Cedric Martel

Guillaume Carrie

Nicolas Capet

Sébastien Rougerie

François Vincent

Thierry Deloues

Franck Letestu

Jérôme Sokoloff

Olivier Pascal

Cédric Martel

Multi path and jam resistant antennas for GNSS receivers

Keywords: GNSS, CRPA, mutual coupling, antenna, HIS

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

Introduction

In Global Navigation Satellite Systems (GNSS) applications, multipath errors are still one of the major error sources in conventional receivers. Furthermore, due to low SNR, single antenna GNSS receivers are susceptible to jamming threats. In the last two decades, the use of adaptive antenna arrays to mitigate the detrimental effects of jamming upon GNSS receivers has gained significant interest in the signal processing community. More recently, antenna array algorithms have also shown their ability in mitigating the GNSS multipaths. Nevertheless, the performances of antenna array are strongly affected by technological defects such as channel mismatch or mutual coupling. The first part of this paper will therefore analyze the impact of mutual coupling, on antenna array performances. These defects can be partially mitigated by signal processing techniques. But the induced computational load can quickly become crippling.

The second part of the paper will focused on the reduction of the mutual coupling at the RF level. Mutual coupling between closely spaced elements is complex and may come from various contributions such as the direct radiation through the air, surface waves propagating on the antenna ground plane and also waves propagating inside dielectric materials. High impedance surfaces (HIS) are proposed to isolate the radiating elements of the array. The integration of the HIS is optimised in an array of cavity backed patch elements. The decoupling capability of the HIS is validated experimentally with the manufacture and test of a demonstrator.

Impact of mutual coupling on antenna array performances

Data Model

Consider an array of m sensors illuminated by one useful and one jamming signals (respectively 'u' and 'j'), the output can be written:

) ( ) ( . ) ( . ) ( t t j t u t j u n a a y    (1)
where n(t) is a White Gaussian Noise. a u , a j denote the steering vectors of the array. Single sensor radiation patterns are included into the steering vectors.

The complex weight vector w is controlled in phase and amplitude by the array processor to provide the Controlled Radiation Pattern Antenna (CRPA) output y s (t):

) ( ) ( t t y H s y w  (2)
where the superscript . H stands for Hermitian. All the incoming signals are assumed uncorrelated, the autocorrelation matrix R of the array is then defined as follows:

  I a a a a yy R . . . 2 u u σ P P H j j j H u H      (3)
where P u , P j denote the powers of the incoming signals, 2 σ is the noise power and I the m order identity matrix.

Consider now that mutual coupling exists between the array sensors. Under the narrow band assumption, the array output model is modified as:

  ) ( ) ( . ) ( . ) ( t t j t u t j u n a a C y     (4)
where C is the coupling matrix. The coefficients of C are a function of the load impedance, single sensor impedance and array mutual impedance [START_REF] Svantesson | Modeling and Estimation of Mutual Coupling in a Uniform Linear Array of Dipoles[END_REF]. Usually, the diagonal coefficients of C are close to unity, and the non diagonal coefficients are very close to the ones of the Scattering matrix S.

To analyze the impact of (4) on the array performances, a 2×2 square antenna array with half-wavelength spaced patches has been simulated with HFSS. From Figure 2, it is clear that the narrow band assumption is no more valid on a 20 MHz bandwidth and the array output model (4) has to be replaced by the wide band model:

  ) ( ) ( ). ( ) ( ). ( ) ( ) ( f f j f f u f f f j u n a a C y     (5)

Mitigating Jammers

Due to low SNR (lower than -30dB with a 20MHz bandwidth for GPS BPSK(10) signals), R is assumed to contain interference plus noise only. Thus we can write:

I a a R . . 2 σ P H j j j   (6)
Classically in the GNSS context, adaptive CRPA are designed so as to receive signals from as many satellites as possible while rejecting strong interferers. The most widely used solutions consist in linearly constrained power minimization techniques. One aims at minimizing the array output power while trying to maintain the interference free case beampattern:

1 / min 0 H  w w Rw w w H , (7)
where 0 w is the weight vector design for the desired beampattern. The well known solution is given by:

0 1 0 0 1 w R w w R w    H . ( 8
)
Due to their low power, GNSS signals will not be cancelled by this technique.

The equation ( 8) can either be used to design beamformers if the useful signal Direction Of Arrival (DOA) are known a priori, or nullformers if no prior information is available. The most popular application of this latest technique, named Power Inversion [START_REF] Moelker | An Optimal Array Processor for GPS Interference Cancellation[END_REF] [3], is obtained with w 0 = [1 0 … 0] T and simply consists in minimizing the output power of the array while not weighting a reference sensor. Note that in the interference free case, the radiation pattern of the array is the same as the reference sensor. Based on ( 5), a scenario involving 3 broadband strong jammers has been simulated. The Capon space power spectrum estimated by the array is illustrated on Figure 3 with or without taking into account the mutual coupling. The stars on Figure 3 represent the jammers DOA. Without the coupling effect, the maximum power DOA estimated by the array corresponds to the true jammers DOA. But when coupling is present, the maximum power DOA are biased with respect to the jammers DOA. Furthermore, the jammers appear to arrive from a spread area and not from a discrete DOA. This latest effect contributes to considerably increase the rank of the subspace of a powerful jammer, and then reduces the rejection capacity of the array. This effect has already been observed and detailed in [START_REF] Carrie | Optimal STAP Algorithms to GNSS Receivers[END_REF] when technological defects except coupling are simulated.

To counteract the increase of the jammers subspace rank, a choice has been made in [START_REF] Carrie | Optimal STAP Algorithms to GNSS Receivers[END_REF] to add temporal degrees of freedom to the array (see Figure 4). The optimal Space Time Adaptive Processor (STAP) for GNSS applications has also been derived in [START_REF] Carrie | Optimal STAP Algorithms to GNSS Receivers[END_REF], it enables to partially compensate the technological defects including mutual coupling, but the price is a significant increase of the algorithm computational load.

Mitigating GNSS Multipaths

As already mentioned, in the GNSS context, the power of the incoming signals is usually much lower than the receiver noise level. Thus, if we want to be able to see the GNSS signals and especially their multipaths, we first need to correlate the incoming signal with the reference local code.

Classical adaptive algorithms, such as the ones of previous chapter, have tested on post-correlation incoming signals. The results obtained with these solutions are quite mitigated [5] [6]. Indeed, on the one hand, the small size of the array implies a low space resolution, and on the other hand, the correlation between the Line Of Sight (LOS) signal and the multipaths strongly degrade the performance of classical adaptive high resolution algorithms. This latest effect being the most significant one.

Then, in [START_REF] Rougerie | Comparison of SAGE and Classical Multi-Antenna Algorithms for Multipath Mitigation in Real-World Environment[END_REF] a Space Alternated Generalized Expectation (SAGE) algorithm has been tested with an antenna array in order to reduce the estimation error of the delay of the LOS signal. With 2 more dimensions (Doppler and timedelay), SAGE is able to discriminate the sources even in the presence of close multipaths. However, SAGE is hardly implemented in real time due to the memory requirements and computational cost of the algorithm.

In order to take the advantages of SAGE with more reasonable hardware requirements, a new implementation of SAGE has been proposed and detailed in [START_REF] Rougerie | Multipath mitigation methods based on antenna array[END_REF]. This new implementation is mainly based on the STAP multicorrelators array (see Figure 5). With this method, we can strongly reduce the size of the signal and thus, the complexity of the algorithm. Monte Carlo simulations tend to show similar performance between classical SAGE implementation and the SAGE / STAP multicorrelators algorithm.

 ) . . 2 exp( ) ( t j t c        ) . . 2 exp( ) ( t j t c       int T int T int T int T  int T int T  int T int T ) . . 2 exp( ) ( t j pCs t c        ) . . 2 exp( ) ( t j pCs t c       

Figure 5: STAP multi-correlators

Nevertheless, when mutual coupling is introduced in the simulations, the algorithm performances degrade as the estimated DOA are biased. Such an example, Figure 6 depicts a comparison between LOS DOA and estimated DOA through the SAGE / STAP multi-correlators algorithm. The DOA biases can be higher when considering GNSS signal multipaths whose received power is even weaker.

Once again, the mutual coupling effect can be partially mitigated by signal processing calibration techniques (e.g.: based on a coupling matrix model and prior knowledge of LOS DOA) but the price is still an increase on the computational load and complexity of the algorithm.

Mitigation of inter element mutual coupling

It has been shown in previous section that antenna array algorithms are a possible solution to both GNSS multipaths and jamming threats. Nevertheless, the performances of such techniques are degraded due to inter sensor mutual coupling. Signal processing techniques can be envisioned to mitigate these effects but the price is an increase of the computational load of already computational demanding algorithms. Then, the aim of this section is to propose an array genuinely designed to minimize inter sensor mutual coupling.

Metamaterials as isolating structures

Metamaterials structures such as EBG or others artificial magnetic conductors are attractive to reduce mutual coupling between the elements of an array [START_REF] Svantesson | Modeling and Estimation of Mutual Coupling in a Uniform Linear Array of Dipoles[END_REF].

The metamaterial structure we propose to investigate is made of a thin dielectric slab on which small patch conducting elements are printed on both bottom and top faces. The close proximity of the patches and the overlapping surface produced by the lateral layout of the patches generate a strong capacitive effect. The dielectric slab is placed above a ground plane. Some dielectric material separates the dielectric slab and the ground plane.

Metallized vias connecting the printed elements to the ground plane are commonly used. Although these vias may in some cases enhance the isolating performance of the metamaterial surface, their presence is not always necessary to achieve a decoupling characteristic.

With via

Without via The EM simulation of small GNSS array with HIS resonator is a difficult task due to the multi scale element features: the size of the radiating element is about half a wavelength whereas the size of the HIS unit cell is the order of 1/30 of a wavelength. Moreover, the large number of HIS cells imposes large amount of computational power for a full wave analysis to take place. For this reason, only a partial theoretical design optimisation of the array has been carried out with the HIS. A complete optimisation has been performed experimentally as explained in the following sections.

Description of demonstrator

The radiating elements of the array consist of patches in cavities. Each patch has two vertical probes, which can be fed in phase quadrature to generate the appropriate circularly polarised radiation patterns. A metallic support with cavities for the radiating elements and the HIS resonator is used. On the metallic support, the cavities have no bottom ground planes. With this setup, both the radiating elements and the HIS resonators can be inserted from underneath the structure in a modular fashion.

To obtain sufficient space to insert the HIS resonator between the radiating elements, it is necessary to use a material with a relatively high dielectric constant as a substrate for the patches. For this reason, a ceramic material (DK~8.2) has been chosen as the patch substrate.

For a practical implementation of the manufacture process, the patch has been printed on a thin substrate (0.127 microns) and then glued on the ceramic block. A small ground plane is present at the base of the ceramic block.

The HIS resonator has been made by printing electrically patches on either side of a thin low loss PTFE substrate of relative permittivity of 2.2. The thickness of the substrate is 0.127 mm. The HIS substrate has been fixed with nylon screws on the top of the metallic support at the dedicated locations. A metallic plate can be added to the metallic support from underneath to form the HIS bottom ground plane. The dielectric material inside the HIS cavity is air.

In addition, some adjusting parts in the shape of a ring can be inserted between the metallic support and the HIS bottom ground plane. With a set of these adjusting parts of different sizes, the height of the HIS resonator can be changed. Experimental tuning of the HIS resonant frequency position can then be performed by adding or subtracting the adjusting parts. 11). In the reference case, the array is well matched around the frequency 1.207 GHz with a -10 dB bandwidth of 37 MHz. However, coupling levels with a maximum value of -14 dB can be observed in the operating band.

In the case where the HIS is added around the radiating elements, strong interactions take place between the patches and the HIS due to the short distance separating them. In order to overcome the lack of accurate simulation analysis and the tolerance of manufacture, an optimisation of the array has been carried out experimentally. The optimisation has consisted of tuning the HIS resonant frequency to make it coincide with the patch resonance. This was achieved by varying the cavity height of the HIS resonator with the help of some adjusting ring metallic parts.

Optimum performance have been found for a HIS cavity height of 17 mm. The addition of the HIS resonator has a remarkable effect on the S parameter results (Figure 12). First of all, the patch resonant frequency is shifted 17 MHz toward higher frequencies: 1.224 GHz versus 1.207 GHz in the reference case. Secondly, the matching of the elements has been degraded in presence of the HIS. Although a slight an unbalance between the S11 and S33 is noticeable, the matching of the elements remains acceptable for a significantly large frequency band. The -10 dB bandwidth is similar with and without the HIS resonator. Thirdly, the element mutual coupling has been significantly reduced. With the HIS, the maximum coupling level is about -17 dB and can be as low as -37 dB in some part of the operating band. The impact of the HIS is noticeable with the resonance effect at 1.21 GHz.

Conclusion

In GNSS applications, multipath errors and jamming threats are still one of the major error sources or service unavailability in conventional receivers. Antenna array algorithms are introduced as a possible solution to both these threats. Nevertheless, the performances of such techniques are degraded when considering technological defects and especially mutual coupling. To face this limitation, a HIS based antenna array design is presented.

Figure 1 :

 1 Figure 1: Square array configuration and patch design The array configuration and the patch design are depicted on Figure 1. The obtained mutual coupling coefficients for the L1 GNSS band are illustrated on Figure 2.

Figure 2 :

 2 Figure 2: Array mutual coupling coefficients

Figure 3 :

 3 Figure 3: Capon space power spectrum estimated by a coupled array (top) and an uncoupled array (bottom)

Figure 4 :

 4 Figure 4: STAP array principle

SAGE

Figure 6 :

 6 Figure 6: DOA finding with a coupled array

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: High impedance surface unit cell

  The dimensions of the array are as follows: distance inter element: 112 mm (centre to centre)  cavity size: 60 × 60 × 18 mm  patch size: 35.5 × 35.5 mm  HIS width: 24 mm  HIS cavity height: adjustable from 12 mm onwards  Metallic support: 350 × 240 mm

Figure 10 :

 10 Figure 10: Array with HIS resonator

Figure 11 :

 11 Figure 11: Measured S parameters of the 2 element array without HIS resonators

This design aims at minimizing the array mutual coupling coefficients while not increasing the size of the array.

Acknowledgement

The authors would like to thank Maryse Lopez (ONERA/DEMR) and Olivier Pigaglio (LAPLACE) for their precious help with the measurement of the antenna prototype.


U Mohan Rao

I Fofana

P Rozga

Abderrahmane Beroual
email: abderrahmane.beroual@ec-lyon.fr

J Malde

R Martin

Feipeng Wang

Edward Casserly
email: edward.casserly@ergon.com

M Pompili

L Calcara
email: luigi.calcara@uniroma1.it

Next Generation Insulating Liquids Prepared by the International working Group of IEEE DEIS Technical Committee on Liquid Dielectrics

Keywords: liquid dielectrics, transformers, synthetic ester, natural ester, insulation. I

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

INTRODUCTION

Lately, ester dielectric fluids are majorly accepted for applications in high-voltage apparatuses and have been of high interest to the dielectricians and transformer engineers [START_REF] Rao | Alternative Dielectric Fluids for Transformer Insulation System: Progress, Challenges, and Future Prospects[END_REF][START_REF] Rozga | A review on synthetic ester liquids for transformer applications[END_REF]. However, with the rapid increase in the technological advancements an increase in electric power ratings is foreseen with high predicted load. The electric grid is encountered to a high severity in terms of loading and working profiles. Also, with the inclusion of the distributed power generators and deregulated conditions, this severity has triggered the need to prepare the apparatus on the power network to a wide range of vulnerable operating conditions. Hence, the power and distribution transformers are expected to be prepared for supporting such a foreseen high workability in the near future. Thus, there is a need to focus on the next generation insulating liquids, which poses a high performance to sustain a reliable operation of the high-voltage transformers.

Insulating liquids are used as an impregnant, insulant, coolant, and diagnostic media for various equipment used in the power transmission and distribution network. However, the eco-friendly nature, sustainability, and strict technical (electrical and thermal) requirements of dielectric liquids are of key concern to the utilities. Alternative liquids (ester-based) are foreseen as promising candidates to lead the future of liquid dielectrics and transform the transformer technology, in particular [START_REF] Rao | Alternative Dielectric Fluids for Transformer Insulation System: Progress, Challenges, and Future Prospects[END_REF][START_REF] Rozga | A review on synthetic ester liquids for transformer applications[END_REF]. Thus, a technical committee on liquid dielectrics is developed by the IEEE DEIS society, the focus of which is performance of insulating liquids, condition monitoring of transformers, and insulation aging phenomena [3,[START_REF] Fofana | Reports on the activities of the DEIS Technical Committee on Liquid Dielectrics[END_REF]. Inline of the above concerns, a working group is focused on to understand and summarize the intrinsic nature, behavioral aspects, and challenges of using ester liquids in liquid-filled high-voltage apparatus. The objective of finetuning the ester liquids as high-performance insulating liquids with enhanced workability is the focus of the working group. The objective is to emphasize and accumulate knowledge on the essential properties, challenges, and shortfalls of the ester liquids. The potential concerns include but not limited to additives, viscosity, ionization resistance, oxidation stability, gelling issues, thermal conductivity, regeneration, diagnosticprognostic challenges, and operation in cold regions.

In addition to the above, solid insulation plays a vital role and governs the liquid insulator's service performance. Therefore, the compatibility of the high-performance liquids with upgraded solid insulants will also be a potential subject of the working group. In this line, the primary focus of the researchers and manufacturers remains inclined towards finetuning new alternative liquids to serve as high-performance insulating liquids in the coming generations. To deal with this, it is important to recall the following [START_REF] Rao | Alternative Dielectric Fluids for Transformer Insulation System: Progress, Challenges, and Future Prospects[END_REF][START_REF] Rozga | A review on synthetic ester liquids for transformer applications[END_REF][START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF].

• The rate of degradation of ester liquids is lesser than mineral insulating liquids.

• Ester dielectric fluids proved the possible longevity of solid insulation.

• Moisture being the number one enemy, esters are hydrophilic while mineral oils are hydrophobic.

• Esters have higher fire and flash points.

• The viscosity of natural ester liquids is challenging.

• Esters have high permittivity and less ionization resistance.

• Ester liquids are colloidal stable liquids as compared to that of the mineral oils.

The primary cause for a transformer to meet the condition of failure is overheating, which accelerates the degradation rate of the composite oil-paper insulation [START_REF]Effect of particles on transformer dielectric strength[END_REF]. Thus, the heat dissipation ability of the insulating oil and thermal conductivity of the paper are key parameters. This will facilitate rapid heat dissipation and aids to achieve a control on the decay particles and other colloidal particles in the insulating liquid [START_REF]Effect of particles on transformer dielectric strength[END_REF]. On the other side, the withstand voltage of the insulating liquid also plays an important role that needs to be considered for new, aged, and heavily loaded transformers. This withstand voltage is largely attributable to the decay particles and moisture within the insulation system [START_REF]Dielectric performance of insulating liquids for transformers[END_REF]. Thus, the stability of the insulating towards the aging byproducts (water and decay particles/colloids) and heat dissipation properties are the parameters that need to be considered for a high-performance insulation system. Both of the said factors will act in extending the workability of a transformer insulation system.

Meanwhile, nanofluids are expected to play a promising role as liquid replacements for mineral oils and in improving the performance of the transformer insulation system. The application of nanofluids to large power transformers is a challenging and promising topic of research. With an effective choice of nanoparticles as additives to the insulating liquids, critical properties like, viscosity, oxidation stability, thermal conductivity (cooling), ionization resistance, dielectric strength, etc. may be significantly controlled/regulated. The performance improvement achieved through nanoparticles could be the orders of the magnitude that leads to extended workability and service life of a transformer. So, application of nanofluids may lead to drastic improvement in electrical, thermal, and chemical properties of the insulating liquid which results in enhanced workability and reduced size of the insulation system.

The future of alternative insulating liquids for transformers being the main subject of the present article, few major properties/aspects that need to be focused for improved performance of the ester liquids are discussed. The present article summarizes the state-of-the-art challenges concerning the streamer behavior in ester liquids, oxidation stability and gelling, and a brief overview on the additives for transformer liquids.

II. STREAMER BEHAVIOR IN ESTER LIQUIDS

Streamer behavior in ester liquids has been studied for many years, wherein a particular increase was observed at the beginning of the twenty-first century when synthetic esters were introduced to the dielectric liquids market for a larger scale [START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF].

One of the most important parameters of a streamer is propagation velocity, which is related to another parameter describing streamers -acceleration voltage [START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF]. For voltages below acceleration voltage, streamer velocity increases marginally, however, above this characteristic indicator propagation velocity of streamers are typically of an order of magnitude higher than these developing below acceleration voltage [START_REF] Liu | Streamer characteristic and breakdown in synthetic and natural ester transformer liquids under standard lightning impulse voltage[END_REF][START_REF] Lesaint | Prebreakdown phenomena in liquids: propagation modes and basic physical properties[END_REF]. Independently of the liquid type, streamers are generally classified as slow (1st and 2nd modes) or fast (3rd and 4th modes). Their velocities are from some mm/µs for slow developing streamers up to 10-100 mm/µs for fast developing ones. Either slow or fast propagating streamers may lead to breakdown in dielectric liquids. However, breakdown is usually noticed by a 2nd mode (slow) and/or 3rd mode (fast). The change of the streamer nature from slow to fast may be explained on the basis of the field stress at that instance. If the electric field stress is in the range of less than 10 MV/cm, it may be considered as slow streamers (in 1st mode or 2nd mode). If the electric field stress is in the range of 10 MV/cm to even 100 MV/cm, it may be considered as fast streamers (in 3rd mode or 4th mode) [START_REF] Rozga | Prebreakdown and Breakdown Phenomena in Ester Dielectric Liquids[END_REF]. However, the above-quoted values may differ when taking into account the liquids of different chemical composition, electrode geometry and type of voltage.

The current state-of-the-art in this field shows that in the case of some streamer characteristics the differences between mineral oil and esters are of minor while some other are remarkable. The first group includes among others the streamer inception voltage. However, it is reported that, in a non-uniform field, the streamers in esters (natural and synthetic) and mineral oils are initiated at similar magnitudes of testing voltage. Furthermore, the influence of the radius of curvature of the needle electrode on streamer inception in esters and mineral oils was assessed to be of the same order [START_REF] Dang | Investigations on streamers phenomena in mineral, synthetic and natural ester oils under lightning impulse voltage[END_REF]. Minor difference also concerns streamer shape. The photographically recorded streamer shapes in general do not differ from each other whether ester or mineral oil is considered [START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF][START_REF] Rozga | Prebreakdown and Breakdown Phenomena in Ester Dielectric Liquids[END_REF]. A significant parameter of streamers is their stopping length. Both for positive and negative voltage polarity, the stopping length of streamers propagating in ester liquids was identified to be longer than that of the streamers in mineral oil. In addition, having in mind that the stopping length increases with applied voltage, this should be pointed out that this increase is observed as faster for esters. The major difference between streamers developing in esters and mineral oils is acceleration voltage. Especially for longer gaps (> 25 mm) the fast streamers in ester liquids appear at much lower testing voltage than in the case mineral oil. Thus, this may be said that transition from slow (2nd mode) to fast (3rd or 4th mode) streamers in esters occurs relatively faster than mineral oils. This may be attributable to the chemical structure of esters, which contain the polar molecules (ester groups) that are characterized by lower ionization potential than the hydrocarbon groups in mineral oil [START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF][START_REF] Rozga | Prebreakdown and Breakdown Phenomena in Ester Dielectric Liquids[END_REF][START_REF] Dang | Investigations on streamers phenomena in mineral, synthetic and natural ester oils under lightning impulse voltage[END_REF]. Thus, concerning the streamer behavior in ester liquids, the further challenges may be focused on further analyzing the streamers in esters to find the way allowing for improvement their properties, especially in the field of partial discharge inception voltage and acceleration voltage. The influence of degradation aspects on the streamer behaviour is still a posing challenge to the researchers. Importantly, the influence of additives (added to improve the performance) also needs to be investigated.

III. OXIDATION STABILITY AND GELLING

Ester dielectric fluids are divided into two main types namely synthetic esters and natural esters. Both types of esters offer distinct advantages over mineral insulating oils. Synthetic esters are chemically made from a reaction between and alcohol and short chain fatty acids, whilst natural esters are extracted from naturally grown seed crops such as soybean and rapeseed/canola. This leads to both ester types having different chemical structures which impact their oxidation stability. Given the structural differences in the two types of esters, standards have been adopted and modified to test the oxidation stability of each ester type. The oxidation stability tests are typically performed to check the quality of the ester liquid rather than determine the deterioration rate of the ester liquid over a prolonged period. The details of which are discussed below:

A. Synthetic esters

The main oxidation stability test referred to in the ester specifications is IEC 61125 Method C (1992, originally developed for hydrocarbon oils). In this method, a sample of the ester is maintained at 120°C for 164 hours whilst air is bubbled through the sample in the presence of a copper catalyst. At the end of the test, the following are determined and reported:

• Sludge (%w/w more appropriate to mineral oils)

• Soluble acidity (acids dissolved in the ester)

• Volatile acidity (acids formed which exit the sample as vapor, captured in a separate flask)

• Total acidity (sum of volatile and soluble acidity)

• Dielectric Dissipation Factor @ 90° (DDF, post oxidation)

However, there is an option to test at 500 hours to allow one to understand the extra stability of synthetic esters, but this is not mandatory as per the above said specification.

B. Natural Esters

Whilst drafting the specification for unused natural esters (IEC 62770, 2013), the experts found that it was impossible to set sensible oxidation parameter limits for natural esters based on IEC 61125 Method C, as the results of the round robin tests from several laboratories showed too much variation and poor repeatability. Ultimately, the solution was to modify Method C to only 48 hours instead of 164 hours. This gave a test method which was acceptable and more robust for natural esters. As with synthetic esters, the same post oxidation parameters are reported, with the addition of viscosity (reported a % increase over the initial un-oxidized viscosity). In 2018, an optional test method variation was added to IEC 61125 as Appendix C. In this method a sample of fluid is sealed in a tube with a copper catalyst, and the sample maintained at 150°C for 164 hours. However, the post oxidation analysis is different as shown below:

• Soluble acidity (acids dissolved in the ester)

• Dielectric dissipation Factor @ 90° (DDF, post oxidation)

• Kinematic viscosity (reported a % increase over the initial un-oxidized viscosity)

It is worth mentioning that this method is purely optional and has not yet been adopted in any ester specifications. The IEEE C57.147-2018 standard lists IEC 61125, Method C as a test that can be performed for oxidation stability, however, it is not a mandated tests as it is not listed or recommended in the ASTM D 6871 standard.

It is important to understand that natural esters tend to polymerize on oxidation making the liquid more viscous. It is quite possible for the natural ester liquid to develop thin films over the large surface of the core winding assembly (solid insulation) and sols and gelling within/over the bulk of the liquid. A view natural ester subjected to thermal stressing under open beaker aging is illustrated in Fig. 1. A detailed impact of the gel formation and its impact on the insulation system will be reported in the further articles of the working group. The formation of which leads to several unavoidable consequences. However, manufacturers have come up with suitable modifications to avoid this peculiar situation. It is reported that natural esters are serving well in the global transformers with an effective condition monitoring and maintenance planning policies..

IV. ADDITIVES IN TRANSFORMER INSULATION

Historically, additives in dielectric liquids are introduced into the liquid to improve the performance and serviceability of the liquid. The typical additives may be seen as two-fold, added by the liquid manufacturer (while producing the liquid) and condition monitoring engineers (on-site). The earlier is helpful to improve the performance of the new insulating liquid and the latter is added to rejuvenate and improve the serviceability of the aged/reconditioned liquid. The additives typically include oxidation inhibitors. Lately, the additives also include, depressants, and dyes along with inhibitors. Despite the above said additives, service life extension and performance improvement are continuous elements that are within the interest of the transformer industry. The potential uses of nanofluids have been reported by many researchers. A review of the works on this topic can be found in [START_REF] Jacob | Review on natural ester and nanofluids as an environmental friendly alternative to transformer mineral oil[END_REF]. Various nanoparticles have been widely used for preparing transformer liquids -based nanofluids. Most of the researchers have investigated Fe3O4, Fe2O3, TiO2, Al2O3, ZnO, CuO2, CuO, SiO2, BN, fullerene (C60), graphene, etc. for improving the properties of transformer liquids (mineral oils and ester liquids). The majority of the researchers have focused on the properties like viscosity, breakdown voltage, flash-point, and degradations aspects using various nanoparticles [START_REF] Jacob | Review on natural ester and nanofluids as an environmental friendly alternative to transformer mineral oil[END_REF]. However, a few researchers have also reported the prebreakdown investigations using nanofluids [START_REF] Beroual | AC and Lightning Impulse Breakdown Voltages of Natural Ester Based Fullerene Nanofluids[END_REF]. It seems that investigations in this field of research on nano-additives are affirmative, which initially showed potential in improvement of the breakdown characteristics of the esters. The propagation of streamers in ester liquids may also be controlled by adopting suitable electronic scavengers [START_REF] Rao | A Review on Pre-Breakdown Phenomena in Ester Fluids: Prepared by the International Study Group of Ieee Deis Liquid Dielectrics Technical Committee[END_REF][START_REF] Rozga | Prebreakdown and Breakdown Phenomena in Ester Dielectric Liquids[END_REF].

Apart from the adding nanoparticles for property enhancements, oxidation inhibitors and pourpoint depressants are also other additives that need to be investigated. Despite being the fact that, pourpoint and oxidation stability are challenging aspects for natural esters, a very limited literature is available on the these aspects. The critical challenge in this line concerning alternative liquids include finding suitable antioxidants and pourpoint depressants. Recently, few manufacturers have come up with low pourpoint insulating liquids.

Meanwhile, high thermal conductivity and stability of the cellulose material is another important parameter. It is possible to achieve high performance solid insulation (paper/pressboard) with involvement of suitable nanoparticles. Also, it is to be recalled that transformer life is generally designed for few decades and a huge number of transformers are connected to the power grid. Thus, a huge potential of challenge and research scope is underlined within the investigation of suitable additives that will rejuvenate the insulation system.

V. CONCLUSION

In this paper, key concerns foreseen to improve the workability of ester liquids are discussed. The main concerns include characterizing, comparing and studying the various stabilization against oxidation by additives (antioxidants) or improving the thermal and electrical properties by nanostructuring (considered as the object of the next industrial revolution). The behaviour at low temperatures is crucial too. These alternative oils being more viscous, their capacities as coolants as well as the consequences on the conceptual aspect of transformers are to be evaluated. In fault diagnosis of power transformers filled with biodegradable liquids is another actual limiting factor.

The proposed activities under the umbrella of the TC-LD is consequently, expected to prepare utilities for the wide use of esters in the next generation of transformers, which is of industrial / economic importance and of critical academic significance. The planned activities may also have an impact on international standards.

Fig. 1 .

 1 Fig. 1. Gelling of natural ester under open beaker aging conditions in laboratory.


Matias Kühnau
email: matiaskuhnau@hotmail.com

Akylas Stratigakos
email: akylas.stratigakos@minesparis.psl.eu

Simon Camal
email: simon.camal@minesparis.psl.eu

Samuel Chevalier

George Kariniotakis
email: georges.kariniotakis@minesparis.psl.eu

Resilient Feature-driven Trading of Renewable Energy with Missing Data

Keywords: energy trading, data-driven optimization, missing data, renewable energy sources, robust optimization

Advanced data-driven methods can facilitate the participation of renewable energy sources in competitive electricity markets by leveraging available contextual information, such as weather and market conditions. However, the underpinning assumption is that data will always be available in an operational setting, which is not always the case in industrial applications. In this work, we present a feature-driven method that both directly forecasts the trading decisions of a renewable producer participating in a day-ahead market, and is resilient to missing data in an operational setting. Specifically, we leverage robust optimization to formulate a feature-driven method that minimizes the worst-case trading cost when a subset of features used during model training is missing at test time. The proposed approach is validated in numerical experiments against impute-then-regress benchmarks, with the results showcasing that it leads to improved trading performance when data are missing.

I. INTRODUCTION

The integration of renewable energy sources, such as wind and solar, in deregulated electricity markets necessitates dealing with their inherent variability and uncertainty. At the same time, as power systems are becoming increasingly data-centric [START_REF] Union | Digitalisation of the energy system[END_REF], advanced data-driven methods, based on machine learning and optimization, enable improved decisions and help mitigate uncertainty.

The implicit assumption underpinning most applications of data-driven methods is that data will always be available when needed in an operational setting, i.e., at test time. Nonetheless, in real-world applications, data can go missing for a multitude of reasons. Network latency, equipment failures, or cyberattacks, can render input data unavailable at test time and compromise decision quality. For instance, an EC survey [START_REF]A review of the ENTSO-E transparency platform[END_REF] on the timeliness of data published on the ENTSO-E This work was supported in part by the Smart4RES Project (Grant No 864337) funded under the Horizon 2020 Framework Program and in part by the Carnot M.I.N.E.S project Flexi4Value (Grant No 220000499) supported by ANR.

Transparency Platform1 finds that "for every data domain, fewer than 40% of users reported that data were always there when needed." To ensure that the outcome of data-driven decision-making processes remains consistent, it is pivotal to develop methods that are resilient to missing data.

A. Related work

Leveraging data to optimize the participation of renewable energy sources in electricity markets has received a lot of attention over the last two decades. Typically, available data comes in the form of historical production data and associated contextual information (or features) related to weather and market conditions. Earlier works focus on leveraging contextual information to generate probabilistic forecasts of uncertain renewable production, which are subsequently used to derive optimal trading decisions-see, e.g., [START_REF] Pinson | Trading wind generation from short-term probabilistic forecasts of wind power[END_REF]. Recently, the focus is placed on data-driven optimization methods that directly map contextual information to decisions. We highlight [START_REF] Mazzi | Purely data-driven approaches to trading of renewable energy generation[END_REF], where meta-optimization is utilized to derive trading decisions, [START_REF] Stratigakos | Prescriptive trees for integrated forecasting and optimization applied in trading of renewable energy[END_REF], where decision trees are trained to directly optimize for trading cost, and [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF], where the trading decisions are directly forecast from contextual information.

Missing data at test time, i.e., the case when a subset of features used for model training becomes unavailable in an operational setting, poses a major challenge to model performance. This problem has been primarily examined w.r.t. forecasting applications. In [START_REF] Tawn | Missing data in wind farm time series: Properties and effect on forecasts[END_REF], two approaches are proposed to handle missing data in wind power forecasting, namely retraining without the missing features and impute-then-regress. Retraining typically outperforms impute-then-regress methods, however, the number of additional models required might render it impractical. To address this challenge, [START_REF] Stratigakos | Towards resilient energy forecasting: A robust optimization approach[END_REF] develops a robust optimization approach to enable model resilience while also maintaining practicality for energy forecasting applications. Nonetheless, missing data at test time for renewable trading applications have not been thoroughly examined. One exception is [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF], where separate models are trained to deal with potential delays in data acquisition, effectively retraining without missing features.

B. Aim and Contribution

In this work, inspired by [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF] and [START_REF] Stratigakos | Towards resilient energy forecasting: A robust optimization approach[END_REF], we develop a featuredriven model to directly forecast the trading decisions of a renewable producer participating in a day-ahead electricity market that is resilient to missing data. We leverage robust optimization to formulate a model that optimizes for the worstcase trading cost when a subset of features is missing at test time. The proposed approach is agnostic to the missingness mechanism (whereas [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF] only addresses a specific missingness pattern due to data acquisition delays), maintains consistent performance, and remains practical. For validation, we examine an aggregation of wind power plants participating in a day-ahead market and compare the proposed approach against state-of-the-art methods coupled with imputation. The results illustrate that the proposed approach maintains consistent performance and outperforms impute-then-regress benchmarks.

The rest of this work is organized as follows. Section II formulates the problem of trading in a day-ahead market. Section III develops the proposed methodology. Section IV presents the experimental setup and discusses numerical results. Finally, Section V provides conclusions and perspective on future work.

II. TRADING PROBLEM FORMULATION

In this section, we formulate the problem of trading renewable production in a day-ahead market.

A. Trading in a Day-ahead Market

We assume a renewable producer, e.g., wind power producer, participating as a price-taker in a day-ahead market with a dual-price balancing mechanism. For each clearing period t of the day-ahead market, the producer submits an energy offer E da t 2 . As temporal constraints do not apply, we drop subscript t for simplicity. During real-time operation, the system operator activates balancing reserves to maintain the equilibrium between demand and supply. To recover the cost of reserve activation, all participants have to buy back (sell) the amount of energy shortage (surplus), thus balancing their individual position.

Let E be the uncertain renewable production, λ da be the day-ahead clearing price, and λ u , λ d be the marginal cost of activating upward and downward reserves, respectively. We assume that if the total demand exceeds the total supply, i.e., the system requires upward regulation, then λ u ≥ λ da and λ d = λ da . Conversely, if the total demand exceeds the total supply, i.e., the system requires downward regulation, then λ d ≤ λ da and λ u = λ da . If E da > E, i.e., the real-time production is less than the contracted amount, then 2 We assume that the offer is always accepted at zero marginal cost.

the producer buys back the difference at price λ u , while if E da > E sells the excess production at price λ d . Hence, if the producer deviates from the contracted offer in the direction that helps decrease the amount of regulation required, no financial penalty is incurred. We further define ψ u = λ u -λ da and ψ d = λ da -λ d to be the non-negative upward and downward unit regulation costs.

The uncertain profit for a single period t can be concisely written as

ρ = λ da E -ψ u (E da -E) + + ψ d (E -E da ) + , (1)
where (t) + = max(0, t). Evidently, both the renewable production E and all the market quantities λ da , ψ u , ψ d are unknown; thus, the trading profit ρ is also stochastic. The first term in [START_REF] Union | Digitalisation of the energy system[END_REF] relates to the revenue from the day-ahead market, while the term in the bracket is the non-negative imbalance cost. A risk-neutral producer aims to maximize the expected trading profit. Since the first term in (1) does not depend on trading decisions, maximizing trading profit is equivalent to minimizing the imbalance cost. The problem of minimizing the imbalance cost is given by min

E da E max ψ u (E da -E), ψ d (E -E da ) (2a) s.t. 0 ≤ E da ≤ E, (2b)
where the expectation is taken w.r.t. the joint distribution of uncertain parameters, constraint (2b) ensures that the trading decision is non-negative and respects the nominal capacity E. Problem ( 2) is an instance of the well-known newsvendor problem. Assuming we have access to the probability distribution of E and we know the true values of ψ u , ψ d , the optimal offer is derived analytically from

E da * = F -1 E ψ d ψ d + ψ u , (3)
where

F -1 E
is the inverse of the cumulative distribution function of E-see [START_REF] Pinson | Trading wind generation from short-term probabilistic forecasts of wind power[END_REF] for a proof. Thus, the optimal offer equals the τ -th quantile of the probability distribution, where τ = ψ d ψ d +ψ u . Clearly, as both the probability distribution of E or the actual values of ψ u , ψ d are unknown and need to be forecast.

B. Data-driven Approaches

We assume to have access to a training data set

D = {(E i , ψ u i , ψ d i , x i )} n i=1
of n observations of joint realizations of renewable production, unit regulation costs, and associated contextual information x. Here, x denotes a p-size feature vector, e.g., weather forecasts, historical production data, and market-related data, all of which are available to the producer prior to submitting the offer E da . Data set D can be used to approximate the solution to the original problem [START_REF]A review of the ENTSO-E transparency platform[END_REF].

A first data-driven approximation is given by min

E da 1 n n i=1 max ψ u i (E da -E i ), ψ d i (E i -E da ) (4a) s.t. 0 ≤ E da ≤ E, i = 1, . . . , n, (4b)
which defines the so-called Sample Average Approximation (SAA) of [START_REF]A review of the ENTSO-E transparency platform[END_REF]. By linearizing the maxima terms in (4a), problem (4) can be reformulated into a linear programming (LP) problem. Effectively, the SAA finds the trading decision that minimizes the in-sample imbalance cost given data set D.

While the SAA enjoys several nice theoretical properties, it does not leverage the available features x. Typically, features x can be utilized to train forecasting models that target the uncertain problem parameters. Conversely, we can construct a feature-driven policy that directly maps features to trading decisions and minimizes the imbalance cost, i.e., express E da as a function of x. Following [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF], we consider an affine policy E da (x) = w ⊺ x parametrized by coefficients w 3 . The problem of finding the optimal decision rules is given by

min w 1 n n i=1 max ψ u i (w ⊺ x i -E i ), ψ d i (E i -w ⊺ x i ) (5a) s.t. 0 ≤ w ⊺ x i ≤ E, i = 1, . . . , n, (5b)
where we effectively replaced the decision variable in (4) with the feature-driven policy. For an out-of-sample realization of features, say x 0 , the optimal energy offer is derived directly from w ⊺ x 0 , which bypasses the optimization solver. Evidently, it is possible for an out-of-sample decision to be infeasible; in this case, a projection to the feasible set [0, E] is required. Note that if the unit regulation costs ψ u i and ψ d i are replaced with a point forecast, i.e., their conditional expectation, then ( 5) is equivalent to regression. In this case, we could also estimate the predictive density of E and find the optimal offer from the expected optimal quantile, as shown in (3).

III. RESILIENT DATA DRIVEN ENERGY TRADING

An implicit assumption underpinning the feature-driven solution ( 5) is that the feature vector x will always be available at test time. If a subset of features, however, is missing, then the performance of the feature-driven policy is compromised. In this section, we robustify (5) against missing features.

A. Modeling Uncertainty

To model uncertainty w.r.t. feature availability, we assume that Γ (integer) features, at most, could be missing from each observation. Following [START_REF] Stratigakos | Towards resilient energy forecasting: A robust optimization approach[END_REF], we introduce binary variables α i ∈ {0, 1} p that model feature availability for the i-th training observation. This is done as x i ⊙ (1 -α i ) where ⊙ is the element-wise multiplication, where α ij = 1 results in the deletion of the j-th feature.

In energy forecasting and trading applications, certain features may not make sense to be deleted, e.g., calendar variables. Moreover, a subset of features might be grouped, e.g., polynomial terms of the same variable or variables received from the same source, and, thus, would go missing together. To model these cases, a set of equality constraints are introduced as Mα i = 0, where M ∈ R m•n models the m additional constraints. For instance, if the first feature cannot be deleted, 3 To model the bias term, we assume a vector of ones is padded to x. then a row vector [START_REF] Union | Digitalisation of the energy system[END_REF]0] is appended to M. Similarly, if

α 1 = α 2 then [-1, 1, 0] is appended to M.
For the i-th observation, we model the uncertainty w.r.t. feature availability with the following discrete uncertainty set

U i = {α i | α i ∈ {0, 1} p , j∈[p] α ij = Γ, Mα i = 0}. (6)
As all vertices of (6) occur at integer values, we can replace (6) with its convex hull, given by ( 7)

A i = {α i | 0 ≤ α i ≤ 1, j∈[p] α ij = Γ, Mα i = 0}, (7)
which is a polyhedral uncertainty set. Therefore, using ( 7) instead of ( 6) results in equivalent solutions.

B. Resilient Feature-Driven Model

In this section, we develop the proposed resilient featuredriven model. First, the feature-driven problem ( 5) is equivalently written as

min t,u,w t, (8a)
s.t. 1 n n i=1 u i ≤ t, (8b)
ψ u i (-E i + w ⊺ x i ) ≤ u i , i ∈ [n] (8c)
ψ d i (E i -w ⊺ x i ) ≤ u i , i ∈ [n] (8d) 0 ≤ w ⊺ x i ≤ Ē, i ∈ [n] (8e)
where we effectively linearized the inner maxima terms, with t, u i denoting auxiliary variables, u being an n-size vector that comprises of u i , and [n] being shorthand for {1, . . . , n}. Using [START_REF] Tawn | Missing data in wind farm time series: Properties and effect on forecasts[END_REF], we robustify the feature-driven models (8) against uncertainty w.r.t. to feature availability. The robust version is given by

min t,u,w t, (9a)
s.t. 1 n n i=1 u i ≤ t, (9b)
ψ u i (-E i + w(x i ⊙ (1 -α i )) ≤ u i , i ∈ [n], ∀α i ∈ A i , (9c)
ψ d i (E i -w(x i ⊙ (1 -α i ))) ≤ u i , i ∈ [n], ∀α i ∈ A i , ( 9d
)
0 ≤ w(x i ⊙ (1 -α i )) ≤ E, i ∈ [n], ∀α i ∈ A i , (9e)
which we term resilient feature-driven model. Effectively, [START_REF] Bertsimas | Robust and adaptive optimization[END_REF] finds the set of linear decision rules that map features to decisions that minimize the worst-case trading cost when Γ features are missing. Constraints (9c)-(9e) are robust constraints. As we are dealing with an LP problem and a polyhedral uncertainty set, ( 9) can be reformulated into a tractable problem using techniques from robust optimization [9, Ch. 2]. For illustration, consider constraint (9c). As this constraint needs to be satisfied for all realizations of α i , it suffices to be satisfied for the worst-case scenario. For the i-th observation, assuming ψ u i > 0, (9c) is equivalently written as

-E i + wx i -max αi∈Ai (w ⊙ x i ) ⊺ α i ≤ u i 1 ψ u i . ( 10
)
The inner max problem is given by

max αi (w ⊙ x i ) ⊺ α i , s.t. α i ≤ 1 : µ i ≥ 0, j∈[p] α ij = Γ : ζ i , M α i = 0 : π i , α i ≥ 0,
where µ i , π i , γ i are dual variables of appropriate size. As this problem is linear in α i , it can be replaced in (10) by its dual,

min µi,γiπi j∈[p] µ ij + γ i Γ, (12a)
µ i + γ i + Mπ i ≥ x i ⊙ w, (12b) µ i ≥ 0, ( 12c
)
where the min operator becomes redundant. The rest of the constraints are reformulated similarly. Note that [START_REF] Bertsimas | Robust and adaptive optimization[END_REF] ensures that in-sample trading decisions are always feasible. Nonetheless, it is possible for out-of-sample decisions to be infeasible. In this case, the inferred decision is projected onto the feasible set [0, E].

IV. NUMERICAL EXPERIMENTS

This section presents the experimental setup used to validate the proposed model and discusses our numerical results.

A. Experimental Setup and Input Data

First, we formulate different data-driven methods to derive trading decisions. Next, we examined their performance when features are missing. The following models are tested:

• SAA: the SAA solution (4) that does not account for contextual information and is not affected by missing features.

• FD: the feature-driven method (5).

• EV: this involves offering the expected production value, derived from standard forecasting. For simplicity, EV is the same as FD assuming ψ u and ψ d are equal. • RF: a random forest model that maps data to decisions, as described in [START_REF] Stratigakos | Prescriptive trees for integrated forecasting and optimization applied in trading of renewable energy[END_REF], using the same features as FD. • ResFD: the proposed resilient (i.e., robust) version of FD. Initial experiments showed that including the realizations of ψ u i , ψ d i in the feature-driven models led to worse performance. In fact, the best results were obtained using the in-sample means, e.g., replacing ψ u i with 1 n i∈[n] ψ u i . This is also the case in [START_REF] Munoz | Feature-driven improvement of renewable energy forecasting and trading[END_REF] and effectively amounts to using the in-sample mean of unit regulation costs as the respective point forecast.

For models that cannot handle missing data, i.e., FD, EV, and RF, we follow the impute-then-regress approach with /RQJLWXGH /DWLWXGH :LQG 1:3 The power production data comprise an aggregation of 13 wind farms located in mid-west France with a total nominal capacity of 120MW. The measured power production is effectively our target and represents the "perfect" day-ahead offer for each observation.

The NWPs are used to create the feature vector x. The NWPs consist of weather forecasts in a spatial grid with a half-hourly resolution, namely wind speed, wind direction, solar irradiance, and cloud coverage. To model the nonlinear relationship between wind speed and wind power production, we further add a quadratic and cubic transformation of the wind speed variable. For each of the 13 wind farms, we find the NWP grid point that is closest in terms of Euclidean distance and append it to the feature vector, resulting in a total of 52 features. Fig. 1 illustrates the geographical distribution of wind farms and respective NWPs. Each grid point determines a group of variables, i.e., all the variables go missing together (13 groups in total). Hence, the possible combinations of missing features are 13 k=0 13 k . In turn, this number renders the retraining method [START_REF] Tawn | Missing data in wind farm time series: Properties and effect on forecasts[END_REF] impractical.

For market data, we consider data from the French electricity market. The market data are used to estimate the insample mean of the balancing penalties and for out-of-sample validation.

B. Results

First, we briefly compare baseline performance without missing features. Table I presents the mean imbalance cost in EUR/MWh. Overall, all feature-driven models significantly outperform SAA. Specifically, EV, FD, and ResFD are the best-performing models with an expected cost of 0.72 EUR/MWh, closely followed by RF. Next, we evaluate performance when features are missing at test time. For each test observation, we randomly remove Γ NWP features, repeating the experiment 10 times.

Fig. 2 plots the mean imbalance cost over all the experiments versus the different values of missing features. Note that for zero missing features, all models perform according to Table I. As expected the overall trading performance decreases when data are missing at the time. The proposed ResFD is the best-performing model when data are missing and showcases fairly consistent performance with a relatively small increase in imbalance cost. The only exception is for the extreme case when Γ = 13, when SAA slightly outperforms ResFD. Regarding the impute-then-regress benchmarks, RF is fairly resilient for up to 3 missing feature groups, while both EV and FD are considerably worse for even a single missing feature group. Overall, ResFD leads to approximately 25% improved trading performance against the second-best model when data are missing.

We further conduct a sensitivity analysis w.r.t. the number of test observations that incur missing data. To this end, we sample a percentage of observations with missing data, and for each observation, we sample a number of missing feature groups, repeating the experiment 10 times.

Table II summarizes the results of the sensitivity analysis. Overall, ResFD outperforms the impute-then-regress methods, and the effect is more pronounced as the percentage of observations with missing data increases. Specifically, the average improvement over the second-best method, RF, is approximately 10%. rier

V. CONCLUSION

In this work, we developed a feature-driven model that directly forecasts the trading decisions for a renewable producer participating in a day-ahead market and is resilient to missing features. Specifically, we formulated a model that minimizes the worst-case trading cost when a subset of features is missing at test time. To validate the proposed approach, we compared performance against a number of impute-then-regress benchmarks considering a day-ahead market with a dual-price balancing mechanism. Overall, the proposed approach led to approximately 25% reduction in trading costs compared to the best impute-then-regress method when data were missing from the test set. A sensitivity analysis examined performance when a subset of the test set had missing data and further validated the efficacy of the proposed method. Future work could focus on trading applications in electricity markets with a single-price balancing mechanism. Further, the proposed methodology could also be applied to robustify the data-driven methods that aim at forecasting the solutions to constrained optimization problems given associated features.

Fig. 1 .

 1 Fig. 1. Location of wind farms and NWP grid points.

Fig. 2 .

 2 Fig. 2. Performance versus the number of missing feature groups.

TABLE II MEAN

 II IMBALANCE COST (EUR/MWH) VERSUS PERCENTAGE OF OBSERVATIONS WITH MISSING FEATURES.

	Percentage SAA RF	EV	FD	ResFD
	5	2.33	0.82 0.86 0.86	0.76
	10	2.33	0.88 1.01 1.01	0.81
	25	2.33	1.03 1.44 1.44	0.94

Data on generation, transportation and consumption of electricity including information for the pan-European market


Francis Bloch

Bhaskar Dutta

Greqam

Francis Bloch Bhaskar

Dutta Mai

Document de Travail n°2006-23 CORRELATED EQUILIBRIA, IN COMPLETE INFORMATION AND COALITIONAL DEVIATIONS

Keywords: JEL Classification Numbers: C72 correlated equilibrium, coalitions, information sharing, games with positive externalities

This paper proposes new concepts of strong and coalition-proof correlated equilibria where agents form coalitions at the interim stage and share information about their recommendations in a credible way. When players deviate at the interim stage, coalition-proof correlated equilibria may fail to exist for two-player games. However, coalitionproof correlated equilibria always exist in dominance-solvable games and in games with positive externalities and binary actions.

* Dutta gratefully acknowledges support from ESRC Grant RES-000-22-0341.


Thermodynamic study of working fluid pairs for an absorption refrigeration process

1 Université Clermont Auvergne, CNRS, ICCF, F-63000 CLERMONT-FERRAND, France

A. Hajlaoui 1 , J-M. Andanson 1 , A. Dequidt 1 , L. Rodier 1 , Y. Coulier 1

Vapor-liquid equilibrium measurements

Acknowledgement References

Vapor absorption refrigeration (VAR) technology is a good alternative to conventional compression refrigeration. However, in addition to the low coefficient of performance of this technology, the fluids used have specific drawbacks that prevent wider use.

To promote this technology, the present work investigates new potential natural refrigerant/absorbent pairs. The detailed knowledge of the thermodynamic properties of the working fluid pairs is crucial to retrofit and modify VAR process.

In this study, carbon dioxide (CO 2 ) has been selected as the natural refrigerant and the absorbent is a bio-based solvent (BBS) with low volatility and strong affinity to the former. Four BBSs are investigated namely cyrene (CYR), propylene carbonate (PC), anisole (ANI) and ϒ-valerolactone (GVL). New experimental vapor-liquid-equilibria data for {CO2+BBS} systems are determined, using a calorimetric and a spectroscopic technique. The experimental results are then used to develop an appropriate thermodynamic model capable of reproducing phase equilibria in order to compare the CO 2 solubility in the different BBSs at the operating conditions of VAR cycle in the future.

Enthalpies of solution of CO 2 in the BBS are determined using a customized mixing flow unit adapted to a SETARAM C80 heat conduction differential calorimeter at constant temperature and pressure.

Calorimetric technique

Thermodynamic modeling ❑ The calorimetric technique allowed to determine simultaneously the solubility of CO 2 in BBS and the heat required to separate it from the absorbent. ❑ To avoid using large amounts of solvent and time consuming solubility measurements, an original approach combining the calorimetric technique and IR spectroscopy is used to rapidly evaluate BBS for VAR process. ❑ At 333,15 K and 4,0 MPa, the highest CO 2 solubility is observed for GVL and the lowest energy required to separate CO 2 from the solvent is obtained with anisole. ❑ SRK equation of state is capable of correlating experimental values and can be used process simulation.

Conclusion

The authors acknowledge financial support from the French projects ANR-20-CE05-0004 AWARE and PAI, supported by the French government ant the Auvergne Rhone Alpes region. The authors also thank Ms. Marianne Kerleaux for her participation in the generation of some experimental data.

Spectroscopic measurements are performed at constant temperatures and pressures.

Knowing CO 2 solubility in the solvent at a given pressure and temperature, the spectroscopic technique allows us to determine the solubility at that same temperature and at various pressures.

Results

Results

Results

  cyrene α (mol CO 2 / mol cyrene) α * : loading charge related to the CO 2 solubility limit in cyrene at 313.15 K and 4CO 2 / mol cyrene) -Δ sol H : Heat of desorption of CO 2 Enthalpy of solution (-Δ sol H ) of CO 2 in cyrene vs loading charge (-Liquid-Equilibrium (VLE) data for CO 2 -BBS system are described using the Soave-Redlich-Kwong (SRK) equation of state [1,2] : Soave alpha function 𝛼(𝑇) ❑ Van der Walls mixing rules and binary interaction parameters : 𝑗 (𝑎 𝛼) 𝑖 𝑎 𝛼 𝑗 (1 -𝑘 𝑖𝑗 ) b = σ 𝑖 𝑧 𝑖 𝑏 𝑖 One parameter to determine using "Simulis Thermodynamic Software" customized high pressure IR reactor Part of the IR spectra of the binary mixture (cyrene + CO 2 ) at 313,15 K Solubility of CO 2 in cyrene at 313,15 K Enthalpy of solution (-Δ sol H ) of CO 2 in four different BBS vs loading charge (α) at 333,15 K and 4,0 MPa [1] Redlich, O. ,Kwong, J.N.S., 1949, Chem. Rev., 44(1), 233-244 [2] Soave, G. , 1972, Chem. Eng. Sci., 27(6), 1197-1203 [3] Hongling, L . , et al., 2011, J. Chem. Eng. Data, 56 (4), 1148-1157 [4] Williams, L. L. , et al., 2002, J. Chem. Eng. Data, 47(2and those from the literature [3-8] are used for model parameter estimation.

  allow us to obtain two information; CO 2 solubility limit in the BBS and the heat of desorption.

  Murrieta-Guevara, F. , et al., 1988, Fluid Phase Equilib., 44(1), 105-115 [6] Mantor, P. D. , et al., 1982, J. Chem. Eng. Data, 27(3), 243-245 [7] Kim, C.-H. , et al., 1989, J. G. Trans. Inst. Chem. Eng, 34, 391-395 [8] Park, S. D., et al., 1991, J. Chem. Eng. Data, vol. 36(1), 80-84