverter. Then, an experimental validation will be presented including a study of the impact of the operating point on the generated EMI and thus on the model.

II. EM MODELING OF AN INVERTER

The system under test has two access ports and a ground port on the network side. In the context of this study, we aim to represent accurately electromagnetic interferences of an existing converter, considered from the DC network, owing to an experimental procedure and without knowledge of the internal electrical architecture. In this EMC modeling approach, a ZDM impedance and an IDM source are placed between the two input lines to represent the differential mode (DM) model. The CM one is represented by two identical ZCM impedances connected to a voltage source VCM placed between each of the lines and the ground (Fig. 1). Fig. 1.

Global converter EMC model

This EMC model can be identified, using a simple measurement protocol based on the separation of propagation modes, in order to allow the optimized design of two filtering structures (CM and DM) dedicated to each mode of disturbance. The support of our study is a 4 kW drive chain including a DC / AC converter (MOSFET based) feeding an RL load that emulates an electric actuator.

A. Impedance Modeling

The equipment under test is modeled as seen from the network side. It has two ports and a ground connection. Therefore, the impedance matrix can be calculated as follows:

[V] = [Z][I] => [ V 1 V 2 ] = [ Z 11 Z 12 Z 21 Z 22 ] [ I 1 I 2 ] 
The indices in (1) correspond to the access numbers in Fig. 1. The relations allowing to compute the CM and DM sources from voltages and currents in structure are:

V CM = V 1 +V 2 2 ; V DM = V 1 -V 2   i CM = I 1 + I 2 ; i DM = I 1 -I 2 2   [ V CM V DM ] = [ 1 2 ⁄ 1 2 ⁄ 1 -1 ] [ V 1 V 2 ]   [ I 1 I 2 ] = [ 1 2 ⁄ 1 1 2 ⁄ -1 ] [ i CM i DM ]  => [ V CM V DM ] = [P][Z][P] t [ i CM i DM ]vecP = [ 1 2 ⁄ 1 2 ⁄ 1 -1 ]  => [ V CM V DM ] =[ Z 11 +Z 12 +Z 21 +Z 22 4 Z 11 -Z 12 +Z 21 -Z 22 2 Z 11 +Z 12 -Z 21 -Z 22 2 Z 11 -Z 12 -Z 21 + Z 22 ][ i CM i DM ]  => [ V CM V DM ] = [ Z CM Z′ conversion Z conversion Z DM ] [ i CM i DM ] 
In our case, four impedances are measured directly with an impedance analyzer when the converter is off and connected to the load. The procedure is as follows:

-Z11 is measured between line 1 and ground, -Z22 is measured between line 2 and ground, -ZCM is measured between the two short-circuited lines on one side and the ground on the other side, -ZDM is measured between line 1 and line 2 (converter placed far from the ground).

This modules of impedances are shown in Fig. 2:

Fig. 2.

Measured input impedances

From these measurements, the impedances Z11 and Z22 are clearly identical leading to the same CM impedance. This allows us to propose a simple electrical model with only three localized impedances (see in Fig. 3).

The proposed model consists of two parallel RLC circuits placed between each line and the ground.

The DM impedance is also represented by two parallel RLC circuits placed between the two input lines. Measurements and identifications led to the determination of the value of each element of the model [START_REF] Tarateeraseth | Accurate Extraction of Noise Source Impedance of an SMPS Under Operating Conditions[END_REF]. These values are shown in Fig. 3.The theoretical impedances corresponding to each measurement configuration have been calculated from this model.

Comparison between measured and calculated input impedances

The experimental results presented in Fig. 4 are clearly in good agreement with simulated results up to 30 MHz.

B. CM and DM Sources Model

The EMC model must be completed by two equivalent CM and DM sources. The symmetry of the line impedances limits the phenomena of mode conversion, which simplifies the extraction of the equivalent sources.

In order to realize this extraction, the power converter is fed through two identical line impedance stabilization networks (LISN) placed on each line between the network and the system under test. They have two main functions. Firstly, they prevent incoming CM and DM disturbances from the supply network. Secondly, they maintain the specified impedance at the system under test over the working frequency range. This impedance is standardized by DO160F. The equivalent EMC model corresponding to this measurement configuration is shown in Erreur ! Source du renvoi introuvable.. According to the complete model given in Fig. 1, the equivalent sources can be extracted by measuring the two line currents (between LISNs and the converter). The equivalent CM voltage source and the DM current source can be calculated by:

I 1 + I 2 = I MC => V MC = (I 1 + I 2 ) ( Z RSIL +2Z MC 2 )   I 1 -I 2 = 2 ( 4Z MC Z MD 4Z MC + Z MD ) I MD (2Z RSIL + 4Z MC Z MD 4Z MC + Z MD )  => I MD = (I 1 -I 2 ) ( Z RSIL 4Z MC + Z RSIL Z MD + 1 2
)  Before measuring the currents I1+I2 and I1-I2, it is necessary to determine the "black box" model of the LISN. The characterization of the LISN is performed as close as possible to the boxes after adding a 50 ohm in parallel with the resistance R2 of the LISN (see in Fig. 6).

C. EMC Model Validation

In order to verify that the proposed model can correctly represent the levels of EMI generated by the converter, we have calculated the LISN voltages from the model shown in Erreur ! Source du renvoi introuvable.. Meanwhile, these LISN voltages have been measured directly using the oscilloscope. Experimental and calculated LISN voltage spectrum corresponding to VRlisn1 and VRlisn2 are shown in Fig. 9. We observe a good matching of the two spectra which proves that the proposed model as well as the method of identification of its parameters are correct. This EMC model, constituted by disturbance equivalent sources and CM and DM equivalent impedances, provides a relevant way to calculate and to optimize the EMC filter for a given operating point. In the next section, the impact of a change in the operating point on our model will be shown.

III. IMPACT OF THE OPERATING POINT

To be able to change the operating point of the PWM inverter under test, we used a three-phase Star-connected balanced RL load whose resistance is variable. We chose 4 operating points: the first corresponds to 5% of the total power that can dissipate the load that corresponds to 200 W, the second 30% that corresponds to 1.2 kW, the third 60% that corresponds to 2.4 kW and the last 90% that corresponds to 3.6 kW.

The first step consists to observe the influence of different operating points on the impedance measurements. We followed the same procedure described in paragraph II.A to measure the CM and DM impedances for the different points; the following curves have been obtained: These measurements show that the variation of the load has no measurable impact on the equivalent DM & CM impedances. These results look to be logical because the physical structure of the system under test has not changed with the operating point. This allows us to maintain the same impedance model presented in Fig. 3 for the different operating points.

The second step consists to observe the impact of the load variation on the current measurements. We followed the same steps as described in Section II.2 to obtain the equivalent CM and DM disturbance sources spectrum; they are shown in Fig. 11 and Fig. 12. Considering these spectra, we note that the change of operating point has no impact on the CM source, only the DM one is impacted. These results look to be logical because the CM current mainly depends on the variation of the voltage in the switching cells [START_REF] Jeannin | Le transistor MOSFET en commutation: Application aux associations série et parallèle de composants à grille isolée[END_REF], [START_REF] Duc | Circuit de pilotage intégré pour transistor de puissance[END_REF], [START_REF] The Van Nguyen | Circuit générique de commandes rapprochées pour l'électronique de puissance[END_REF]. The value of this CM current can be evaluated by the equation:

𝐼 𝐶𝑀 = 𝐶 𝑑𝑉 𝑑𝑡  
However, the variation of the switching voltage depends mainly on the gate resistance RG: Therefore, a low RG gate resistance causes fast switching, with higher dv/dt dynamics and, consequently, the CM current is more important. However, a large RG gate resistance value causes slow switching, with smaller dv/dt dynamics and hence a low CM current [START_REF] Duc | Circuit de pilotage intégré pour transistor de puissance[END_REF]. In our case, when we change the operating point of the power converter, we change the power and subsequently the current dissipated by the load. However, the control of the converter (gate resistance, gate voltage) remains the same. Therefore, the CM current does not change significantly.

On the other hand, we observe a difference between the curves of the different operating points from the switching frequency up to 2 MHz for the DM source. These results look to be logical because the DM current mainly depends on the load current and on the diode recovery mechanism [START_REF] Loudière | Modélisation d'une chaine d'entrainement électrique d'un point de vue CEM -Impact de la Température[END_REF]. The increase in the load current increases also the maximum recovery current and thus the disturbance levels. The overcurrent propagates in the network and its influence can be observed in the DM current measured at the network level.

To clearly observe the evolution of the DM current as a function of load variations, we have firstly represented the curves of IDM for which the envelope of the DM source is the largest for three operating points (see Fig. 14). Then, we have represented these curves in 3D to observe the evolution of the DM current according to the frequencies and the variation of load power at the same time (see Fig. 15). Finally, the (Fig. 16) presents the DM current as a function of the load power variation for different frequencies. From these curves, we note that the DM current evolves quite linearly with the variation of load power. This type of behavior is very interesting because the evaluation of the DM current in the frequency domain can be done for any operating point even if it has not been identified during the measurements. A few operating points (at least 2) are sufficient to inform the model parameters.

In our case, we have chosen to identify the model in the worst case (the operating point for which the harmonics of the current are maximum) in order to size the filter. Consequently, it will be valid for the other operating points at lower output power. It would be possible to do the same for the modeling of an electrical distribution network (system approach) on which several converters would be connected.

IV. CONCLUSION

Reducing the volume and improving the efficiency of EMC filters of power equipment are strategic issues, particularly in the case of complex systems such as those found in aerospace or automotive industries. In this paper, a characterization procedure for a DC-AC converter has been presented. Then, a study of the impact of its operating point was carried out, in order to evaluate the evolution of the parameters of the EMC model with the load current. This method makes it possible to build a behavioral EMC model in the frequency domain of an existing converter. Note that this type of model can be built from an extensive model (predictive), this reduces the runtime of the model.

Fig. 3 .

 3 Fig. 3.Converter equivalent model of impedance for EMI analysis

Fig. 4 .

 4 Fig. 4.Comparison between measured and calculated input impedances

Fig. 5 .

 5 Fig. 5.CM and DM sources associated with the power converter equivalent model of impedance

Fig. 6 .

 6 Fig. 6. Equivalent model of the LISN In order to improve the accuracy in the calculation of the two equivalent sources, currents I1+I2 and I1-I2 have been directly measured on the test bench using two identical current probes (Pearson Model 6595, 0.25 V/A, 50 Ω) for a given operating point (VDC = 540V et IDC = 3,6A). The acquisitions at this step are performed using an oscilloscope LeCroy (WaveRunner HRO 64 Zi 400 MHz 12bit 2gs/s). The FFT of the time domain measurements are then processed. The frequency domain data are corrected by applying the inverse transfer function of the sensors, leading to the current values, i.e. I1+I2 and I1-I2. The CM and DM sources are then calculated using (9) and (10). The calculated spectrum corresponding to CM and DM sources are shown in Fig. 8 and Fig. 8.

  .

Fig. 7 .Fig. 8 .

 78 Fig. 7. Calculated spectrum of CM sources (FFT)

Fig. 9 .

 9 Fig. 9. Calculated and measured LISN voltages

Fig. 10 .

 10 Fig. 10. Measured impedances for four operating points

Fig. 11 .

 11 Fig. 11. CM source for different operating points

Fig. 13 .

 13 Fig.13. Electromagnetic disturbance due to the CM[START_REF] The Van Nguyen | Circuit générique de commandes rapprochées pour l'électronique de puissance[END_REF]

Fig. 14 .

 14 Fig. 14. DM current spectra for different operating points

ACKNOWLEDGMENTS

The authors would like to express their thanks to DGAC that funds the project.


Alain Alcouffe
email: alain.alcouffe@ut-capitole.fr

Fanny Coulomb
email: fanny.coulomb@sciencespo-grenoble.fr

FROM PEACE THROUGH FREE TRADE TO INTERVENTIONISM FOR THE PEACE : THE DEVELOPMENT OF J.M. KEYNES' THOUGHT FROM THE FIRST TO THE SECOND WORLD WAR

Keynes became advisor at the British Treasury in 1915 and was then directly interested in the question of war finance. He vainly opposed the option of total war for Great Britain, even being on the brink of resigning from the Treasury when the British government called for conscription.

After the German capitulation, his book The Economic Consequences of the Peace (ECP) published in 1919 had a great impact, as a major criticism of the conditions of the Treaty of Versailles. Keynes then plays a central role in the debate on the economic future of Germany

The link between Keynes and the economy of war is rather paradoxical. On the one hand Keynes played a significant role in his country's war effort during the two world conflicts and recognized the stimulus on growth induced by European rearmament, on the other hand, his inclination towards different dimensions of pacifism manifested itself on many occasions in his life, as in the conclusion of the General Theory, which ends with hope that the envisaged reforms of capitalism will lead to a world without war. We propose to show that Keynes, even if he has not developed a true economic theory of war and peace, was nevertheless constantly concerned to promote solutions. The expression "military Keynesianism", which was widely used during the US-Soviet arms race, helped to raise suspicions of Keynes' sympathy for rearmament or war as privileged economic stimulus, or even for authoritarian regimes. His hostility to trade liberalism in times of crisis also fostered the image of a Keynes who was a proponent of economic nationalism, the latter often paired with the praise of militarism, as illustrated by the case of Friedrich List in his National System of Political Economy (1840). A reading of Keynes' writings can however undermine these assumptions. Indeed, pacifism has clearly marked his thinking and his concern to resolve the economic crisis is directly linked to his hostility to war, which he considers nevertheless inevitable in the absence of a return to prosperity.

After presenting the asserted pacifism of young Keynes (part I), we will explain that his thinking of the 1920s is marked by a radical critique of the spirit of revenge and the call for international cooperation (Part II). Finally, we will show how, in the 1930s, when the economic crisis is prolonged and rearmament accelerated, he will propose an original economic theory, in breach with liberalism, whose validity can be tested by the increase in European military expenditure (Part III).

I -FROM THE WAR OF THE BOERS TO THE FIRST WORLD WAR: A STATED PACIFISM

The young Keynes is marked by the liberal thought and pacifism, which was widely spread among economists at the beginning of the 20th century 1 . Close to conscientious objectors, he is in accordance with the ideas developed by the future Nobel Peace Prize winner Norman Angell, who calls for European disarmament.

Hostility to Recruitment and Militarism

It was during the Second Boer War (October 1899-May 1902) that Keynes, who had been studying at Eton College since September 1897, was confronted directly with the war 2 . Indeed, at the beginning of the year 1900, the directors of the college informed the students that it was their duty to join the ranks of the volunteers 3 . Keynes asks his parents for advice and, faced with their lack of enthusiasm, decides to stay away, exposing in a letter to his parents his lucidity about the patriotic outbursts:

« About the volunteers-I have not joined .. . I think that without your letter which amounted to a refusal I should have been compelled to be engulfed in this marvellous martial ardour that has seized the school. Some say that patriotism requires one to join the useless Eton shooters, but it seems to me to be the sort of patriotism that requires one to wave the Union Jack. » (ibidem). These reservations of a very young man towards the recruitment do not necessarily translate a conviction pacifist but four years later he reaffirms, in his essay on Burke: « there is a certain minimum of personal freedom which should never be infringed » . 4 This hostility to the recruitment is reflected in his memoir My early beliefs where he writes that he and his friends believe « in a continuing moral progress by virtue of which the human race already consists of reliable, rational, decent people, influenced by truth and objective standards, who can be safely released from the outward restraints of convention and traditional standards and inflexible rules of conduct, and left, from now onwards, to their own sensible devices, pure motives and reliable intuitions of the good. ».

5 and is found in the letter he sent to the tribunal in February 28, 1916, for the recognition of his conscientious objection :

« I claim complete exemption because I have a conscientious objection to surrendering my liberty of judgement on so vital a question as undertaking military service. I do not say that there are not conceivable grounds on which I should voluntarily offer myself for military service. But having regard to the actually existing circumstances, I am certain that it is not my duty to so offer myself, and I solemnly assert to the Tribunal that my objection to submit to authority in this matter is truly conscientious. I am not prepared on such an issue as this to surrender my right of decision, as to what is or is not my duty, to any other person, and I should think it morally wrong to do so. »

6

This document distances itself from a pacifism of principle since it admits circumstances in which he could engage in war. The positions he asserts are no doubt the result of multiple influences, but the hesitations to conscientious objection show his concern not to be cut off from his pacifist friends. In fact, as soon as the war broke out, Keynes put his talents at the service of the finances of his country, as shown in his September 1914 article in the Economic Journal

7

, then in his analyzes of German finances (1915) and his activities at the Treasury, which suffice to exempt him from conscription. In fact there is a coherence between Keynes's positions in his works devoted to the First World War, as shown in particular in his article on the German war economy. Keynes, editor of the Economic Journal, published reviews of German economic literature. The publication in December 1914, January and March 1915 of special issues of the journal Archiv für Sozialwissenschaft und Sozialpolitik devoted to the economic analysis of war could not escape him. He first analyzes the articles dealing with financial problems and notes the analogies with the situation in the United Kingdom, which he studied in his September article:

« The state of credit in Germany during the first two or three weeks of war seems to have resembled that which we experienced here much more closely than we have known hitherto. ». This issue also contains an article by Professor Brentano on the war and on the avoidance of its recurrence. Professor Brentano bases his hopes on the internationalisation of the sea and of the establishment in the future of such union between civilised peoples as may make impossible the recurrence of such misfortunes as those which the world now suffers. »

Keynes also reports an article by E. Bernstein on the failure of the general strike to oppose the war. He explains in detail how the German Social-Democracy found itself trapped by the chain of events by which each country could consider that it was a defensive war for him and Keynes takes advantage of it to attack all the warmongers: « It is the horrible paradox of war and the perpetual scourge of peace parties in all countries, that when once war is joined it is for all nations a war of defencea scourge of which those who, I will not say love war, but hate lovers of peace, do not spare the use. » On the other hand, he is pleased with Bernstein's manifestation of an undiminished love of internationalism and international amity, which he contrasts with an article by Edgar Jaffe (March, 1915) on the militarization of economic life. This is an opportunity for Keynes to denounce the 'German myth', shared by a minority of the German population, which recommends economic isolationism, 'permanent war' (and thus economic militarization), and the refusal of individualism, with a regulation system based on the State apparatus, aiming at an increasing effectiveness rather than human welfare. On this point he speaks of an 'hideous doctrine' 8 . However, according to him, such theories are largely minority among German economists.

« What is the general impression produced on the mind of an English reader of these Kriegshefte? Principally, I think, that Germany and Germans are not so different from the rest of the world as our daily Press would hypnotise us into believing. The German myth, which is currently offered for our belief, is of a superhuman machine driven by inhuman hands. The machine is a good one, but has by no means moved with such uncanny smoothness, as we come too easily to believe when it is hidden from us by a curtain of silence. Nor are the drivers, after all, so changed from what before the war we used to think them. In spite of Professor Jaffe, the general note is of moderation, sobriety, accuracy, reasonableness, and truth. » This respectful analysis of the German economists must be compared with the recommendations contained in the letter to the Times of A. Marshall of August 22, 1914, in which, if the latter accepted the idea of a defensive war in view of the risks to England posed by the installation of the Germans in Antwerp, he asserted his esteem for the German people (exceptionally conscientious and upright, sensitive to the calls of duty, tender in their family affections, true and trusty in friendship) . « It is therefore our interest as well as our duty to respect them and

8

"The Economics of War in Germany", The Economic Journal, Vol. 25, No. 99 (September 1915), pp. 443-452. make clear that we desire their friendship, but yet to fight them with all our might. » 9 . This letter from Marshall provoked strong criticism. In a period of patriotic excitement, Marshall's love for Germany was taken as a provocation and unleashed anger against him.

The influence of Norman Angell's active pacifism

The peaceful liberal movement was very active on the eve of the First World War, especially in Great Britain. Thus, in 1910, in a real best-seller, The Great Illusion, a revised edition of a 1903 book, the British economist Norman Angell 10 asks governments to realize that the time for wars is over and to proceed to disarmament. Over and above strictly economic considerations, Angell portrays the progress of civilization as bringing about a reduction in the psychological influence of physical force, and human combativity tending more and more to transfer to the intellectual level. Thus, intellectual strength would gradually replace physical force, including at the level of the army, whose effectiveness increasingly depends on engineers and technicians 11

. Psychological progress must also lead to collective accountability, awareness of the need to renounce the use of physical force between states, thereby destroying the "homogeneous personality of States."

12 Denouncing the international arms race 13 , The idea that "the prosperity of nations depends on their political power, from which flows the necessity of protecting themselves against the aggression of other nations, desirous of diminishing our power to their advantage" is a "Gross and dangerous mis-conception", an "illusion" 14 . He wants to show that "no other nation could gain any advantage from the conquest of the British colonies, and Great Britain could not suffer material damage by their loss ..." He also opposes the idea that a nation can be enriched by war, since the war indemnity itself fails to cover all the costs generated by the conflict and its preparation 16 . On this point, Angell expresses ideas as early as 1910 which will be found in Keynes' The Economic Consequences of Peace (ECP). According to Angell, it is useless to impose a war indemnity, which could only be paid by exports to the recipient country, causing a social crisis due to the "dumping" represented by these devalued foreign products. The negative effect of the receipt by Germany of the payment by France of 225 billion gold marks at the end of the war of 1870-71 is an illustration of the undesirable nature of the reparations. , 1997 and 2007. 20 Op.cit.

. It was finally William Ashley who was asked to write a report on this subject. Ashley then asks the assistance of Keynes, who accepts in June. At the instigation of Ashley is translated a note of Adolph Wagner imputing the difficulties of Germany after the War of 1870 to the indemnity paid by France. Finally, after a few shuttles, the report was delivered by its two authors on 2 January 1917 22 . The memorandum states that the payment of an indemnity generally has negative effects on the economy that pays it and positive ones for the recipient economy. The memorandum estimates that the effect of the indemnities paid by France from 1871 was at first positive for the German economy but was then cancelled out by the speculation that followed. In particular, the report rejects the idea that the indemnity stimulates the trade of the defeated country and is an obstacle to the exports of the winners. The report does not, therefore, exclude the idea of reparations, and in in that regard it constitutes a disavowal of the positions expressed by Angell before the war.

The Ashley report, however, questions Germany's ability to pay and advocates moderation in the demands of the victorious allies. Keynes was to return to this Activities , 1914[START_REF] Dostaler | Le partage de l'Afrique et le traité de paix[END_REF], (London, MacMillan, 1971, pp. 311-330) question at the end of 1918

23

. He sets out as the starting point for negotiation a total income paid by Germany (reparations and others) not exceeding 3 billion pounds : an effective payment of 2 billion, if paid without adverse consequences, was considered as being very satisfactory in all cases.

The criticism of war indemnities The Economic Consequences of the Peace

The criticism of the reparations imposed on Germany by the Treaty of Versailles, developed in ECP ( 1919) 24 , has remained famous. Keynes went so far as to leave the British Treasury to protest against the reparations, and he militated firmly for a revision of the treaty and even for a cancellation of all reparations. He talks in ECP of a "Carthaginian peace", leading to the destruction of Germany, which if forced to pay the expected amount of reparations will plunge definitively into the crisis and even famine. And then the reaction can be terrible. Keynes considers that the economic collapse would exacerbate the risk of a workers' revolution in Germany, making the spread of Bolshevism in that country and its strengthening in Russia likely. But he also evokes another scenario, that of an authoritarian reaction, with the setting up of a regime having the ambition to dominate Europe 25 . In 1921 he reaffirmed that, in Germany, the economic weakening added to the political humiliation of the Treaty of Versailles could lead to a power takeover by the military 26 . The Allies must be wary of an "authoritarian reaction" as much as of Bolshevism, as it could lead to a world war.

On this issue, he was much criticized by American economists, but even more by French economists, notably for his condemnation of the French plan of the occupation of the Ruhr (the French government considering that Germany was voluntarily escaping its payment obligations). In ECP, Keynes is himself very critical of the French demands and the intransigent position of Clemenceau 27 . His opinion is that the French are not aware of the disastrous economic implications of reparations for the whole of Europe. He regretted in 1930 that economic science was not sufficiently disseminated in French academic and official circles 28 . In ECP (1919), he wrote that the main issue is not whether the reparations are morally justified : only economic pragmatism should influence the political choices of Allied governments. And precisely, Keynes puts forward the argument of international economic interdependencies: the political measures aimed at weakening the German economic potential are short-sighted, as an economic recession in Germany would spread to the rest of Europe. Since Germany was the economic heart of Europe before the war, its economic collapse would directly destabilize several Central European countries, which need a strong German economy 29 . To explain the problems with the Treaty of Versailles, Keynes seeks to prove that Germany does not have the capacity to pay. After presenting the three forms of payment of reparations by Germany (transfers of wealth in the form of gold or foreign securities, value of ownership of the ceded territories, annual payments partly in cash and partly in goods such as coal, potash and dyes), he writes :

« It is evident that Germany's pre-war capacity to pay an annual foreign tribute has not been unaffected by the almost total loss of her colonies, her overseas connections, her mercantile marine, and her foreign properties, by the cession of ten per cent of her territory and population, of one-third of her coal and of three-quarters of her iron ore, by two million casualties amongst men in the prime of life, by the starvation of her people for four years, by the burden of a vast war debt, by the depreciation of her currency to less than one-seventh its former value, by the disruption of her allies and their territories, by Revolution at home and Bolshevism on her borders, and by all the unmeasured ruin in strength and hope of four years of all-swallowing war and final defeat. All this, one would have supposed, is evident. Yet most estimates of a great indemnity from Germany depend on the assumption that she is in a position to conduct in the future a vastly greater trade than ever she has had in the past. » Describing the trade imbalances of France and Italy, as well as the disastrous budgetary situation of these two countries, coupled with a galloping inflation, he explains that a weakening of Germany would have an aggravating effect on all European economy 32 . As early as 1918 Keynes had pointed out the possible perverse effects of reparations, which could force Germany to expand its exports, thus worsening international economic competition. Reparations must therefore only absorb the normal German trade surplus, which could have been spent on foreign investment and exports 33 .

In 1920, returning to the question of reparations, he added that it was not justified to hold Germany solely responsible for the war, what was encapsulated within the idea of "reparations". He reproached the injustice of the Treaty of Versailles and its lack of respectability, for it did not respect the fourteen points of Wilson (the basis on which the Germans accepted peace) refusing to pay punitive damages. On this point (already discussed in the first chapter of the ECP), he denounced Wilson's incomprehensible turn, which first had "the words of a philosopher," and then submitted to the "reality of power ". "On some subjects, we have not kept faith. He recalls that there are eternal reasons for man to keep his promises and calls for a "crusade for the inviolability of international commitments" 34 .

Poor reception of Keynes ideas in France

In France, the positions of a Keynes judged "Germanophile" arouse much indignation. After the First World War, France's economy was very weak and the idea that "Germany will pay" was dominant.

Many whose character is warlike and democratic aspirations uncertain. Blondel 36 shares this point of view. André Tardieu (1921,16), Clemenceau's advisor to the Treaty of Versailles and future President of the Council, expressed his fear of the future consequences of the aggressiveness of the German people, considering that in Germany, "university intellectuals" provide the "philosophical justification of a war of rapine ". He is directly opposed to Keynes's argument about the political risks of too high reparations: "You say that there is a risk of provoking a nationalistic irritation in Germany. The German defeat already created this feeling. Hence the need to protect against a risk that will exist anyway. " 37 Germany's banning from the international community is however disputed by some economists. As early as 1921, the French economist Charles Rist, strongly influenced by Keynes, emphasized the concrete difficulties of a practical implementation of some provisions of the Treaty of Versailles. Moreover, German payments to France were very low in the early post-war years, and in 1924 the Dawes Plan decided to suspend them until Germany's financial situation was restored. Charles Rist refused, however, to adhere to the pessimistic predictions of the British economist regarding the non-payment of reparations, and endeavoured to seek the practical means by which these payments could be made obtaining foreign resources by means of an international loan, or, if it is not possible, by internal taxation or borrowing. In the latter two cases, "the resources obtained would be in national currency. To ensure its transfer, Germany would be obliged to seek foreign currency on all financial markets. The price of the German currency in relation to all other currencies would fall, apart from any measure of inflation and without any appreciable variation in its domestic purchasing power. Due to the positive disparity of the mark thus obtained, German exports would be stimulated, to the extent exactly necessary to supply the State with the foreign currencies it would need. In order to procure them, it would only have to buy them in German currency, from its exporters. "

41 Later, Rueff will oppose the principle of the Dawes Plan of the limitation of transferable cash, which may be lower than the tax levy for reparations, in the event of the risk of monetary problems 42 . Rueff's argument, widely endorsed by French liberal economists, brings water to the supporters of the solution of "making Germany pay".

Several years later, the controversy was revived by the Second World War, which naturally gave rise to many reflections on its causes. In his book against ECP published in 1946, "The calumniated peace or the Economic Consequences of Mr. Keynes"

43

, « La paix calomniée ou les Conséquences Economiques de M. Keynes » the French historian Etienne Mantoux accuses Keynes of having caused a polemic on the reparations which contributed to nourishing a feeling of injustice in Germany, creating the conditions of the Second World War. Meanwhile, Mantoux revisits the question of Germany's capacity to pay at the time of the Treaty of Versailles, believing that Keynes' forecasts of his possible economic collapse were unfounded.

Keynes has never spoken beyond the 1920s on this issue, especially since his reflection radically evolved after the publication of the ECPs, which were based on 41 « Les ressources obtenues le seraient en monnaie nationale. Pour en assurer le transfert, l'Allemagne serait amenée à rechercher, sur tous les marchés financiers, les devises étrangères. Le cours de la monnaie allemande par rapport à toutes les autres monnaies baisserait, en dehors de toute mesure d'inflation et sans qu'il y ait variation sensible de son pouvoir d'achat intérieur. Du fait de la disparité positive du mark à l'étranger ainsi réalisée, les exportations allemandes se trouveraient stimulées, dans la mesure exactement nécessaire pour fournir à l'Etat les monnaies étrangères dont il aurait besoin. Pour se les procurer, il ne lui resterait qu'à les acheter en monnaie allemande, à ses exportateurs. »Jacques [START_REF] Rueff | Le change, phénomène naturel, dans Théorie monétaire[END_REF], Le change, phénomène naturel, dans Théorie monétaire, Volume II, Plon, Paris, 1979, pp. 141-180, p. 173. 42 Jacques [START_REF] Rueff | Une erreur économique: l'organisation des transferts, dans Théorie monétaire[END_REF], Une erreur économique: l'organisation des transferts, dans Théorie monétaire, op. cit., pp. 181-200, p. 183. 43 Mantoux, Etienne (2002[START_REF] Mantoux | La paix calomniée ou Les Conséquences économiques de M. Keynes[END_REF]), La paix calomniée ou Les Conséquences économiques de M. Keynes, L'Harmattan, Paris. a liberal theoretical corpus. Thus, according to Turnell 44 : « Keynes was in effect to repudiate much that he wrote on the links between economics and war in ECP. Of course, it is also true that the economics in ECP, and Keynes's analysis of the transfer problem that he was convinced German reparations would impose, was based on an orthodoxy he would spend the rest of his life in trying to change".

As we shall see later, Keynes also distinguished himself in the late 1930s from the determined pacifism of a part of the radical European left.

The call for international economic cooperation to restore economic prosperity

In ECP Keynes asserts that an economist justifies his existence only if he participates in the elaboration of concrete solutions to improve the functioning of the liberal economy. It is therefore necessary to adopt a particular methodology, namely the observation of concrete problems as the basis of an original economic theory. He predicts the revolution in Europe if nothing is done to educate public opinion, whose hidden currents are ultimately more determinant for the survival of the system than the actions of politicians 45 . In 1923 he reaffirmed that war should be avoided as much as possible, not because it involves an economic cost, but because it is contrary to human nature 46 . However, adequate political and economic structures must be put in place. To better reconcile the imperatives of international peace and the defence of national economic interests, Keynes calls for international economic cooperation. In ECP, he links the issue of reparations to the question of interallied debt. The argument is that the danger of a debt crisis is weighing heavily on the world economy and is all the greater because, for the first time, the debt burden is not represented by "true capital" and is not "limited by the system of ownership". According to Keynes, the system of large-scale debt between governments is not natural, nor in accordance with human nature, and no people can long consent to work to fulfil a foreign payment 47 . With the same concern for international co-operation, Keynes requested, as early as 1919, financial assistance from the United States, in the form of an international loan, to help restart an European production penalized by trade deficit and by monetary disturbances 49 . At the 1919 Financial Conference in Amsterdam, he suggested the creation of a new international currency and settlements in the form of barter (compensation). He also proposes a free trade area in Europe to restore the economic efficiency destroyed by the innumerable new European borders and to promote peace 50 .

Keynes' concern to find solutions to pacify international relations was reflected in all his foreign policy proposals. As early as 1921, apart from the cancellation of inter-allied debts 51 , he advocates disarmament, in the belief that national economic structures could benefit from the savings achieved by reducing military spending. 52 It may be noted that the idea of a global disarmament to promote economic recovery was widespread in the aftermath of the war. Thus, the US Treasury, when it refuses to make a new loan to Europe in 1919, advocates disarmament, adequate taxation and domestic loans for the healing of the world. In the framework of the Subcommittee of the League of Nations for the Limitation of Armaments, Keynes proposes the limitation of military expenditure without success

53

.

Finally, at the Genoa Conference for the Stabilization of International Trade in 1922, he called for an international economic agreement. The maintenance of political risks in Europe necessitates this agreement, which must help to organize international relations in terms of peace, commercial freedom and economic wealth, so that "being a citizen of the European Society allows us to enjoy again the old individual liberties" 54 . Keynes adopted a liberal discourse at this period, implicitly linking world peace with the development of international trade regulated by international agreements that prevented protectionism and aggressive trade policies. against Japan would have a positive effect 61 . More generally, he is confident in the effectiveness of economic sanctions:

"I am sure that the world greatly underestimates the effect of economic sanctions. "

62

In 1935 he proposed strengthening the League of Nations by limiting its membership to those countries that truly wished to make peace. He now thinks that it is too late to speak of disarmament, and that, on the contrary, the L.O.N. Must strengthen its economic and military power to be a deterrent to law-breakers. He deplores, with a biting irony, the impasse in this project, because of the refusal of the United States, the keystone of the system, to participate in it.

"However, we are told that it is an impossibility; The United States is too frightened by their ties with the outside world, they are too far from the fate of the rest of the world and even civilization to make the necessary commitments. "

63

Keynes does not believe that the confrontation between democracy and totalitarianism can be resolved by armed conflict. Convinced that the adherence of public opinion can only be achieved by adhering to a moral ideal embodied in particular by a new economic theory, he denounces the uselessness of a new "war of religions".

"I consider with disgust the growing tendency to make the struggle between the two ideologies (or would it be conceded that there are three?) another War of Religion, to believe that the outcome can or will be determined by the strength of arms, and to think that it is our duty to make haste in all the corners of the world where those of our faith are oppressed. Let us suppose that the war is taking place, and suppose for argument that we win. And after ? (...) What are we going to do about the defeated? Are we going to impose on them our favorite ideology (whatever it is by then) in an up-to-date peace treaty, or do we assume that they will adopt it with spontaneous enthusiasm? (...) The defeat would be a complete disaster. Victory, as usual, would be useless, and probably pernicious. It is an illusion to believe that the conscience of guilt in the necessary murder can regulate what is primarily a moral issue. Therefore, and furthermore, I maintain that calls for peace are of the utmost urgency. "

64

A controversy subsequently emerged, with some critics of Keynes accusing him of having fostered sympathies for the Nazi regime, advancing the evocation of totalitarianism as a privileged regime to apply economic interventionism in the preface to the German edition of the General Theory in 1936. Such an argument seems difficult to accept, as Keynes's pacifism and his attachment to democracy and individual freedoms are reflected in all his writings. In addition, the precise authorship of this preface is subject to caution 65 . Keynes strongly criticized the national autarchy policies of the 1930s, with the development of fascism, Nazism and communism. We have already mentioned his rejection of German economic nationalism in the 1910s (see Part I). In his 1933 article on national selfsufficiency 66 , he explained that protectionism should be treated with caution; "For I must not be supposed to be endorsing all those things which are being done in the political world today in the name of economic nationalism. Far from it. " (1933: 244). On the other hand, Markwell recalls that in the mid-1930s Keynes spoke of "brigand powers" about Germany, Italy and Japan. security arising from the international situation and maintaining their calls for peace until the eve of the world conflict. A few months before the Munich Conference of September 29th, 1938, he opposed the wait-and-see policy of the Allied governments, and he advocated against the ineffective "negative pacifism" which Lloyd George and Chamberlain had hitherto opposed to Hitler, A "positive pacifism," which he said was the only bulwark against the "positive militarism" that characterized Germany. He proposed the creation of a new League of Nations, more responsible, with broad powers 70 . As for the Munich agreement between Chamberlain and Hitler, he wrote:

« Neither the Prime Minister nor Herr Hitler ever intended for one moment that the play-acting should devolve into reality. »

71

All his writings leave little doubt as to his lucidity vis-à-vis the events of the period and his hostility to the strongly militarized undemocratic regimes. However, to the end, he will believe in the power of deterrence to avoid open conflict. For example, in 1938 he referred to the creation of a new "European Pact", which would be responsible for both the defence and the prevention of conflicts, through deterrence, and which would ensure the collective security of Europe by channelling forces in the direction of common action 72 . It would have three types of functions: financial assistance, blockade, a total military alliance. Deterrence is the main idea of the Pact. This institution would be mainly devoted to European problems; For the rest of the world, alongside a European League, similar institutions could be set up in America (the "American League"), the Pacific League, or the African League 73 .

The rejection of liberal globalization

The persistence of the economic crisis in Europe and the United States in the interwar years persuaded Keynes of the need to find other solutions to save the capitalist system from external challenge or internal destabilization. In 1934 he wrote that the inadequate economic policy measures taken in response to the economic crisis stemmed from a misunderstanding of economic mechanisms, still based on the same liberal theory as in the eighteenth century, which was henceforth unsuited to contemporary economic realities. Thus, the "standard system" of liberals and Marxists must be abandoned, with the establishment of a new "scientific paradigm"

74

. State policy in the economic field is then presented as the main remedy against the crisis. Despite his closeness to the Liberals in linking peace and economic prosperity, Keynes no longer considers laissez-faire to guarantee either of these two goals, and advocates interventionist economic policies.

In 1938 he strongly reaffirmed that economic theory, in the face of the temptation of totalitarian ideologies, which was harmful to world peace, must be fundamentally an instrument of defence for the liberal regime. The role of the economist is essential, at any rate likely to become decisive.

« We are learning to honour more that formerly the achievements of our predecessors and the Christian civilisation and fundamental laws of conduct which they established in a savage world. We are seeing and enduring events, worse that which have not been seen and endured since man became himself. If we still recognise the difference, not merely between peace and war, but between good and evil, and between right and wrong, we need to rouse up and shake ourselves and offer leadership. »

75

In his famous article of 1933, National Self-sufficiency, Keynes recalls that he was raised in the conviction of the need for free trade:

« I was brought up, like most Englishmen, to respect free trade not only as an economic doctrine which a rational and instructed person could not doubt, but almost as a part of the moral law. » 76 Indeed, in his early writings, Keynes firmly denounced protectionism. In 1910 he made a speech at the Cambridge Union against the reform of customs duties: "Protection is a flower which grows in rank soil. Free trade was a barrier to "internal corruption" generated by protectionism

77

. And in 1922 Keynes called on Great Britain to resist the wave of protectionism in Europe, dismissing the idea that it could solve the problem of unemployment 78 . But in 1933, Keynes's thinking on international trade evolved. According to him, the idea that economic liberalism is synonymous with international peace has not been validated by the facts. He accuses the search for maximum international specialization and international dissemination of capital to be responsible for aggressive national policies conducted in order to protect particular interests, conquer new markets and increase economic imperialism. He has doubts about the economic benefits of an increased reliance on economic structures on foreign capital and external policies. He considers that capital flows (especially when the relationship between marginal efficiency of capital and interest rates is more advantageous abroad) impede the development of sound economic policies and he denounces the separation between ownership and management of firms. While reaffirming his pacifism, he proposed in 1933 a radical break with the orthodox model: « For these strong reasons, therefore, I am inclined to the belief that, after the transition is accomplished, a greater measure of national self-sufficiency and economic isolation between countries that existed in 1914 may tend to serve the cause of peace, rather than otherwise. At any rate, the age of economic internationalism was not particularly successful in avoiding war; and if its friends retort, that the imperfection of its success never gave it a fair chance, it is reasonable to point out that a greater success is scarcely probable in the coming years.» 79

The economic revival through European rearmament, a validation test for the General Theory Regarding military expenditure as a specific public expenditure, Keynes is aware of their possible effectiveness against unemployment 80 . The stimulation of economic activities linked to arms production should lead to an increase in employment and wages, which could lead to an increase in demand and hence in production. Assuming a 150 million pounds increase in military spending, Keynes concluded that this could help overcome unemployment, because of their multiplier effect:

« If so, the direct effect of the armament expenditure may be to take 300,000 men of the dole. I fancy that even the pessimists would reckon that a fairly conservative figure. » 81 However, the increase in armaments production is only a second-best solution, which does not meet social needs and which will slow down the productive potential of the economy in the long term. Military expenditures are the most unproductive (but useful for national defence) of public expenditures. Weapons are either stocked or destroyed in the event of war; therefore their production does not The successes of German industry have in fact been observed by the European democratic countries since the mid-1930s. In 1938 Keynes explains this growth by the fact that the country makes full use of its production capacity and he advises Great Britain to refer to this example. He is in fact favourable to a flexible state planning 83 . But even if he rehabilitates interventionism, it does not mean that he favors a militarization of economic life.

In fact, Keynes regrets that the only public expenditure of his time that helps to absorb unemployment is made in the military sector. Without the military spending, the British government would never have issued public loans. However, military expenditure's effectiveness in overcoming the economic crisis can only be temporary. A policy based on other public spending would be more effective.

Moreover, in 1942, in his analysis on the European economies after the Second World War, Keynes highlighted the lack of productivity of military spending and the burden it represents on the national economy, while public finances should be used to encourage national productive capacities. He denounced Germany's ban on keeping an army and an armaments industry, saying that these measures were a means of ridding the country of a considerable economic burden 84 . The growth of German exports, at the expense of other countries, such as Great Britain, bearing the burden of international security, will be inevitable. Keynes suggests that, in compensation, Germany contributes to the financing of peacekeeping in the world. When the Second World War seemed inevitable, Keynes began to study the best way to finance the conflict without weakening the British economy. In a short paper published in 1940, How to pay for the war 85 , he applied the principles expressed in the General Theory. Public action is needed to reduce inflation, in a context of economic recovery due to the conflict. Keynes recommends an equal distribution of the financial burden between social classes, for justice and social peace. The economic recovery generates a large private saving, half of which must be levied by the tax, in order to finance the war. Additional financial needs could be met by delaying individual consumption, through state measures in agreement with trade unions, such as deferred payments. These measures could stop inflation by restricting purchasing power.

Conclusion

After the Second World War, Keynes wrote an open letter to President Roosevelt 86 , but despite the seemingly Keynesian character of New Deal economic measures, there was deep dissension between the two men. If Roosevelt favoured social measures, he remained hostile to the organization of a state interventionism financed by borrowing, a solution nevertheless advocated by Keynes, to revive purchasing power.

Keynes's renewed appeal to international economic solidarity after the Second World War was also not directly followed up: his project to control commodity prices was discarded, as well as that of an international currency managed by a global institution ensuring the equilibrium of all balances of payments through capital transfers from surplus countries to deficit countries. Keynes' thought was deeply nourished by his participation in international negotiations during the inter-world wars, in particular as adviser at the British Treasury. If he could perceive war as a laboratory of experimentation to test the validity of his theory of the effect of public stimulus spending, he can in no way be considered as a supporter of "military Keynesianism", i.e. the use of military spending as a privileged instrument of economic policy (justified notably by the expected technological spin-offs from military investments). Keynes was openly hostile to militarism and war, convinced that they were contrary to human nature. All his proposals for the reorganization of global governance were aimed at ensuring the absence of international conflicts related to economic or monetary issues. His disappointment at the failure of international economic sanctions as an effective instrument of peace was great. In September 1945 he called on Europe, as soon as the economies were restored, to return to trade liberalism, with the help of the United States, for the benefit of all in the long term. Even if this may seem contradictory to his declarations of the 1930s in favour of a return to more selfsufficiency, this shows precisely the primacy of his preoccupation with the pacification of international relations. According to him, global peace should be consolidated by international institutions guaranteeing the prosperity of all countries through interventionist measures based on the principle of international solidarity. .

BIBLIOGRAPHY

21

  Le détail de la préparation du Memorandum se trouve dans Markwell Donald (2006), John Maynard Keynes and International Relations: Economic Paths to War and Peace, Oxford & New York: Oxford University Press. (pp. 55-58), Voir aussi W. Barber Barber W. J. (1991), 'Attempts to comprehend the nature of war, 1910-20', in Goodwin C.D. (ed), Economics and national security, History of Political Economy, annual supplement, 23, Duke University Press, 75-76.. 22

  in-chief of the Journal des Economistes, calls for moderation in the demands of the victors, in order not to maintain the spirit of revenge among the defeated. However, he advocates the dismantling of the German Empire and of the Austro-Hungarian Empire as a basis for future peace. Charles Gide 40 , on the other hand, criticizes the confiscation of Germany's colonies, what will penalize the restoration of its industry and encourage the country to conquer South American and Asian markets.For Jacques Rueff, on the other hand, in opposition to the principles of the Dawes plan and defender of the mechanism of an automatic balance of payments' adjustment, there is no problem of transfer. Germany may secure the transfer by 36Georges Blondel (1920), La situations économique et financière de l'Allemagne, Revue Economique Internationale, pp. 443-460.37« Vous dites qu'on risque de provoquer en Allemagne une irritation nationaliste. La défaite allemande a créé déjà ce sentiment. D'où le besoin de se protéger contre un risque qui existera de toute façon. » Ibid, p. 445. -38 See Alcouffe A. (1997), Keynes and the French guardians of the Say's law, ESHET, 1st conference, Marseille et Coulomb F. (2000), « Charles Rist, la revanche de l'économie sur l'esprit de revanche », in Dockès P. (dir.) : « Les traditions économiques françaises 1848-1939 », Editions du CNRS, Paris. 39 Yves Guyot (1921), L'année du Traité de Versailles, Journal des Economistes, pp. 3-27. 40 Charles Gide (1919), Le partage de l'Afrique et le traité de paix, Revue d'Economie Politique, pp. 754-765.

  85Keynes J.M.(1940), How to pay for the war. A radical plan for the Chancellor of the Exchequer, London, Macmillan. 86An Open Letter to President Roosevelt John Maynard Keynes, 1933. Publishing Information.

-

  The Collected Writings of John Maynard KEYNES, edited by Donald Moggridge, Macmillan, St Martin's Press, for the Royal Economic Society.  Volume II, The economic consequences of peace (1919), 1971.  Volume XI, Economic articles and correspondence, 1983.  Volume XIII, The General Theory and after. Part I: Preparation, 1973.  Volume XVI, Activities 1914-19: The Treasury and Versailles, 1971.  Volume XVII, Activities 1920-2: Treaty revision and reconstruction, 1977.  Volume XIX, Activities 1922-9: The return to gold and industrial policy, 1981.  Volume XX, Activities 1929-31: Rethinking employment and unemployment policies, 1981.  Volume XXI, Activities 1931-9: World crises and policies in Britain and America, 1982.  Volume XXIV, Activities 1944-6: The transition to peace, 1979.  Volume XXVI, Activities1941-6: Shaping the post-war world: Bretton Woods and reparations, 1980.  Volume XXVII, Activities 1940-6: Shaping the post-war world: employment and commodities, 1980. -Alcouffe A. (1997), Keynes and the French guardians of the Say's law, ESHET, 1st conference, Marseille. -Angell N. (1910), La grande illusion, Hachette et Cie, Paris. -Angell N. (1921), The fruits of victory -Barber W. J. (1991), 'Attempts to comprehend the nature of war, 1910-20', in Goodwin C.D. (ed), Economics and national security, History of Political Economy, annual supplement, 23, Duke University Press, 75-76.. -Biard d'Aunet (1923), Le règlement des réparations, Revue d'Economie Internationale, pp. 4-40. Blondel G. (1920), La situations économique et financière de l'Allemagne, Revue Economique Internationale, pp. 443-460 -Coulomb F., Hartley K.,Intriligator M.D. (2008), 'Pacifism in the economic analysis : a historical perspective', Defence and Peace Economics, Fall.

  Keynes also supported Angell in 1931 when he successfully applied for the Nobel Peace Prize. However, as we shall see later, Keynes later departed from the pacifism of the left, considered as too wait-and-see, of which Angell was one of the representatives. New World Orders of the Twentieth Century, Failed Imagination? Anglo -American New World Orders from Wilson to Bush, Manchester University Press

	Keynes endorsed Angell's ideas, and even invited him to speak at the Political
	Economy Club in February 1912 17 .
	Norman Angell will consider in 1921 18 that the Keynesian criticism of the Treaty of
	Versailles in ECP validates his theory of 1910 on war indemnities as detrimental to
	the basic interests of the victorious countries. He admitted, however, an error in his
	book The Great Illusion: the claim that international economic interdependencies
	would prevent the occurrence of prolonged hostilities because of the risk of
	economic collapse. However, Germany had been able to endure four years of
	economic isolation and blockade without collapsing. But according to Angell, the
	German economy would not have resisted beyond this, in the absence of a declared
	conflict 19 .
	Keynes warmly welcomed Angell's book of 1921, calling it (according to Angell)
	"extremely good"
		16	Ibid., pp. 77-92.
	17	Turnell Sean (2002), Keynes, economics and war, Macquarie Economics Research Papers, 7,
	Sydney.
	18	Norman Angell (1921), The fruits of victory
	19	Williams Andrew J. Failed Imagination?:

20

.

  « Lenin was certainly right. There is no subtler, no surer means of overturning the existing basis of society than to debauch the currency. (…) The profiteers are a consequence and not a cause of rising prices. By combining a popular hatred of the class of entrepreneurs with the blow already given to social security by the violent and arbitrary disturbance of contract and of the established equilibrium of wealth which is the inevitable result of inflation, these Governments are fast rendering impossible a continuance of the social

		and economic order of the nineteenth century. But they have no plan for
		replacing it. » 31
	29	ECP, 145.
	30	ECP, chapter V.

30

He is particularly worried about the risk of inflation in Europe, which can undermine the very foundations of capitalism, preparing for a social revolution :

  French economists support the Treaty of Versailles. For Biard d'Aunet 35 , the reparations are too weak, in view of the "complicity of the German nation", Memorandum by the Treasury on the Indemnity Payable by the Ennemy Powers for Reparation and other Claims, 1918, Volume 16 p 380.

	31	ECP, chapter VI.
	32	ECP, chapter VI.
	33

34 Everybody's Magazine, September 1920, Volume 17 p 51. 35 Biard d'Aunet (1923), Le règlement des réparations, Revue d'Economie Internationale, pp. 4-40.

  The cancellation of Allied debts would be a sign of solidarity between the Allies and would contribute to the establishment of a new Europe

		48 .
	44	Turnell Sean (2002), Keynes, economics and war: a liberal dose of realism, Macquarie economic
	research papers, Sydney, p. 2.
	45	ECP, p 243.
	46	The Manchester Guardian Commercial, 4 January 1923, Volume 17 p 450.
	47	Op.cit., p 178.
	48	Scheme for the rehabilitation of European credit and for financing relief and reconstruction,
	1919, Volume 16 p 429.

  an intertemporal transfer of purchasing power, but a definitive exclusion of the factors of production from the economic process. It is unproductive consumption. Preparation for war, more than the war itself, is a temporary incentive for economic activity. But Keynes disapproved of what is now known as "military Keynesianism," because it could only have short-term effects. It is better to direct public investment towards the building sector or civil engineering, which are socially useful.« I have a special extra reason for hoping that trade-unionists will do what they can to make this big transition to fuller employment work smoothly. I began by saying that the grand experiment has begun. If it works, if expenditure on armament really cure unemployment, I predict that we shall never go back all the way to the old state of affairs. I we can cure unemployment for the wasted purpose of armaments, we can cure it for the productive purposes of peace.»

		82
	79	"National Self-Sufficiency," op.cit.
	80	'Will Rearmament Cure Unemployment?', The Listener, 1st June 1939, 528-532.
	81	Op.cit., 531.

represent

III -THE KEYNESIAN THOUGHT AS A RESULT OF A REFLECTION ON THE ECONOMIC STAKES OF WAR

As Turnell

55 recalled, Skidelsky

56

, who wrote a biography of Keynes, believed that Keynes had little competence in international relations, particularly because of his distance from the ruling circles of British foreign policy. Turnell contests this point of view, arguing that if Keynes had been able to launch a few "bad calls" 57 , his thought nevertheless evolved during the interwar period, following a thorough reflection on these subjects. Indeed, the "Keynesian revolution" has undoubtedly been largely fueled by empirical reflections on the causes of war and the conditions of world peace 58 . In Keynes's mind, the war, no matter how detestable, brings about new solutions of economic policy and international cooperation through international institutions.

Advocating for the use of economic weapons as a means of preventing a world war During the 1930s, Keynes made several proposals for political cooperation for the defence of peace, convinced of the potential effectiveness of the economic weapon. In 1929 he participated in a Draft Convention for Financial Assistance by the League of Nations which proposed the future prevention of conflicts by the by means of the financial weapon. The prospect of financial support to one of the two belligerent parties in a potential conflict would deter the other party from starting hostilities 59 . A few years later, in 1935, he defended the principle of economic sanctions against Italy, refuting the argument that this country might declare war. For Keynes, such an act would not fall within a reasonable calculation 60 . Finally, in 1937, he said that there were nine out of ten chances that the threat of sanctions

Support for the rearmament of democracies

In the interwar years, Keynes worked for peace and was reluctant until 1938 to recognize the war with Germany as inevitable. In 1937 he wrote :

« I maintain that the claims for peace are paramount. It is our duty to prolong peace, hour by hour, day by day, for as long as we can. We do not know what the future will bring, except that it will be quite different from anything we could predict." 68 However, his pacifism is not unlimited. As early as 1936, with the rise of military aggressions by the "brigand powers" and the outbreak of the Spanish Civil War, he recognized the necessity of the rearmament of European democracies, with a dissuasive aim

69

. His health problems probably prevented him from multiplying the writings on this issue in 1937-1938. But on the eve of the Second World War he was opposed to the policy of the Socialists who were considering calling on people to refuse to fight, pay taxes or work in case of war. He will even accuse the liberal anti-war economists before World War I, including Angell, of having served the interests of their country by refusing to recognize the reality of a danger to national 65 Sur ce point, voir Hagemann [START_REF] Harald | The German edition of Keynes's General Theory: Controversies on the Preface[END_REF], The German edition of Keynes's General Theory: Controversies on the Preface, Universität Hohenheim , conférence ESHET, mai 2013. 66 "National Self-Sufficiency," The Yale Review, Vol. 22, no. 4 (June 1933), pp. 755-769 67 [START_REF] Markwell | idealism and the economic bases of peace[END_REF], J.M. Keynes, idealism and the economic bases of peace, in Long David, Wilson Peter, "Thinkers of the twenty years ' crisis",Clarendon Press,Oxford,p. 197. 68 The New Statesman and Nation, 10 July 1937. 69 Letter to MSN, 13 July 1936, reproduced in CW XXVIII, p.47.