domain: spi phys

text:
T Hamoudi

M Guionie

X Delen

J.-B Dherbecourt

J.-M Melkonian

A Godard

M Raybaut

P Georges

High power, tunable hybrid fiber/bulk laser at 1030 nm and parametric frequency conversion in the near and mid-infrared

We report on a high power, wavelength-agile, hybrid fiber-bulk laser, emitting up to 3.5 mJ 15 ns pulses at 1030 nm at 5 kHz repetition rate. This hybrid laser is composed of a commercial seed single-frequency fiber laser, tunable over several GHz, amplified in a dualstage single-mode Ytterbium doped fiber amplifier, followed by two Yb:YAG solid crystal amplifiers. In order to illustrate the potential of such an architecture for future differential absorption lidar emitters, this hybrid source was used to pump an optical parametric amplifier (OPA) seeded by the idler at 3 µm generated by a fixed nested cavity optical parametric oscillator (NesCOPO). The wavelength agility of the pump is transferred to the OPA signal beam in order to reach a fine tunability in the 1.5 µm region. This fine tuning was used to measure the transmission spectrum of an acetylene gas cell.

Introduction

Pulsed, kHz repetition rate, nanosecond coherent sources, emitting energies in the hundreds of microjoules to millijoules levels are of high interest for remote sensing applications. In the frame of Differential Absorption LI-DAR (DIAL) developments for remote quantification of trace gases, single-frequency sources, with specific tuning capabilities are required. The DIAL measurement method is based on the comparison of the optical transmission of the atmosphere for at least two consecutive wavelengths usually designated as λ ON , chosen in coincidence or close to a targeted gas species absorption line, and λ OF F , emitted out of resonance and close to an absorption minimum. For applications such as atmospheric greenhouse gases concentration measurements, these wavelengths are generally required to be emitted in the near or mid-infrared with a sufficiently narrow ⋆ E-mail: xavier.delen@institutoptique.fr linewidth typically smaller than a few 100 MHz in order to sample the pressure broadened absorption lines [1,[START_REF] Orr | Encyclopedia of Analytical Chemistry[END_REF].

Another highly sought and hard to achieve key property is the capacity to switch rapidly from one wavelength to another in order to satisfy the assumption of equivalent backscattering coefficient for the consecutive emitted wavelengths used in the DIAL method. Ultimately the wavelength switching sequence should be faster than the coherence time of the atmosphere limited by the turbulence, which ranges in the few milliseconds [START_REF] Menyuk | [END_REF]. The strategies to reach kHz or higher wavelength switching rates are essentially constrained by the technological choices for the laser source and the method used to interleave the different wavelengths. For DIALs the most general approaches consist in either using high energy (> 10 mJ) -low repetition rate (< few 100 Hz) Q-switched lasers with multiple pulses (corresponding to ON and OFF DiAL wavelength), or in using lower energies (< few mJ) -high repetition rate (> 1 kHz) sources. In the first case, peak powers can exceed hundreds of kilowatts which is very favorable for long range applications, but the spectral versatility is mostly limited to a very few number of wavelengths [4][5][START_REF] Refaat | IGARSS 2019[END_REF]. In the high repetition rate case, a lower peak power can be balanced by more versatility like fast multiple wavelengths operation [START_REF] Abshire | [END_REF]8]. One important motivation for multiplewavelengths sources is to get access to multiple molecules concentrations within the same observation frame, which can be useful to limit biases on mixing ratio retrievals [5,9], or to increase the retrieval sensitivity by probing the absorption line of interest with a larger number of samples [10,11].

For the laser sources, multi-wavelengths schemes are based on a variety of techniques consisting in either combining single frequency lasers with electro-optic switches in order to injection seed pulsed laser and OPO cavities [4,12,13], current scanning of a single seeder [START_REF] Abshire | [END_REF][14][15][16], electro-optic modulation and filtering of a single seeder [9], or no-seeder cavity mode filtering controlling with piezo-electric transducers [START_REF] Godard | Encyclopedia of Analytical Chemistry[END_REF]. These last two approaches consequently result in relatively simplified and robust implementations. Single seeder scanning approaches also allows reduced alignment and cavity length control requirements [15,[START_REF] Stephen | Components and Packaging for Laser Systems IV. 10513[END_REF].

In this paper we propose to use parametric conversion for pulsed multiple wavelengths generation in the mid-infrared, with a slightly different approach where fast tunability is provided by a high energy narrow linewidth near infrared pump laser. The principle is to use this tunable laser to pump an OPA stage seeded by a broadly but slowly tunable pulsed OPO. The fast tunability of the pump laser can then be directly transferred to the signal or idler wave generated by the OPA and thus unlock DIAL system functionalities such as adaptation of the emitted wavelength sequence to varying environments or gas concentration ranges and dynamics [START_REF] Hamperl | [END_REF]20].

Pump tuned parametric devices have been previously demonstrated and used for spectrometry application but essentially in CW or quasi-CW regimes [21][22][23], less suitable for remote sensing. In order to go beyond and be able to carry out DIAL measurement over 1 km distance, it is required to have a rapidly tunable high peak-power pulsed laser with narrow spectral linewidth.

As a pre-requisite for a future DIAL system, we have specifically developed a pump laser providing tunable multi-mJ pulses (up to 3.5 mJ) at a high repetition frequency (5 kHz), and with a narrow spectral linewidth. This pump laser is based on a hybrid Master Oscillator Power Amplifier (MOPA) architecture, which combines fibered and bulk ytterbium amplifiers. The hybrid architecture takes advantage of i) the wavelength agility of fiber amplifiers, where fast wavelength scanning is supported by an externally modulated seed laser, ii) the pulse and repetition frequency active shaping with acousto-optic intensity modulation, and iii) the high energy storage capability provided by solid state lasers amplifiers. Taking advantage of the large emission bandwidth of Ytterbium doped laser materials, the high power pulsed output could be continuously tuned across a broad frequency range (up to 230 GHz in the reported set-up) through seeder modulation at the MOPA input. Finally, the high output beam quality makes this MOPA laser source ideal for down conversion pumping application in order to reach the 1.5 -4 µm range. To illustrate this purpose the hybrid source was used to pump an OPA seeded with the 3 µm idler beam from a nested cavity optical parametric oscillator. The pump wavelength tuning was then directly transferred to the signal wave at the OPA output, which could be used to probe the absorption spectrum of an acetylene gas cell.

2 Hybrid MOPA fiber/bulk laser A hybrid fiber/bulk architecture, in MOPA (Master Oscillator Power Amplifier) configuration, based on ytterbium-doped materials has been developed (Fig. 1). As in [24,25], the driving idea is to take benefit from the advantages of both fiber and bulk architectures. Singlemode fiber amplifiers are particularly interesting because they offer the possibility of high gain, high compactness and robustness. Nevertheless, fiber amplifiers are largely limited by Stimulated Brillouin Scaterring (SBS) with high peak powers pulses and narrow emission linewidth (< 50 MHz). Thus, amplification in single-mode optical fibers has been maximized until the onset of this nonlinear effect. The last amplification stages were based on Yb:YAG crystals, as this material has a much higher Brillouin threshold mainly due to the larger beam sizes, the shorter propagation length and the lower Brillouin gain coefficient.

In this system, the seed source is a commercial fiber laser (Koheras BASIK Y10) at 1.03 µm delivering a continuous wave (CW) power of 10 mW. Its fast tunability can reach 10 GHz with 3 dB cut-off at 200 Hz, and its slow one can reach 0.70 nm (200 GHz). The linewidth of this laser has been verified with a Fabry-Perot cavity with a finesse of 140 and FSR = 2 GHz, so ∆λ < 14 MHz. An optical isolator is placed for safety directly at the output of this laser. A double pass configuration is used for the first fiber amplifier. After passing through a circulator, the signal is first amplified in CW in a 6 µm core diameter, 1.3 m long single mode Yb-doped fiber. The pump diodes used are single mode, fiber Bragg grating stabilized, laser diodes delivering up to 700 mW at 976 nm. At the output of the first amplifying fiber, an acoustooptic modulator (AOM) is placed to generate pulses and a mirror is used to get a second pass. The opening time of the AOM, in double pass configuration, is defined by the round-trip time of the photons between the first and the second pass through the AOM, i.e. t AR = 2d/nc + τ , with τ the desired pulse duration, d the propagation distance, n the refractive index of the fiber and c the light velocity.

With the duty cycle, the average signal power after the AOM is strongly reduced by about 47 dB, including transmission losses. An average power in the µW range returns to the fiber amplifier at the input of the second pass. The signal is extracted by the circulator after two passes in the active fiber. The average input power being much lower at the second pass than at the first pass, the return of the pulse signal back into the fiber does not change the inversion of population along the fiber. Thus, the gain in the fiber is established by the first pass and is reused by the second pass. Figure 2 shows the output pulse energy and peak power versus the seed power in CW. The energy at the output of the double pass increases when the seed power is reduced, which might seem counterintuitive at first sight. This is due to the lower gain saturation in the first pass where the seed is CW, leading to a higher pulse energy at the overall two-pass amplifier output. With this configuration, energies of 20 nJ have been obtained, i.e. a peak power of 1.3 W. The second fiber amplifier is expected to be limited by SBS. For a fiber, the power threshold for SBS can be written as [26][START_REF]Nonlinear fiber optics. Fifth edition[END_REF][START_REF] Yeniay | [END_REF]:

P th = 21 A ef f g b L ef f (1)
where A ef f is the effective fiber area, g b the Brillouin gain and L ef f the effective fiber length. This effective length depends on the gain or losses in the fiber. For a gain G much larger than 1 in an amplifying fiber, the relation (1) can be rewritten [29]:

P th = 21 A ef f ln (G) g b L (2)
Although this result is approximate because it assumes a constant gain along the fiber, it shows that a higher SBS threshold can be obtained with a high gain short fiber. The output peak power can thus be maximized by searching for the optimum fiber length. If too long, the low linear gain close to the fiber output will result in a lower SBS threshold. However, if the fiber is too short, the overall gain might be too small to reach the SBS limit. To search for the optimum fiber length experimentally, we monitored the back scattered SBS pulse using a fiber coupler. Figure 3 shows the evolution of the output signal energy (Fig. 3b) and the power backscattered by the SBS and amplified in the fiber (Fig. 3a) as a function of the pump power. The results are presented for four different fiber lengths and allow to adjust experimentally the optimal length. The length of the fibers differed by only 12 cm between the maximum and minimum length. With the 1.06 m fiber (blue squares), the signal and SBS power increase very rapidly, up to pumping powers of 300 mW (limited at this level for component safety). The output energy is then of 3.2 µJ. By shortening the fiber by 10 cm (black rounds), the saturation of the pump absorption results in a saturation of the output power versus the pump power. It indicates that the linear gain is still high at the output, which results in a higher SBS threshold. It is then possible to obtain an output energy of 5 µJ corresponding to a peak power of over 330 W. This study shows the importance of taking into account the SBS to evaluate the optimum fiber length for the amplifier.

To reach higher energies of the order of mJ, two Yb:YAG based amplification stages were placed at the output of these fiber stages. The crystals used have a doping of 1.5% and dimensions of 2 × 8 × 17 mm 3 . Similar double-pass amplifier architectures have already been described in previous work [START_REF] Lesparre | SPIE LASE -Solid State Lasers XXIV: Technology and Devices[END_REF][START_REF] Lesparre | [END_REF][32][START_REF] Lesparre | Advanced Solid State Lasers[END_REF].The first stage is optimized to maximize the gain whereas the second stage is designed to extract high output energies.

Figures 4a) and b)

show the evolution of the output energy of the first and second amplifiers versus the pump power for different seed power levels. An output energy per pulse of 3 mJ at 5 kHz was measured without significant degradation of the beam profile, with an M 2 < 1.2 (Fig. 5a). The anti-reflective coating of the crystal of the second amplifier was damaged at an output energy higher than 3.7 mJ for a signal beam waist of 200 µm. The Fig. 5b) shows the temporal profile of the 15 ns pulse at the laser output. We observe a slight modulation in the pulse (at around 500 MHz) due to parasitic reflections into the first stage of fiber amplification. The temporal profile has a flat top shape because pulses are generated with temporal gate from the AOM into the CW laser seed (see Fig. 1).

Finally, spectrum measurement have been made with an Optical Spectrum Analyser (OSA). Nevertheless, the resolution of this detector was insufficient (2 GHz). A second measurement of the spectrum with a Fabry-Perot cavity (finesse of 140 and FSR = 2 GHz), showed that a lateral line was present. This indicates that the lateral line due to the modulation at 500 MHz is very weak (Fig. 5b).

Parametric conversion towards near and mid-infrared and potential for future DIAL emitters

To show the potential of this laser for frequency conversion to the near or mid-infrared by a non-linear stage, we used this hybrid laser as a pump for an OPA stage.

The OPA is seeded by a Nested Cavity OPO (NesCOPO) pumped by a wavelength-controllable, commercial fiber laser emitting at 1.064 µm [START_REF] Barria | [END_REF]35]. The pump pulses have a duration of 320 ns at a repetition rate of 5 kHz, with a maximum energy of 30 µJ. The NesCOPO crystal is a 4 mm long periodically-poled Lithium Niobate (PPLN) crystal. Several gratings are inscribed on the crystal, allowing to produce radiation in the 3.2 -3.7 µm range. For the following tests, the idler emission of the OPO was set in the 3.2 µm range. The maximum measured energy for the single frequency idler wave is 0.7 µJ at 3.2 µm.

The wavelength of the NesCOPO is measured using a wavemeter, and a control loop is implemented to lock its emitted frequency.

The overall experiment is shown in Fig. 6. The idler wave is injected into a sequence of 3 PPLN crystals of 34, 25 and 13 mm length respectively, pumped by the hybrid laser described above. The 2 mm thickness of these crystals limits the maximum incident pump energy to 1 mJ, to avoid damages on the crystal facets. Such limitation could be loosened by the use of higher aperture PPLN crystals, which are now commercially available. With a maximum pump energy of 1 mJ and a waist of 350 µm for each beam, a maximum energy of 150 µJ was measured for the signal (λ s OPA) at the output of the OPA. Further improvement of the OPA set-up, with longer crystals, an idler filtering between the crystals and with a better temporal overlap of the pump and idler waves, could lead to an even higher conversion efficiency.

Given that the OPO idler beam seeds the OPA at a fixed and stabilized wavelength, the signal wavelength of the OPA (λ s OPA) can be tuned by tuning the pump laser wavelength. This is shown on Fig. 7 for two different modes of operation of the hybrid pump laser. In a first slow mode (Fig. 7a), the temperature of the 1030 nm fiber seed laser is adjusted to cover a 0.7 nm range (200 GHz) to tune the λ s OPA over around 1.7 nm. In a faster and finer wavelength tuning mode (Fig. 7b), the hybrid pump laser wavelength seeder is adjusted by playing on the voltage through a piezoelectric Thus, it can generate rapid tuning scan over 10 GHz, which could be of high interest for DIAL, allowing to generate λ ON /λ OF F wavelengths sequences, with arbitrary positions over an absorption line.

Such a prospect is illustrated by scanning the absorption lines of an acetylene cell. The signal wave generated in these OPA stages is partially coupled in a 50/50 fibered beam splitter. One part is directly connected to a photodiode, the other part passes through the acetylenefilled cell and is coupled to another photodiode, allowing to measure the cell transmission. The cell is 3 cm long and filled with 100% acetylene at a pressure of 200 Torr. The points in Fig. 8 are experimental data and the line is a theoretical curve calculated using HITRAN database [36]. For each point, we observe a signal fluctuation of 3 to 5% represented by the vertical error bars. The wavelength error is mostly dominated by the resolution of the 200 MHz lambdameter and those of the pump wavelength, i.e. an error of 0.02 nm.

In the future, using the NesCOPO signal instead of the idler, and combining the coarse wavelength tuning of the NesCOPO over several hundreds of nm, with the fine and easy tuning of the hybrid pump laser, could thus allow scanning of absorption lines in the 3.2 -3.7 µm region, where most molecules of interest display strong absorption lines. Such OPA scheme could also be envisioned using other seed devices than a NesCOPO, as a single frequency laser diode, which would reduce the tunability, but further simplify the set-up. Since this OPO is single frequency [START_REF] Barria | [END_REF], it could be interesting to mea-sure the signal linewidth by beating it with a narrow linewidth laser diode. Then, the idler linewidth would be deduced from the pump and signal linewidths, knowing the difficulty to find a single frequency laser diode in this spectral range.

Conclusion

We reported on a high power, 5 kHz, 1030 nm, wavelength agile hybrid fiber-bulk laser, emitting up to 3 mJ of pulse energy in the nanosecond regime. This hybrid laser is based on a commercial seed fiber laser, whose wavelength is tunable over several GHz, amplified in a dual-stages Ytterbium doped single-mode fiber amplifier, followed by two Yb:YAG bulk amplifiers. In order to demonstrate the potential of such an architecture for future differential absorption LIDAR emitters, this hybrid source was used to amplify a single frequency, nested cavity, optical parametric oscillator, emitting a fixed wavelength in the 3 µm region, within an OPA set-up. Wavelength tuning of the hybrid pump allowed to tune the emission of the OPA generated signal wave, showing interesting potential for DIAL emitters. Such a pump laser tuning scheme could also be envisioned using other seed devices, as a laser diode, in order to further simplify the set-up. This pump could also be used for direct pumping of innovative OPO architectures, such as backward OPO, which also allows no-seeder narrow linewidth emission in the pulsed regime and could benefit from pump tuning.

Fig. 1

 1 Photodiode

Fig. 2

 2 Fig.2Amplified laser pulse energy (left axis) and peak power (right axis) versus incident pump power.

Fig. 3

 3 Fig. 3 For different lengths of doped fiber. (a) Brillouin power measured at the oscilloscope as a function of pump power. (b) Signal energy as a function of pump power.

Fig. 4 (

 4 Fig. 4 (a) Signal energy at the output of the first amplification stage and beam profile. (b) Signal energy at the output the second amplifier stage.

Fig. 5 (

 5 Fig. 5 (a) Measurement of the caustic and the spatial profile of the signal beam at full power at the output of the double pass amplification. (b) Time measurement recorded for a laser output power of 3W.

Fig. 6

 6 Fig. 6 Overall parametric frequency conversion set-up. DM, dichroic mirror; M, mirror.

Fig. 7

 7 Fig. 7 Wavelength of the signal wave (recorded with a wavemeter) from the OPA as a function of the wavelength of the OPA pump, for a fixed wavelength OPO idler wave. The left figure (a) corresponds to a coarse, slow temperature tuning of the hybrid laser seeder. The right figure (b) is obtained for a fast tuning of the hybrid laser seeder (1Hz frequency tuning speed, limited by the wavemeter acquisition time).

Fig. 8

 8 Fig.8Absorption measurement of acetylene as function of the emitted OPA signal wavelength, tuned by tuning the frequency of the hybrid pump laser. Points represent experimental data and the line is a calculation from HITRAN database[36].

Acknowledgement

The authors are thankful to Kjell Martin Mølster from Department of Applied Physics, Royal Institute of Technology (KTH), Stockholm (Sweden), for helpful discussion about the laser linewidth.

Funding

This work is supported by "Investissements d'Avenir" LabEx PALM (ANR-10-LABX-0039-PALM)


Student Member, IEEE Han Deng

Member, IEEE Jingyang Fang

Member, IEEE Yang Qi

Senior Member, IEEE Yi Tang

Senior Member, IEEE Vincent Debusschere

A Generic Voltage Control for Grid-Forming Converters with Improved Power Loop Dynamics

Keywords: Grid-forming converter, LCL filter, power control, stability, voltage control

Grid-forming converters (GFMCs) provide voltage support and other grid services, such as inertia and droop, in more-electronics power systems. The GFMCs with small droop coefficients or connected to the grid through a small impedance may have a fast power loop, which leads to serious conflicts with the inner ac voltage control loop, resulting in power oscillations or even instability. To address this problem, an extremely fast ac voltage loop should be expected for GFMCs. On top of that, GFMCs must operate well in both stand-alone and grid-tied conditions. Through detailed analysis, this paper first reveals that the grid-tied operation of a dual loop controlled GFMC with an LCL output filter features a much slower voltage control loop than that of the stand-alone mode. To improve the dynamics of GFMCs, we propose a generic voltage control scheme with a high-pass filter in the current feedback loop. The generic voltage controller has fast voltage tracking performances under both grid-tied and stand-alone operations, thereby enabling GFMCs to achieve better power regulation and grid service. The improvements in both voltage tracking and power control are verified via experiments.

I. INTRODUCTION

T O reduce the carbon footprint, power electronics device- interfaced distributed generations (DGs) are widely integrated into modern power grids [START_REF] Olivares | Trends in microgrid control[END_REF]. Conventionally, DGs are coupled to the grid through current vector-controlled voltage source converters (VSCs) with phase-locked loops (PLLs), which are also named as grid-feeding converters (GFDCs) [START_REF] Rocabert | Control of power converters in ac microgrids[END_REF]. Although GFDCs have a fast response and can limit current easily, GFDCs cannot provide voltage support, which is necessary in power electronics-dominated power systems. Furthermore, the negative influence of the PLL introduces instability under weak grids [START_REF] Ashabani | Integrating vscs to weak grids by nonlinear power damping controller with self-synchronization capability[END_REF]. To address these challenges, voltage vector-controlled grid-forming converters (GFMCs) emerge as a new trend [START_REF] Lasseter | Grid-forming inverters: A critical asset for the power grid[END_REF].

GFMCs use vector voltage control to operate as a stiff ac voltage source. With the same control structure, GFMCs can work regardless of whether they are connected to the grid. The control of GFMCs includes an outer power control loop and an inner voltage control loop. For the power control loop, droop control is applied in parallel-connected UPS systems to achieve autonomous power sharing [START_REF] Guerrero | Control strategy for flexible microgrid based on parallel line-interactive ups systems[END_REF]. The powersynchronization control utilizes the inherent synchronization mechanism in an ac system to synchronize the VSC with the grid [START_REF] Zhang | Interconnection of two very weak ac systems by vsc-hvdc links using power-synchronization control[END_REF]. Virtual synchronous generators (VSGs) or virtual synchronverters further simulate the inertia and droop mechanism Part of this paper has been presented at the Energy Conversion Congress and Exposition (ECCE), 2021. This submission is a much extended version.

of synchronous generators to achieve power sharing and grid frequency support [START_REF] Qing-Chang | Selfsynchronized synchronverters: Inverters without a dedicated synchronization unit[END_REF]. The low-pass-filtered power feedback combined with the droop control is proved to be equivalent with the VSG control in view of the virtual inertia provided to the grid [START_REF] Suul | Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids[END_REF]. The dynamic characteristics of different droop mechanism and virtual inertia implementation strategies in GFMCs are evaluated in [START_REF] Liu | Comparison of dynamic characteristics between virtual synchronous generator and droop control in inverterbased distributed generators[END_REF]. The coupling effects of active and reactive control and related stability issues are studied in [START_REF] Wu | Small-signal modeling and parameters design for virtual synchronous generators[END_REF]. [START_REF] Qi | Synthetic inertia control of gridconnected inverter considering the synchronization dynamics[END_REF] analyzes the synchronization stability of GFMCs with capacitors as dc power supplies. [START_REF] Fang | Statespace modeling and control of grid-tied power converters with capacitive/battery energy storage and grid-supportive services[END_REF] investigates the complete small signal models of GFMCs with capacitors and batteries dc sources. To address the low-frequency oscillation problem in microgrids with interconnected GFMCs, a passivity-based design criteria for individual GFMCs is proposed in [START_REF] Qi | Passivity-based synchronization stability analysis for power-electronic-interfaced distributed generations[END_REF], and a converter-based power system stabilizer is proposed in [START_REF] Guo | A converter based power system stabilizer for stability enhancement of droop controlled islanded microgrids[END_REF].

Voltage vector linear control strategies have been comprehensively investigated as well. To provide a pure voltage output with small switching harmonics, LC filters are usually employed in VSC applications [START_REF] Loh | Analysis of multiloop control strategies for lc/cl/lcl-filtered voltage-source and current-source inverters[END_REF]. For grid-tied applications, the inverter-side LC filter and the grid-side impedance will form an LCL filter. Although LC/LCL filters can effectively suppress switching harmonics with a reduced size, LC/LCL resonances will occur and bring stability issues. Generally, multi-loop control structures are utilized in the LC/LCL systems to achieve reference tracking and stability improvement simultaneously [START_REF] Loh | Analysis of multiloop control strategies for lc/cl/lcl-filtered voltage-source and current-source inverters[END_REF]. The principle for inverter-side or capacitor-side current feedback damping is elaborated in [START_REF] Tang | Exploring inherent damping characteristic of lcl-filters for three-phase grid-connected voltage source inverters[END_REF]- [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF]. Capacitor voltage feedback active damping can also contribute to the stability and the waveform quality of LC filtered grid-tied systems [START_REF] Gonzalez | A robust controller for a grid-tied inverter connected through an lcl filter[END_REF]- [START_REF] Geng | Stability analysis and key parameters design for grid-connected current-source inverter with capacitor-voltage feedback active damping[END_REF]. These active damping methods are widely employed in LC/LCL filtered GFMCs with dual-loop voltage controllers. To reduce sensor cost, [START_REF] Wang | Stability analysis and controller synthesis for single-loop voltage-controlled vsis[END_REF] utilizes the internal phase lag of the digital controller computation delay and integrator dominant controllers to avoid stability problems without damping when using a single-loop voltage controller. Apart from the stability and power quality concerns, several voltage control schemes with high control bandwidth are evaluated in [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF].

When designing the outer power control loop structures and parameters, it is usually assumed that the power is much slower than the inner voltage control loop for simplicity. As a result, the voltage control loop can be regarded as a unit constant. This is a simple and effective approach in the multiloop controller design.

However, the inner voltage loop of GFMCs is usually designed under the stand-alone mode [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF], [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF]. This paper reveals that the voltage tracking dynamic response of a dualloop voltage controlled GFMC under the grid-tied mode is much slower than that under the stand-alone mode. Further-more, under cases of strong grids with small grid impedance or DGs with small capacities and low droop coefficients, the power loop bandwidths will be high [START_REF] Arco | Small-signal modelling and parametric sensitivity of a virtual synchronous machine[END_REF]. This means the hypothesis of the control bandwidth decoupling might not stand even when the voltage control loop is designed to be fast under the stand-alone mode. The voltage loop with a slow dynamic response will cause coupling between the outer power control loop and the inner voltage control loop, which may lead to unsatisfied power control dynamics and even instability. To address this issue, a high-pass filtered current feedback method is proposed, which has little influence on the operation of GFMCs under the stand-alone mode. With the proposed controller, GFMCs can achieve fast and stable voltage tracking performance under both stand-alone and grid-tied modes without any controller alternations. More importantly, the power control stability is also improved especially when the power control bandwidth is high.

The rest of the paper is organized as follows. Section II presents dynamic analysis of the voltage control loop under stand-alone and grid-tied modes and reveals the poor voltage tracking dynamic response of GFMCs under the grid-tied mode. Section III introduces the power control loops of GFMCs along with their interaction with the voltage control loops. Section IV proposes a generic voltage controller to improve the voltage tracking dynamics under the grid-tied mode. Section V discusses the similar problem of worsened dynamic responses in other controllers and other possible solutions. Experimental results are provided in Section VI for verification. Finally, Section VII concludes the paper.

II. DYNAMIC ANALYSIS OF VOLTAGE CONTROL LOOP

Although lots of research has studied the control of LC filtered voltage controlled VSCs, and several control strategies with fast response have been proposed, most of them considered the scenario when the VSC is disconnected from the grid. When the VSC is connected to the grid, the grid-side current is usually considered as a disturbance to the system when analyzing the voltage loop dynamics. This section will introduce the schematic of the studied GFMC system firstly, then show that the dynamic responses of the GFMCs will deteriorate when they are connected to the grid with the conventional dual loop control.

A. System Schematic

Fig. 1 shows the general system schematic diagram of a GFMC. The GFMC is connected to the grid through an LC filter composed of L i and C f . L g represents the grid impedance. Note that if an LCL filter is applied, then L g will also include the grid-side inductance of the LCL filter. The dc source v dc can be either a dc voltage source or a capacitor, representing a battery or the output of another converter in a multi-stage system [START_REF] Fang | On the inertia of future moreelectronics power systems[END_REF].

Generally, the active power P and the reactive power Q flowing from the GFMC to the grid v g can be controlled by tuning the phase reference θ oref = ω oref dt and the amplitude reference V oref of the capacitor voltage v o , where ω oref denotes the output frequency reference of the GFMC. The subscript ref represents the reference signals, abc stand for the three The voltage over the capacitor v o can be controlled under either the stationary αβ frame or the synchronous dq frame, with the controller G v (s) being a proportional-resonance (PR) controller or a proportional-integral (PI) controller, respectively. The current controller G c (s) uses a proportional (P) controller, which feeds back the inverter-side current i i to damp the LC resonance [START_REF] Tang | Exploring inherent damping characteristic of lcl-filters for three-phase grid-connected voltage source inverters[END_REF]. i g and i c represent the grid current and the capacitor current, respectively. The current feedback filter G if (s) is bypassed in the conventional dualloop voltage control.

L i i iabc v oabc i gabc v gabc C f Grid v dc P, Q L g Power Calcu. i iabc v gabc Q P P ref V oref θ oref v iabc SPWM PWM 1-6 Q ref APC RPC αβ abc αβ abc v oabc v oβref G v (s) G v (s)

B. Voltage Control Loop Modeling

Fig. 2 demonstrates the control scheme of an LCL-filtered GFMC in the αβ frame. The solid part represents the standalone mode, and the grid-tied mode should include the dotted part. Note that the analysis can be similarly applied for the PI controller in the synchronous dq frame, as the equivalence has been proved in [START_REF] Zou | Stationary frame equivalent model of proportional-integral controller in dq synchronous frame[END_REF]. In the αβ frame, the voltage controller G v (s) is usually a proportional resonant (PR) controller, i.e.

G v (s) = K vp + 2K vr s s 2 + 2ζ r ω 1 s + ω 1 2 , (1)
where K vp is the proportional gain, K vr is the resonant gain, and ω 1 is the nominal grid frequency. ζ r is the damping factor of the PR controller, whose value is selected as 0.01 to ensure satisfying tracking performance within the range of of (1 ± 0.01)ω 1 . The current controller G c (s) is chosen as

G c (s) = K cp , (2)
where K cp is the current loop gain. The computation and PWM is modeled as a delay function, i.e.

G d (s) = e -sT d , (3)
where the delay time is T d = 1.5T s . T s = T sw is the sampling or switching time period. G if (s) is the current feedback filter, which is a unit gain or bypassed in the conventional dual-loop voltage control.

The plant model from the inverter output voltage v i to the capacitor voltage v o and the inverter-side current i i in the LC system can be derived as

T vLC (s) = v o (s) v i (s) = 1 s 2 L i C f + 1 , (4)

Voltage controller Plant

G v (s) G c (s) G d (s) 1/sL i 1/sC f v oref v o v i i i 1/sL g i g v g v oe G if (s) i c
T iLC (s) = i i (s) v i (s) = sC f s 2 L i C f + 1 . (5)
The open loop transfer function of the voltage loop at standalone mode is then derived as

G LC (s) = vo(s) v oref (s)-vo(s) = Gv(s)Gc(s)G d (s)TvLC(s) 1+Gc(s)G d (s)TiLC(s) = Gv(s)KcpG d (s) s 2 LiC f +sKcpG d (s)C f +1 . ( 6
)
When the GFMC is connected to the grid, the plant model of the LCL system is derived as

T vLCL (s) = v o (s) v i (s) = L g s 2 L i C f L g + L i + L g , (7)
T iLCL (s) = i i (s) v i (s) = s 2 L g C f + 1 s 3 L i C f L g + s(L i + L g ) , (8)
then the open loop transfer function will become

G LCL (s) = vo(s) v oref (s)-vo(s) = Gv(s)Gc(s)G d (s)TvLCL(s) 1+Gc(s)G d (s)TiLCL(s) = Gv(s)KcpG d (s)sLg s 3 C f LiLg+KcpG d (s)s 2 C f Lg+s(Lg+Li)+KcpG d (s) . (9)
Note that the current feedback filter G if (s) in Fig. 2 is bypassed in ( 6) and [START_REF] Liu | Comparison of dynamic characteristics between virtual synchronous generator and droop control in inverterbased distributed generators[END_REF].

C. Voltage Loop Dynamics Evaluation

According to ( 6) and ( 9), the open loop transfer functions are different under LC and LCL modes. More specifically, the low frequency characteristics have changed a lot because the current models ( 5) and ( 8) under the LC and LCL cases are in different formats. Compared to T iLC (s) in ( 5), T iLCL (s) in ( 8) has one more pole at s = 0, which will reduce the type number (TN) of the system G LCL (s) by 1. The TN refers to the number of poles at the origin of the s plane [START_REF] Franklin | Feedback Control of Dynamic Systems[END_REF].

The bode diagrams of G LC (s) and G LCL (s) are shown in Fig. 3. The system parameters and control parameters are shown in TABLEs I and II, respectively. As observed, G LC (s) and G LCL (s) have similar high frequency response. However, during the low frequency range, the slopes of G LC (s) and G LCL (s) are 20 dB dec -1 and 40 dB dec -1 , respectively, which results from the TN difference between the two cases. Similarly, because of the TN difference, the initial phase of G LCL (s) is also 90 • larger than that of G LC (s). The high slope of the LCL (grid-tied) open loop transfer function G LCL (s) at the low frequency range makes the low frequency loop gain poor. Also, the stability margin requirements put limitation on increasing the low frequency loop gain by tuning controller gains. Hence, the voltage tracking speed under the grid-tied mode will be much slower than the stand-alone operation. Note that the controllers and parameters follow the common design guidelines for dual loop voltage controllers for LC/LCL converters to have optimized resonance damping and 3 dB stability margin [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF], [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF]. Fig. 4 shows the simulated waveforms of the voltage step response under the stand-alone and grid-tied modes with same control parameters. As observed, the grid-tied mode response has larger overshoot and settling time than the stand-alone mode, which agrees with the previous analysis. Furthermore, the reduced dynamic response of the voltage control loop is also revealed and analyzed in the synchronous dq frame [START_REF] Deng | Dynamic performance limitation and enhancement of grid-forming converters[END_REF]. This paper is an extension of [START_REF] Deng | Dynamic performance limitation and enhancement of grid-forming converters[END_REF].

Voltage (p.u.) v od v oq V oref M p = 6.0% M p = 22.2% t p = 16 ms t p = 30ms

D. Approximate Amplitude and Phase Transfer Functions

The voltage controller in a GFMC tracks the voltage amplitude reference V oref and phase reference θ oref , which are dc characteristics of the ac signal. Assuming that the ac voltage amplitude over C f is V o , the phase angle of the GFMC is θ o . The transfer functions from the phase and voltage amplitude references to the relative outputs can be represented as where the hatˆdenotes the small perturbation. Because of the symmetry between the d and q axis control, we use the same transfer function T v (s) to represent T p (s) and T a (s). T v (s) can not be written in a simple form directly based on the system and control parameters even in the synchronous dq frame for the complicated coupling relationships on the LCL components [START_REF] Deng | Dynamic performance limitation and enhancement of grid-forming converters[END_REF]. Alternatively, we approximate T v (s) with a second order transfer function from the over shoot M p and the peak time t p of the step responses in Fig. 4 [27], i.e.

T a (s) = Vo (s) Voref (s) ≈ vod (s) vodref (s) , (10)
T p (s) = θo (s) θoref (s) ≈ V o sin θo (s) V o sin θoref (s) = voq (s) voqref (s) , (11)
T p (s) ≈ T a (s) ≈ T v (s) = ω v 2 s 2 + 2ζ v ω v s + ω v 2 , ( 12
)
where T v (s) is the second order approximate voltage loop transfer function,

ζ v = (1/((ln M p )/π) 2 + 1) -1/2
represents the damping factor of the second order system, and

ω v = π(1 -ζ v 2
) -1/2 /t p denotes the natural frequency or the phase or amplitude tracking bandwidth [START_REF] Franklin | Feedback Control of Dynamic Systems[END_REF]. For the LC mode, ω v = 264.5 rad s -1 , and ζ v = 0.67. Yet, for the LCL mode, ω v = 114.5 rad s -1 , and ζ v = 0.40. To summarize, the voltage control under grid-tied mode is slower and more poorly-damped than that under the stand-alone mode. The dotted line in Fig. 4 shows the approximated second order responses for the stand-alone and grid-tied cases.

III. POWER LOOP MODEL AND VOLTAGE LOOP INFLUENCE

This section will model the power loop of a GFMC firstly, then emphasize the influence of the voltage control dynamics on the overall stability of the power control.

A. Power Loop Modeling

For the system in Fig. 1, assume that the grid phase angle is θ g . The power angle between the GFMC and the grid is

δ = θ o -θ g = (ω o -ω g )dt, (13)
where ω o and ω g represents the GFMC and the grid frequency, respectively. Then the active and reactive power flowing from the GFMC to the grid is given by

P = 3Vo[(Vo-Vg cos δ0)Rg+VgXg sin δ0] 2(Rg 2 +Xg 2 ) Q = 3Vo[(Vo-Vg cos δ0)Xg-VgRg sin δ0] 2(Rg 2 +Xg 2 ) , (14)
where δ 0 denotes the steady-state power angle operating point between the GFMC and the grid, X g = ω 1 L g represents the line reactance. Taking partial differential of ( 14) yields the small signal model of the power control plant, i.e.

P = K pδ δ + K pv Vo Q = K qδ δ + K qv Vo , (15)
where the coefficients K pδ , K pv , K qδ and K qv are expressed as

             K pδ = 3VoVg(Rg sin δ0+Xg cos δ0) 2(R 2 g +X 2 g ) K pv = 3(2VoRg+Vg(-Rg cos δ0+Xg sin δ0)) 2(R 2 g +X 2 g ) K qδ = 3VoVg(-Rg cos δ0+Xg sin δ0) 2(R 2 g +X 2 g ) K qv = 3(2VoXg+Vg(-Rg sin δ0-Xg cos δ0)) 2(R 2 g +X 2 g ) . ( 16
)
Noted that for an inductive transmission line where X g ≫ R g , K pv and K qδ are much smaller than K pδ and K qv . The power controllers feed back P and Q through a low pass filter (LPF) G f (s) = ω f /(s + ω f ), which can be written in the form of

P mpu (s) = G f (s)P (s)/S n Q mpu (s) = G f (s)Q(s)/S n , (17)
where ω f is the cutoff frequency, S n is the rated capacity of the GFMC, P mpu and Q mpu are the per unit value of the measured P and Q, respectively. In the active power controller (APC), the frequency reference ω oref of the GFMC is designed as

ω oref = (P ref -P mpu )ω 1 /D p + ω 1 , (18)
where D p represents the active power droop gain in per unit form. Similarly, droop control is also used in the reactive power controller (RPC). The voltage reference V oref for the GFMC is chosen as

V oref = (Q ref -Q mpu )V n /D q + V n ( 19
)
where V n is the nominal voltage, D q stands for the per unit reactive power droop gain. The active and reactive power controller G p (s) and G q (s) can be written in the form of

G p (s) = θ oref (s) P ref (s)-Pmpu(s) = ω1 Dps G q (s) = V oref (s) Q ref (s)-Qmpu(s) = Vn Dq . ( 20
)
Combining the small signal model of the plant and the controllers will finally yield the power loop control block diagram illustrated in Fig. 5, where Pf and Qf represent the per unit filtered active and reactive power, respectively. The active power open loop transfer function G p open (s) is defined as the product of the forward and feedback path transfer functions between Pref and P in Fig. 5, which can be derived as In ( 21), K pv and K qδ are very small compared to K pδ , so G p open (s) can be further approximated as

G p open (s) = Pf (s) Pref (s)-Pf (s) = Gp(s)Tv(s)G f (s) Sn • (K pδ + K qδ KpvGq(s)Tv(s)G f (s)/Sn 1+KqvGq(s)Tv(s)G f (s)/Sn ) . ( 21
)
P ref Q ref 1 / D p ω 1 / s ω g t K pδ 1 / D q T v (s) T v (s) V g K qv K pv K qδ ω f / (s+ω f ) ω f / (s+ω f ) P Q RPC V-P Plant Model VCL δ ΔV V oref ω oref t ω o t V n 1 / S n 1 / S n APC V o P f Q f
G p open (s) ≈ G p (s)T v (s)G f (s)K pδ /S n (22)
From ( 22), the power loop bandwidth is inversely proportional to the droop coefficient D p , the converter capacity S n and the grid impedance X g , where D p associates with G p (s) in [START_REF] Zhang | Capacitor voltage full feedback scheme for lcl-type grid-connected inverter to suppress current distortion due to grid voltage harmonics[END_REF] and X g relates to the coefficients K pδ . Specifically, GFMCs with small capacities or droop coefficients, or connected to the grid through small impedance will have fast power loops.

The power loop bandwidth may vary from 0.1 Hz to 10 Hz [START_REF] Qing-Chang | Selfsynchronized synchronverters: Inverters without a dedicated synchronization unit[END_REF], [START_REF] Ramezani | Seamless transition of synchronous inverters using synchronizing virtual torque and flux linkage[END_REF].

B. Interaction between Power Loop and Voltage Loop

Fig. 6 plots the active power step response of the ideal active power loop with T v (s) = 1 and the actual power loop response when T v (s) is represented with [START_REF] Fang | Statespace modeling and control of grid-tied power converters with capacitive/battery energy storage and grid-supportive services[END_REF]. With the open loop transfer function in [START_REF] Geng | Stability analysis and key parameters design for grid-connected current-source inverter with capacitor-voltage feedback active damping[END_REF] and the parameters in TABLEs I and II, the ideal active power loop bandwidth can be calculated as 8.8 Hz with a phase margin of 85 • . As illustrated in Fig. 6, the designed power loop response is fast and smooth. Nonetheless, the actual active power response oscillates and takes more time to settle. As analyzed in Section II-D, when the GFMC is connected to the grid, the damping ratio drops to ζ v = 0.40, and the voltage control bandwidth decreases to ω v = 114.5 rad s -1 , i.e. 18.2 Hz. Because the voltage loop is poorly damped, and its bandwidth is close to the power loop bandwidth, the power loop response gets undesirable. To reduce the interaction between the power loop and the voltage loop, we design a voltage controller with improved dynamics.

IV. DYNAMIC ENHANCED VOLTAGE CONTROLLER

This section will propose a generic control scheme to improve the voltage reference tracking performance. Then the overall control parameter design guideline will be given.

A. Control Architecture Design

It can be concluded from the previous analysis based on [START_REF] Zhang | Interconnection of two very weak ac systems by vsc-hvdc links using power-synchronization control[END_REF] and ( 9) that the reduced dynamic response of the GFMC under the grid-tied mode mainly results from the TN changing, i.e. the zero of G LCL (s) at the origin introduced by the current feedback loop when the grid impedance is considered. In order to achieve similar satisfied dynamics under both stand-alone and grid-tied modes, we need to cancel the pole of T iLCL (s) at s = 0, i.e. add a zero in the current feedback loop in the low frequency range. Moreover, the resonance damping function in the high frequency range should be kept simultaneously. As a solution, we add a high-pass filter (HPF) G if (s) to the current feedback path as shown in Fig. 1 and Fig. 2. The HPF is implemented in the stationary frame, which is expressed as

G if (s) = s/(s + ω if ), (23)
where ω if is the cutoff frequency of the HPF. After employing the HPF, the voltage control open loop transfer function under LC and LCL modes will become

G LC (s) = vo(s) v oref (s)-vo(s) = Gv(s)Gc(s)G d (s)TvLC(s) 1+Gc(s)G d (s)G if (s)TiLC(s) = Gv(s)KcpG d (s) s 2 LiC f +s 2 G d (s)KcpC f /(s+ω if )+1 , (24)
G LCL (s) = vo(s) v oref (s)-vo(s) = Gv(s)Gc(s)G d (s)TvLCL(s) 1+Gc(s)G d (s)G if (s)TiLCL(s) = Gv(s)KcpG d (s)Lg s 2 C f LiLg+ KcpG d (s)s 2 C f Lg (s+ω if ) +(Lg+Li)+ Kcp G d (s) (s+ω if ) . (25)
As observed in [START_REF] Fang | On the inertia of future moreelectronics power systems[END_REF], the zero at s = 0 in (9) introduced by the current loop is canceled by introducing the HPF G if (s). As a result, the TN of G LC (s) and G LCL (s) becomes the same after employing the HPF in the current feedback path.

The bode diagram of the voltage control open loop gain with the filtered current feedback under stand-alone (LC) and grid-tied (LCL) mode is shown in Fig. 7. As observed, the HPF mitigates the current loop influence on the low frequency range while retaining the resonance frequency damping effect at the higher frequency range. The loop gain under the gridtied (LCL) case is improved significantly especially in the low frequency range when compared to the conventional dual loop voltage controlled GFMC under the grid-tied (LCL) case. At the same time, the open loop gain under the stand-alone (LC) case and the high frequency part of the grid-tied (LCL) case have no serious difference after adding the HPF.

Fig. 8 shows the comparison of the voltage reference tracking dynamics simulation under the grid-tied operation with and without applying the proposed HPF in the current feedback loop. Except for the HPF G if (s), the control structures and parameters are the same. As observed, after applying the HPF, the settling time t s corresponding to a ± 5 % error band is reduced from 71 ms to 17 ms, and the overshoot M p is also reduced from 22.2 % to 5.6 %, showing a great improvement in the voltage reference tracking performance with faster speed and less oscillations.

Note that although the HPF in the feedback path will make it harder to add current saturation directly in the cascaded control path, yet the current information can still be used to limit the current. [START_REF] Qoria | Grid-forming control to achieve a 100power transmission systems[END_REF] proposed a current limiting method that can be applied to any forms of voltage controllers even without a current loop. Hence, the proposed HPF in the feedback path will not impede current limiting in GFMCs.

B. Control Parameter Design

The design of LCL parameters in dual loop voltage control schemes basically follows the rule of not introducing negative virtual resistances to the system, i.e. f r < f s /6 [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF], [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF].

The main control parameters to be designed are the voltage controller gains K vp and K vr , the current controller gain K cp , and the filter cutoff frequency ω if .

Firstly, K cp , K vp and K vr are designed based on the dual loop voltage control structure without considering G if (s). As mentioned before, the function of the current loop is to damp the LC/LCL resonance. Because of the computation and PWM delay, there is a boundary value and an optimized value for K cp [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF]. K cp should be smaller than the boundary value to achieve positive damping. At the optimized K cp , the damping performance is the best. The boundary and optimized K cp values for LC and LCL cases are different. In this paper, K cp is chosen as 6.7 Ω to achieve a satisfied damping effect under both LC and LCL cases [START_REF] Geng | Parameters design and optimization for lc-type off-grid inverters with inductorcurrent feedback active damping[END_REF].

The R controller is chosen because it has a wider control bandwidth than the PR controller in the dual loop voltage control under the same stability requirements [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF]. As a result, K vp is assigned with 0 S, and K vi is selected as 150 S s -1 to achieve a 3 dB gain margin.

Then the HPF cutoff frequency ω if is designed. To ensure enough resonance damping at the resonance peak, we require the current loop gain has at least 0.9 of the original loop gain at the LC resonance frequency, which yields

|G if (jω rLC )| = ω rLC / ω rLC 2 + ω if 2 ≥ 0.9, (26)
where ω rLC = 1/ √ L i C f is the LC resonance frequency. Based on the system parameters given in TABLE I, ω cf is assigned with 2393 rad s -1 .

C. Physical Explanation of HPF in Current Feedback Path

As revealed in Section II, the deteriorated dynamic response of GFMCs under the grid-tied mode is caused by the inverterside current model changing when connected to the grid, which worsens the low-frequency response. On the contrary, the model of the capacitor current i c will not change its form under both grid-tied and stand-alone modes, whose models under the stand-alone and grid-tied modes are expressed as

T icLC (s) = i c (s) v i (s) = sC f s 2 L i C f + 1 , (27)
T icLCL (s) = i c (s) v i (s) = sL g C f s 2 L i C f L g + (L i + L g ) , (28)
respectively. As a result, using the capacitor current feedback damping [START_REF] Pan | Capacitor-currentfeedback active damping with reduced computation delay for improving robustness of lcl-type grid-connected inverter[END_REF] will not worsen the dynamic response of the voltage control under the grid-tied mode. The LCL filter smooths the output current i g , which means the high-frequency signals in the inverter-side current i i go to the capacitor. Consequently, using a HPF in the inverter-side current can be interpreted as emulating the capacitor current feedback active damping. Compared to using the capacitor current as the inner loop, using a HPF in the converter output current can improve the voltage control dynamics without introducing extra current sensors.

D. Controller Discretization

Although the system modeling and the controller design are conducted in the continuous-time domain, the implementation of the controllers is conducted in the discrete-time domain. The controllers in the s domain are mapped to the z domain using the Tustin's method, i.e. the substitution of

s ← 2 T s z -1 z + 1 . (29)
Since the voltage control bandwidth is much lower than the switching frequency, the approximate continuous-time analysis conclusions can be applied to the actual discrete-time system.

V. DISCUSSION

This section first reveals the similar dynamic response worsening problem in different voltage control schemes, then reviews other commonly used state feedback control methods for LC/LCL filtered power converters. The conclusion is that some of these methods are equivalent and may also be applied to solve the problem discovered in this paper.

A. Dynamic Response Reduction Problem in Other Control Schemes

The reduced dynamic response not only exists in the dual loop voltage control scheme discussed in Section II, but also exists in other control schemes whenever the inverter-side current is utilized for resonance damping, even when highbandwidth controllers are designed under stand-alone cases. With the same dual loop control structure in Fig. 2, the voltage controller G v (s) can take other forms, such as proportional resonance with integral (PR-I) controller and integral plus resonance (IR) controllers [START_REF] Liao | Passivity-based analysis and design of linear voltage controllers for voltage-source converters[END_REF]. These I-based controllers have a wider bandwidth than the R or PR controllers used in this paper, while the similar TN changing and reduced dynamic response also exists because of the same control structures. Another widely used control structure is the dual loop voltage control with voltage feedback decoupling. This control structure has extremely fast response under the stand-alone mode, while the TN will decrease by two when the GFMC is connected to the grid, which will also lead to the reduced dynamic response. The proposed filtered current feedback method can also improve the voltage tracking dynamics in the above-mentioned control structures.

B. Review of State Feedback Control for LC/LCL Filtered Power Converters

This paper proposed a filtered current feedback voltage control strategy to improve the voltage control dynamics for grid-forming converters under the grid-tied operation. In addition to the proposed method, there are many other state feedback control with filtering methods to improve the system performances. Here we summarized these control methods as shown in TABLE III. Besides the summarized methods for special functions, [START_REF] Wang | Virtual-Impedance-Based Control for Voltage-Source and Current-Source Converters[END_REF] also gives a thorough review of virtual impedance implementation with different control variables and filter combinations for aims of damping, disturbance rejection and auxiliary services. Most of the summarized methods study the stability problems of the grid-feeding converters, and aiming at improving the LC/LCL resonance damping, widening the stable region of the resonance frequency, and mitigating the computation delay. This paper fills the gap of improving the low-frequency response of the dual-loop voltage controller under the grid-tied mode. Note that the poor dynamic response of GFMCs under the grid-tied mode can be improved through various state feedback control methods, such as capacitor current feedback method mentioned in section IV-C and emulating the capacitor current by taking the derivative of capacitor voltage as shown in Table III. The grid-side virtual impedance can also relieve this problem to some extent, but the voltage deviation caused by the grid-side virtual impedance will put stress on the voltage quality and the power regulation. Despite various solutions, the contributions of this paper focus more on revealing the poor voltage tracking dynamic problem when implementing the dual-loop voltage control with the inverter-side current as the inner loop and providing a possible solution.

VI. EXPERIMENTAL RESULTS

We compared the performance of the conventional dual-loop voltage controller and the proposed filtered current feedback voltage controller under the cases of stand-alone/grid-tied modes and with/without power control loop modes based on the schematic diagram shown in Fig. 1. The system and control parameters are listed in TABLE I and II. The experimental results verify that the proposed controller can improve the voltage tracking dynamics and the power control stability when the grid-forming converter is connected to the grid. An off-grid RL load step increase test indicates that the proposed controller will not influence the normal operation under the stand-alone mode.

Fig. 9 shows the photo of the experimental setup. The gridemulator Chroma 61830 was connected to the grid-forming converter through grid inductors. The grid-forming converter was fed by a constant voltage dc power supply and controlled by dSPACE (microlabbox) control platform. An 8-channel oscilloscope was employed to capture the experimental waveforms. The experimental results are shown in Fig. 1011121314.

A. Voltage Reference Tracking Test

To illustrate the difference of the voltage reference tracking dynamics when the GFMC is under stand-alone and gridtied modes, and to show improvements in voltage tracking with the proposed controller under the grid-tied mode, voltage reference step experiments were conducted. As known, severe voltage differences between two voltage sources will cause large short-circuit current. In order to show the voltage tracking dynamics changing clearly without causing over-current, in voltage reference step experiments, the peak voltage was set at 20 V. The GFMC and the grid emulator were intentionally designed to have the same frequency and phase angle. Fig. 10 illustrates the voltage waveforms when the conventional dual loop voltage controlled GFMC undergoing a 20 % (4 V) voltage reference step change. As observed, in Fig. 10(a), when the GFMC was disconnected from the grid, the voltage v oabc tracked the voltage reference quickly and arrived at a steady state in less than 10 ms. However, in Fig. 10(b), when the GFMC with the conventional dual loop voltage controller was connected to the grid, the transient of the voltage reference tracking became oscillatory and slow. An overshoot existed and the voltage arrived at the steady state in longer than 50 ms, which was much slower than the stand-alone case and conformed to the analysis in Section III.

Fig. 11 shows the voltage waveforms when the GFMC with the proposed controller undergoes a 20 % (4 V) voltage reference step change. The stand-alone voltage tracking performance presented in Fig. 11(a) was as fast as that with the conventional dual-loop controller in Fig. 10(a). Contrary to Fig. 10(b), when connected to the grid, the GFMC with the proposed controller tracked the reference much faster and arrived at the steady state within 10 ms as shown in Fig. 11(b), validating the effectiveness of the proposed controller. The experimental waveforms of the grid-tied operation without and with the proposed controller conformed to the simulation results as shown in Fig. 8.

B. Tests Including the Power Loop

Fig. 12 and Fig. 13 tests the controller performances under the close-loop power control condition. The nominal peak voltage and nominal power were selected as 155 V and 3000 W for this set of experiments. The ideal response of the power control loop was a first order response with the bandwidth of 8.8 Hz. In the experiments, the active power reference for the GFMC stepped from 0 W to 900 W. In Fig. 12, the conventional dual-loop voltage controller was employed. As noticed, when the active power reference changed, a strong oscillation existed in the output power and currents waveforms. The peak-to-peak value of the oscillation reached up to 300 W, which was 33.3 % of the step reference value. Also, it took longer time than expected from the designed power loop (settling time of 72 ms) to converge to a steady state. In contrast, in Fig. 13, when the proposed controller was applied, the power waveforms responded to the active power reference change quickly without any overshoot. As observed, the power response corresponded with the designed power loop characteristic with the designed faster voltage controller.

When the power references change, there was a small amount of reactive power change in both Fig. 12 and Fig. 13. The reactive power deviation was caused by the cross-coupling between the active and reactive power as shown in Fig. 5.

As the reactive power control used droop control instead of the non-steady-state-error control in this work, the reactive power would not converge to 0 VAR, while the droop control would tend to reduce the reactive power deviation by tuning the voltage amplitude reference. Fig. 14 shows the stand-alone operation of the GFMC with the proposed controller under an RL load step change. The active and reactive power references were set at P ref = 0 W and Q ref = 0 VAR, respectively. The RL load value was chosen as R L = 54 Ω, L L = 171 mH. The waveforms of the output voltage v oabc , load current i gabc , active and reactive power P and Q, and the output frequency f o are shown in Fig. 14. When the RL load was suddenly connected to the converter, the current increased smoothly and the voltage was well regulated. The active and reactive power outputs were around 320 W and 310 VAR, respectively. As the droop control was applied and the output power was different from the setting value of 0 W, the frequency deviated from its nominal value around -0.106 Hz. The RL load step experiment shows that the GFMC with the proposed controller can work well under the stand-alone operation.

VII. CONCLUSION

A well-designed fast voltage loop is important for the control loop decoupling and stable operation of LC/LCLfiltered GFMCs especially when the power loop bandwidth is high. This paper reveals with detailed analysis that the conventional dual-loop voltage control with the inverter-side current feedback as the resonance damping has poorer dynamics when connected to the grid than operating in the stand-alone mode, which may threaten the stability of the power control loop. To improve the voltage tracking dynamics and the power control stability of GFMCs under the grid-tied mode, the dualloop control is enhanced by applying a HPF in the current feedback path. GFMCs with the proposed generic voltage controller have fast and stable responses under both standalone and grid-tied modes. The correctness of the analysis and the effectiveness of the proposed controller is verified by the experiments.

Fig. 1 .

 1 Fig. 1. Topology and control schematic of a GFMC.

Fig. 2 .

 2 Fig. 2. Voltage control block diagram under stand-alone / LC mode (solid) and grid-tied / LCL mode (solid + dotted).

3 .

 3 Bode diagram of G LC (s) and G LCL (s) with the conventional dual loop controller.

Fig. 4 .

 4 Fig. 4. Voltage step responses (solid) and corresponding 2 nd order approximations (dotted) under stand-alone and grid-tied modes.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Power control model.

7 .

 7 Bode diagram of G LC (s) and G LCL (s) with filtered current feedback damping.

Fig. 8 .

 8 Fig. 8. Voltage step responses without and with the current feedback filter G if (s) under the grid-tied operation.

Fig. 9 .

 9 Fig. 9. Experimental Setup.

vFig. 10 .Fig. 11 .

 1011 Fig. 10. Experimental waveforms of the conventional dual loop voltage controller under a 20 % voltage step change. (a). Stand-alone mode. (b). Grid-tied mode.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Experimental waveforms of the conventional dual-loop voltage controller under a 900 W (30 %) power reference step change.

vFig. 14 .

 14 Fig. 14. Experimental waveforms of the dual-loop voltage controller with high-pass filtered current feedback under a step increase of RL load with stand-alone operation.

Vincent Debusschere (Senior Member, IEEE) received the Ph.D. degree in ecodesign of electrical machines from the Ecole Normale Superieure de Cachan, Cachan, France, in 2009. In 2010, he joined the Electrical Engineering Laboratory, Grenoble Institute of Technology, Grenoble, France, as an Associated Professor. His research interests include renewable energy integration, modeling of flexibility levels for smart grids, multicriteria assessment, and optimal design of complex systems.


Pascal Engel

Trust and the doxastic family

Keywords: Belief, Acceptance, Doxastic attitudes, Trust, Reason, K. Lehrer

This article examines Keith Lehrer's distinction between belief and acceptance and how it differs from other accounts of belief and of the family of doxastic attitudes. I sketch a different taxonomy of doxastic attitudes. Lehrer's notion of acceptance is mostly epistemic and at the service of his account of the "loop of reason", whereas for other writers acceptance is mostly a pragmatic attitude. I argue, however, that his account of acceptance underdetermines the role that the attitude of trust plays in his analysis of reason.

anticipated much later work on the difference between belief and other doxastic attitudes of the same family. Lehrer was one of the first, in contemporary analytic philosophy, to rediscover an insight which goes back at least from the Stoics, the idea that there is a variety of forms of assent, among which belief is only one species. 1 Acceptance and belief differ not only psychologically but also epistemologically.2 I would like here to try to assess Lehrer's contribution, to compare it with others similar distinctions, and to suggest a slightly different way of carving the doxastic nature at its joints.

The attitude of acceptance is a central piece in Lehrer's well known internalist and coherentist definition of knowledge. Barring various complexities of the definition, S knows that P iff (i) P is true (ii) S accepts that P (iii) S is personally justified in accepting that P (iv) S is justified in accepting that P on the basis of S's evaluation system at t, and if acceptance of P coheres with that system, and in a way which is undefeated (Lehrer 1990, 26, Lehrer 1997, 44) Why "acceptance" and not "belief" as in the traditional analysis of knowledge?

Because, according to Lehrer, only an attitude which involves more than belief, and the evaluation of one's beliefs, is susceptible to provide the appropriate personal justification required for full blown and undefeated knowledge. Acceptance, like belief, is a functional state, but a second-order one: it has a belief content as its object, but it is not a belief. It is a meta-level state, involving conscious reflection and reasoning. It is defined in terms of some aim or purpose of a subject, namely the purpose is attaining truth and avoiding error, through reasoning and evaluation of one's own justifications. Belief, in contrast may be defined in terms of a purpose (it "aims" at truth, as one says) but not necessarily (many beliefs are not aimed at truth). Belief involves mostly the processing of information at the first-order level that is not necessarily conscious or available directly to reasoning or inference. I may for instance, have the information -and therefore in this sense believe -that your phone number is 41 22 379 70 53, because it is the number that came up to my mind when your name was mentioned to me , but unless I have manifested my capacity to reason from this belief, I am in no position to evaluate that belief. Or I can learn, hence believe, that the person I have met a few minutes ago just died, but I may not accept it. In these examples the content of the belief and the content of the acceptance are the same, but there are many cases of dissociations where one does not accept what one believes. Thus a racist juror may accept that the black man who is in the box is innocent, although he believes that he is guilty (Lehrer 2000, 196).

Lehrer's views on belief and acceptance share a number of features with the views of other authors, but it has also a number of specific features. His distinction between mind and metamind (1990a) resembles [START_REF] Fodor | The modularity of mind[END_REF] distinction between input systems, which are largely automatic, modular and encapsulated and the central system which is slow, holistic and inferentially promiscuous, but it does not coincide with it, since for Fodor it is belief which lends itself to inference and reasoning and to holistic connections, whereas sensory systems and subdoxastic states are not so penetrable to inferences. [START_REF] Stich | Beliefs and subdoxastic states[END_REF] Lehrer does not deny that belief can give rise to inferences, but as he points out (2000,211) inference from beliefs can be largely habitual and automatic, just as higher-order evaluation can be habitual and automatic. The distinction has more to do with the conscious character of acceptance.

Neither should Lehrer's distinction be assimilated to Cohen's (1989[START_REF] Cohen | An essay on belief and acceptance[END_REF]) and [START_REF] Bratman | Practical reasoning and acceptance in a context[END_REF] similar distinction. Cohen and Bratman take acceptance to be essentially occurrent, voluntary, associated to conscious acts of judgments, motivated by pragmatic reasons and contextual in nature. They both emphasize the pragmatic nature of acceptances, which can occur in the absence of belief (which is a mere feeling on Cohen's view), and which can be entertained for purely practical or prudential reasons (the standard example is the lawyer who believes his client guilty but accepts in court that he is not). Bratman defines acceptance as an attitude which one typically takes in the activity of rational planning, and Cohen defines acceptance as a kind of policy: "To accept that p is to have, or to adopt a policy of deeming, positing, or postulating that p -i.e of including that proposition or rule among one's premises for deciding what to do or think in a particular context, whether or not one feels it true that p." (Cohen 1992, 4). The policy can be adopted for ultimately epistemic reasons, as when one takes something for granted for the sake of future argument, but the adoption of the policy is itself provisional and pragmatic in the sense that it is contextually determined. In contrast Lehrer insists that acceptance is a functional state which is distinct from its possible manifestation as a commitment expressed by a speech act of the form: "Accept that P". So acceptance can be dispositional and need not be voluntary, 3 although it is functionally related to preferences. Moreover, for Lehrer, acceptance is essentially an epistemic attitude, not a pragmatic one. For Lehrer the meta-doxastic attitude that on has towards P for practical reasons, hence associated to practical reasoning, is better called preference, which stand to desires in the same relation as the one in which acceptances stand to beliefs (Lehrer 1997, 11-12). Epistemic acceptances are those which play a role in personal justification: S is justified in accepting that P if and only if P coheres with S's evaluation system at t, and if all objections to P are neutralized. Acceptance is evidence sensitive, belief not necessarily. Acceptance is relatively stable, and it is not contextual in the sense in which pragmatic acceptance is according to Cohen or Bratman. « Acceptance that H introduces a kind of rational equilibrium over rational reflection at a point in time in the sense that rational reflection on the present evidence will not change my acceptance of H or the degree of probability of H which is a factor in in the acceptance of H at the present time. Thus if R is the rational consideration of the evidence for H at t , and I accept that H, then the antecedent probability for H is stable on R, i that is Pt (H) = Pt (H/ R)" (Lehrer 2000, 215). This is not true for belief because belief may arise without evaluation of evidence.

Lehrer's distinction is explicitly drawn for epistemological, not primarily psychological, purposes. Acceptance plays an important role in his "loop" of selftrust, upon which he bases his conception of knowledge (1997,16). His "principle of trustworthiness" -I am trustworthy in what I accept -is the "keystone" of his "loop of reason". Is it justified? I would like to argue here that this depends in large part upon how we understand the attitude of acceptance. I shall first try to show that Lehrer's classification of attitudes has to be refined if it is supposed to be descriptively adequate. I shall then try to apply the lessons of the classification to his account of trust as the foundation of knowledge.

Two systems of doxastic attitudes or one?

There are a number of ways one can draw the psychological distinction between belief and acceptance and classify these with the family of doxastic attitudes. One can, on the one hand, take belief to be the most general attitude, of which acceptance is a species. 4 Alternatively one can call "acceptance" the general attitude of holding true or of assenting to a proposition, and take belief to be a subvariety. (Stalnaker 1984, 79-81) But most views imply that belief and acceptance are different functional sates. On the paradigm notion of belief, belief is a dispositional or functional state, involuntary and passive, sensitive to evidence, and inferentially promiscuous, whereas acceptance is most often understood as a conscious mental act, tied to the speech act of assertion, sensitive either to evidence or to pragmatic goals, and expressing a commitment in future doxastic deliberation. Some, however, hold that acceptance can be tacit (Frankish 2004, 94-95). Others give definitions of belief which are actually very similar to the way acceptance is characterised: "To believe a proposition is to accept it with the aim of thereby believing a truth" (Velleman 2000, 251). Where should we draw the line?

A number of mental states share with paradigmatic beliefs some features, but lack others. They seem to deserve the title of "belief" only by courtesy. The list is quite long and its limits are uncertain. Can forgettings, cases in which apparently one loses a belief because it is erased partially from memory, be counted as beliefs? They have sometimes be called "in between believings" [START_REF] Schwitzgebel | In between believing[END_REF]. Can "feelings of knowing", such as the "tip of the tongue phenomenon" [START_REF] Koriat | The Feeling of Knowing[END_REF] be included as well? There has been a lot of discussion about the existence of putative "tacit beliefs" which lay dormant within our mind but which manifest themselves when one mentions them, such as my belief that Elephants do not wear pyjamas in the wild [START_REF] Lycan | Tacit belief[END_REF]. More common examples of implicit beliefs lying at the borderline of the territory are racist beliefs, upon which one often acts but which do not surface in assent, and irrational or delusive "beliefs", like those which appear in self-deception. These states have some belief-like features, but lack others. Thus patients affected by the Capgras delusion have belief-like states (they seem to believe that their friends and family have be replaced by impostors), but their "beliefs" are often unresponsive to evidence and to inference, and do not lead, like ordinary beliefs, to characteristic behaviour or actions. Self deceptive "beliefs" are often associated with explicit disavowal, refusal to consider contrary evidence. Other belief-like states, like fears or phobias, have a strong emotional component and a strong behavioural uptake, although they are not sensitive to evidence and to other beliefs. Cohen (1992, 5) talks of "belief feelings" or of "creedal feelings" to characterise mental states like hope, fear, expectation or surprise. [START_REF] Gendler | Belief and alief[END_REF] has called "alief" the kind of cognitive state mixed with vertigo in which one finds oneself when walking on the transparent road way above the Grand Canyon. Emotions have, on most views, a cognitive component, but it is hard to disentangle it from the emotion, as the "paradox of fiction" exemplifies: watching an horror movie you do not believe that what you see on the screen is real, but you feel the emotion of fear nevertheless, which in turn implies the belief that what you see is real [START_REF] Radford | How can we be moved by the fate of Anna Karenina[END_REF]). Imaginings can be described as kinds of beliefs, but can also be described as very unlike believings [START_REF] Velleman | On the aim of belief[END_REF]. Some descriptions of religious belief make it dubious that they are beliefs at all. Last but not least, where in this list of "beliefs" should we put degrees of beliefs or "partial beliefs"? Are these a genuine kind of mental state or a mere construct of the Bayesian conception of belief as degree of probability? What is their relationship to "full" belief? The list could be extended. In all these cases, it is debated whether these belief-like states are beliefs or whether they are only belief's bedfellows. 5 How can we frame a taxonomy which could take into account all the members of the doxastic family? One important framework in recent cognitive psychology is the "dual system" view according to which there are two distinct systems in human cognition, in particular in reasoning, decision making, and learning, characterised respectively by the following general features (Evans and Frankish 2008, 16): 5 I borrow this expression from Tim Bayne, who suggested it in a talk in January 2011. On the issue whether delusional beliefs are genuine beliefs ( and for a positive answer) see [START_REF] Bayne | In defence of the doxastic conception of delusion[END_REF]. For the opposing view, see [START_REF] Currie | Recreative minds[END_REF]. It is tempting to classify credal feelings, "alief", irrational beliefs and dispositional beliefs within System 1 and acceptances within System 2. But this is too quick. A number of properties of paradigm belief features within System 1 have System 2 features and vice versa: inferentially promiscuous, reflective, evidence based. Belief in many ways is, to take up Scanlon's (1998), a "judgment-sensitive" attitude as much as a dispositional state. System 1 is supposed to be strongly modular and System 2 to be distributed and holistic, but a number of writers have argued that System 2 is modular as well [START_REF] Sperber | In Defense of massive modularity[END_REF]. Even pragmatic acceptance, of which a number of writers claim that it can occur in the absence of belief, actually depends upon pre-existing beliefs. 6 So the distinction between the two systems is moot, and the properties which characterise them may well apply to several subsystems which share only partly these properties. Thus some processes in reasoning fit both System 1 and System 2 (Carruthers, to appear).

6 Thus take [START_REF] Bratman | Practical reasoning and acceptance in a context[END_REF] example of the prospective builder who accepts that the cost of the house he plans to build will be a certain amount, $ 1000 000. But what he accepts here is not that the house will cost $ 1000 000. What he accepts is that the probability of the estimated cost of $ 1000 000 exceeds the estimated cost that he makes. And he makes this estimation just in the way a good Bayesian would do it: by calculating the prior estimated probability, and by taking the conditional probability of the cost given this first estimate. He accepts finally that the probability that the total cost exceeds his prior estimate. But he certainly makes this estimate on the basis of his subjective probabilities. So acceptance here is not identical to a subjective probability, although it is based on one (see on this example [START_REF] Engel | Believing, accepting, and holding true[END_REF].

Although there are strong empirical reasons for the existence of different cognitive subsystems having a number of properties of System 1 and of System 2 respectively, there is no reason to suppose that these subsystems all belong either to System 1 to System 2, hence that the distinction between two types of systems or of processes in the mind corresponds to a natural kind. When it comes to belief, there is reason to think that there is much more continuity between "belief like" and "acceptance like" features than the dual systems view allows. I would like to come back to the alternative -and more classical -picture of a unique, but layered system of doxastic attitudes, with one general category, belief, and a variety of subcategories belonging to that species. This picture has been advanced recent cognitive psychology for a variety of other mental states. Thus it has been proposed that intentions can be divided into "motor intentions" at a low cognitive level, "intentions-in-action" at the time of execution, and "prior" intentions as plans. (Searle 1983[START_REF] Pacherie | The content of intentions[END_REF] Similarly we can propose the idea that there are continuous levels of belief, starting at a low level with creedal feelings and dispositional beliefs, moving upwards to degrees of belief and to assents, with judgments and acceptances at the top level. On such a continuous taxonomy it is not clear that there is a real distinction between states of beliefs and states of acceptance. We can start from a minimal characterisation of belief as a mental attitude governed by a norm of truth:

(NT) A belief is correct if and only if it is true7

Although this norm governs most beliefs when an agent deliberates reflectively on whether to believe that P (Shah and Velleman 2005), it can nevertheless be argued that it governs also irrational or unconscious belief formation. Contrary to what a number of critics of the truth-norm for belief have argued, the norm need not be an explicit prescription which would regulate every step of belief formation at the intentional level. Thus take irrational beliefs, such as delusions. It has been argued that patients who are victims of such pathologies actually do not believe the things that they seem to believe, but only believe that they believe them and imagine them: their doxastic attitudes are more like hallucinations than like belief [START_REF] Currie | Imagination; Delusions and Hallucination[END_REF]. It seems hard to say that the patients with such monothematic "beliefs" have doxastic attitudes which can be put on a par with beliefs. Indeed the beliefs which are characteristic of the Capgras delusion are very unlike ordinary belief, but they nevertheless involve a feeling or and emotion of conviction towards a proposition.

The state in question may be inferentially insulated and insensitive to evidence, but it is sensitive to the norm of truth (Bayne and Pacherie 2005, 183).

On the taxonomy of propositional attitudes sketched here, there is no basic division between belief-like states having features of System 1 and acceptance like states having features of System 2. here is only one category of states, belief, which is a form of assent or of holding true, governed by a norm of truth, which can be, in some cases, a mere creedal feeling, or in others a judgement or an acceptance. This does not mean that there are no distinctions and that there is no room for a rational epistemic attitude such as that which Lehrer describes as acceptance.

Acceptances in turn can be either epistemic, governed by epistemic reasons and for the sake of theoretical reasoning, or pragmatic and contextual and for the sake of prudence. There is a further subdivision: when they are pragmatic, they can either be governed by epistemic or by practical reasons (Fig. 1).

BELIEF CREDAL FEELINGS DISPOSITIONAL ACCEPTANCE EPISTEMIC PRAGMATIC

Epistemic prudential

Fig. 1 In what sense is this taxonomy distinct from Lehrer's? It is distinct in that on the present taxonomy, acceptance is a kind of belief and not a separate doxastic attitude, and in that acceptance comes in two forms, epistemic and pragmatic. For

Lehrer acceptance, which is distinct from belief, has one form and is essentially epistemic. All of this may be a matter of terminology, and a lot of the literature on acceptance has turned on terminological matters. But the issue is not only terminological, as we shall see when it is applied to the problem of the foundation of reason.

Acceptance and Trust

Acceptance is not merely a psychological attitude. It is also, and foremost, an This principle is supposed to be the "keystone" at the basis of knowledge, and because it is self-supporting it is the basis of the "loop of reason". Lehrer (1997, 6) gives us the "acceptance argument":

(1) I accept that P with the aim of truth

(2) Therefore I am trustworthy in accepting that P

(3) Therefore I am reasonable in accepting P with the aim of truth

The argument raises at least two questions: (a) how is my trustworthiness in accepting that P supposed to flow from my accepting that P? (b) What is the nature of the acceptance which gives rise rises to trustworthiness? Let us start with the second question. As we saw, Lehrer characterises acceptance in epistemic terms. It is a functional state which is directed towards truth, and leading to inference and reasoning. But although it is theoretical, Lehrer describes it as an act of reasoned choice which is "the freedom to be guided by my acceptances" (Lehrer 2000, 219), hence as a kind of act and as a practical attitude which is much similar to what I have called above pragmatic acceptance, although it is governed by epistemic reasons (it aims at truth and knowledge). Indeed the fact that trust is the result of a reasoned choice orientated towards reason and truth does not make it a pragmatic acceptance in the prudential sense, for it is a pragmatic acceptance in the epistemic or theoretical sense, according to the above distinctions. But it involves a choice of the agent, and is in this sense pragmatic nevertheless. Trust too, as it figures in premise (2) of the acceptance argument, is a practical attitude. What is characteristic of the attitude of trust is that it is not, as such, evidentially constrained. But trust, like acceptance, can be more or less, rational. The problem is: can it be rational in a sense which would give us the appropriate bootstrapping in the loop of reason? There is bootstrapping in that to accept that P is to take oneself to be trustworthy in one's belief that P. Lehrer's idea is to use acceptance as a minimal epistemic attitude which can serve as a lever to obtain knowledge, but without taking it to involve knowledge, for otherwise the strategy would be circular (David 1991). But one may ask: if acceptance is supposed to be the minimal attitude on the way to knowledge, how does one know that one is reasonable? Whether or not reasonableness is supposed to do the trick in levelling knowledge, the sceptic is entitled to ask: how do we know that we are

reasonable? To answer this question, Lehrer (1990) adds the condition of trust. If I accept that P, I must accept that it is for me more reasonable to believe that P than to believe that not P, and I do that with the aim of believing the true and avoiding the false. But I must also accept that I am trustworthy about P. As Lehrer says :

I must accept that I am trustworthy as well: that when I accept something, that is a good enough reason for thinking it to be true, so that it is at least more reasonable for me to accept it than to accept its denial. (1990,122) The full acceptance argument contains two more steps:

(4) I am reasonable to trust my acceptance that P

(5) I am reasonable to accept P Here the trust condition ("I am worthy of my trust") is a meta-level condition which says that I am trustworthy concerning my acceptances. But, as Lehrer (1997,7) notes, this seems to be the argument from despair: unless I accept my own trustworthiness I cannot respond to the sceptical doubt. It indeed looks perfectly circular: my reasonableness presupposes my trust in my reasonableness, but my trust presupposes that I take myself to be reasonable. But, Lehrer tells us that although the sceptic will never be convinced of ( 2), (2) applies to the acceptance of itself. It is actually a "keystone loop" as he calls it (1997, 9) of my claim to know.

The loop or the circle is virtuous.

Although trust is here a personal matter the attitude of acceptance is motivated by self-trust. But both the sceptic and the dogmatic are here prone to ask: where does this trust come from? Is it the product of a pure act of faith? Could I accept something for reasons which would fall short of being epistemic or could I accept something for no reason at all, or "for nothing"? Clearly not, and for two reasons.

In Lehrer is, like a number of other philosophers who have invoked the notions of trust and of acceptance as the basis for reasonableness, 9 faced to a dilemma: either the notion of acceptance is a pragmatic one, and (unless one sides with pragmatism in holding that theoretical reasons and justifications are reducible to practical ones) it cannot be a basis for reason, or it is an epistemic attitude. If so, it must contain an evidentialist and a reliabilist component. Keith Lehrer admits that acceptance is based on a capacity to accept. (1997,56). What else can this capacity be, if not a disposition to believe that I am trustworthy? Hence my acceptance of my own trustworthiness must be based on my believing myself to be reliable in my acceptances. And this kind of belief cannot bootstrap itself into the status of a keystone of reason.

  epistemological and theoretical attitude. It plays a crucial role in what that Lehrer calls his principle of trustworthiness: (T) I am trustworthy in what I accept with the objective of accepting something just in case it is true

  the first place, I accept only what is worthy of acceptance. So truth is the aim, and it is implicit in Lehrer's argument that one values truth. Lehrer actually argues against instrumentalism (ibid, 56): truth is the ultimate goal and I am not worthy of my trust if I reach truth by pure luck. In the second place I do not ascribe to myself trustworthiness unless I can give reasons for it. As he says: "What makes me trustworthy in what I accept… is that I exercise a capacity to accept what is worth accepting. This requires a capacity to discern that the ends that I pursue in acceptance are worth pursuing." (ibid, 56) But this does not come by magic. The capacities are grounded somewhere. My reasons for being a worthy guide to the truth must have something to do with my reliability. 8

On the different kinds of assent in ancient Stoicism, see e.g.[START_REF] Frede | The Sceptic's two kinds of assent[END_REF],[START_REF] Barnes | The beliefs of a Pyrrhonist[END_REF] . There are indeed a lot of similarities between what the Stoics called sunkatathèsis and what Lehrer calls acceptance.

See Cohen (1989, 1992),[START_REF] Bratman | Practical reasoning and acceptance in a context[END_REF],[START_REF] Engel | Believing, accepting, and holding true[END_REF][START_REF] Engel | Dispositional belief, assent, and acceptance[END_REF][START_REF] Engel | Believing and accepting[END_REF],[START_REF] Frankish | Mind and supermind[END_REF].

Compare here with[START_REF] Frankish | Mind and supermind[END_REF], who also has a category of implicit acceptances.

This is the line taken byDennett (1978, 300-309) who distinguishes belief, a general dispositional state, from "opinion", which is the product of a verbal assent.

See Shah andVelleman (2005) andEngel (2008).

Acknowledgements

I would like to thank Guido Melchior and Martina Fuerst for their invitation at the Lehrer workshop in Graz in 2011 where a version of this article was read, and for their patient editorial work. I thank a referee of this article for his or her very helpful comments, hoping that I have taken them enough into account, and the participants for their reactions. Finally, I thank Keith Lehrer, for his generosity and for his teachings over the years.