Anne-Laure L Perrier

Jean-Marc M Duchamp

Olivier Exshaw

Member, IEEE Robert Harrison

Philippe Ferrari

A Compact Semi-Lumped Tunable Complex Impedance Transformer

Keywords: impedance transformer, microwave, miniaturized device, tunable device, varactors

This article describes the design and performance of a compact tunable impedance transformer. The structure is based on a transmission line loaded by varactor diodes. Using only two pairs of diodes, the circuit is very small with a total length of only λ/10. Both the frequency range and the load impedance can be tuned by varying the varactor bias voltages. Our design provides a tunable operating frequency range of ± 40% and an impedance match ranging from 20 Ω to 90 Ω at 0.8 GHz and from 30 Ω to 170 Ω at 1.5 GHz. In addition, a new approach that considers losses for the simulation and measurement of this impedance transformer was investigated. The measured performance of a 1 GHz prototype design confirmed the validity of this new approach.

I. INTRODUCTION

NE of the most formidable challenges in the field of microwave telecommunications is designing tunable devices. In the near future, more and more applications will require systems that can operate over variable frequency ranges. Tunable impedance transformers are important in applications such as transistor impedance matching over a large tunable bandwidth for maximum gain or minimum noise factor purposes, in designing tunable power dividers, in characterizing MMIC transistors, and in general matching networks.

Microwave impedance transformers and matching networks are based either on transmission line sections having specific characteristic impedances, or on L, T, or Π structures. The most usual section type is the quarter-wave impedance transformer, while L, T, or Π structures can be realized by using lumped elements like CLC (capacitor-inductorcapacitor) devices [START_REF] Sun | Design of Π impedance matching networks[END_REF], or by using single-or double-stub structures.

The overall length of transmission-line based structures can be reduced by loading a high impedance line with capacitors [START_REF] Hirota | Reduced-size branch-line and rat-race hybrids for uniplanar MMIC's[END_REF]. Such devices can also be made tunable if the fixed capacitors are replaced by tunable capacitors, such as varactor diodes or MEMS capacitors.

In the last decade, several authors have proposed the use of transmission lines loaded by switches in series or in parallel to realize tunable impedance transformers [START_REF] Bischof | Variable Impedance Tuner for MMIC's[END_REF][START_REF] Collins | A novel MMIC source impedance tuner for on-wafer microwave noise parameter measurements[END_REF][START_REF] Mcintosh | Novel MMIC source-impedance tuners for on-wafer microwave noiseparameter measurements[END_REF]. These first devices demonstrated the impedance transformer principle but they cover a small part of the Smith chart. Based on a CLC structure and a quarter-wave transformer, a resonant cell topology [START_REF] Sinsky | Design of an electronically tunable microwave impedance transformer[END_REF] has been used in several configurations [START_REF] Jung | Micromachined frequency-variable impedance tuners using resonant unit cells[END_REF][START_REF] Kim | Low-loss analog and digital micromachined impedance tuners at the Ka-band[END_REF][START_REF] De Mingo | An RF electronically controlled impedance tuning network design and its application to an antenna input impedance automatic matching system[END_REF]. In spite of a medium Smith chart coverage, the length of these devices is important because they are realized with a minimum of two quarter-wave transformers.

With the MEMS switches and MEMS varactors development, other designs based on single-stub [START_REF] Vaha-Heikkila | A 20-50 GHz RF MEMS single-stub impedance tuner[END_REF], doublestub [START_REF] Lange | A reconfigurable double-stub tuner using MEMS devices[END_REF][START_REF] Papapolymerou | Reconfigurable double-stub tuners using MEMS switches for intelligent RF front-ends[END_REF][START_REF] Zheng | A monolithic reconfigurable tuner with ohmic contact MEMS switches for efficiency optimization of X-band power amplifiers[END_REF] or triple-stub topologies [START_REF] Vaha-Heikkila | A reconfigurable 6-20 GHz RF MEMS impedance tuner[END_REF][START_REF] Watley | Reconfigurable RF impedance tuner for match control in broadband wireless devices[END_REF] have also been realized. Most of these devices need lots of MEMS switches or varactors complicating the bias commands. Moreover these impedance transformers require large surface to cover, in general, small or medium part of the Smith chart.

Lots of these impedance transformers are demonstrated to be used as tuners and some of them are used to realize antenna [START_REF] De Mingo | An RF electronically controlled impedance tuning network design and its application to an antenna input impedance automatic matching system[END_REF] or transistor [START_REF] Qiao | An intelligently controlled RF power amplifier with a reconfigurable MEMS-varactor tuner[END_REF] matching. Recently, new and improved tunable impedance transformers have been described. A very compact lumped-element CLC impedance transformer with a ±36 % tunable bandwidth for a 50 Ω load was demonstrated [START_REF] Jrad | Design of an ultra compact electronically tunable microwave impedance transformer[END_REF]. Some designs, based on transmission lines with variable characteristic impedance, were also presented [START_REF] Jeong | Tunable impedance transformer using a transmission line with variable characteristic impedance[END_REF][START_REF] Chun | Variable Zc transmission line and its application to a tunable impedance transformer[END_REF]. These devices are original but not lead a large coverage of the Smith chart. A double-slug impedance tuner, based on a distributed MEMS transmission line and employing 80 RF-MEMs switches, could produce 1,954 different complex impedances around the center of the Smith chart [START_REF] Shen | A reconfigurable RF MEMS based double slug impedance tuner[END_REF]. This device is also original but we note the important number of MEMS switches witch complicate the bias commands. A MEMS impedance tuner was realized at 25 GHz [START_REF] Lu | A novel MEMS impedance tuner simultaneously optimized for maximum impedance range and power handling[END_REF]. This design was used with MEMS varactors, and provided continuous impedance coverage. Compared to the other impedance transformers referenced in this paper, this MEMS impedance tuner is optimized in term of surface, variable element number and Smith chart coverage. Moreover, many of these designs were characterized only for a 50 Ω load, so that the insertion loss was known only for that particular case. In some instances, only S11 was measured, resulting in no information on insertion loss.

This article describes how a tunable and compact impedance transformer, using only two pairs of varactors, was designed. Two different methods for the optimization and experimental characterization of the circuit were developed. The first and the simplest method was based on the synthesized impedances representation with a 50 Ω load. In this case, the synthesized impedance was extracted from the S22 parameter. The insertion loss was not measured for complex loads.

The second method was much more complete: the return loss and the insertion loss of the transformer, loaded by a complex impedance, were calculated and measured using an external tuner. We emphasize that the first method is accurate only for lossless devices, as it cannot extract the characteristics of lossy circuits.

In a previous study [START_REF] Perrier | A varactor tunable complex impedance transformer[END_REF][START_REF] Pistono | Hybrid Tunable Microwave Devices Based On Schttky-Diode Varactors[END_REF], we demonstrated the ability of a new topology to design a tunable complex-impedance transformer, operating at 5 GHz. Its length was ~λ /3 and it needed only two varactors. Its key features were a large matching impedance range, from 5 Ω to 300 Ω, and a tunable frequency range of ± 15 %. However, this prototype exhibited significant insertion loss, from 2 to 6 dB, owing to the use of low-Q varactors, resulting in a nonoptimized design.

This article describes the design and performance of a new, more compact, and better optimized configuration than the one presented earlier [START_REF] Perrier | A varactor tunable complex impedance transformer[END_REF]. The design procedure leads to a small λ /10 long transformer with reduced insertion loss.

This article is organized into six sections. Section II details the principle of the impedance transformer. The transformations from 50 Ω to possible complex impedances are illustrated on Smith charts. Section III presents the development of two different simulation approaches: "synthesized impedance" method, and the "matching load" method. The two methods are compared in the case of lossy and lossless circuits. Section IV discusses the design of an impedance transformer for a proof of concept. This device is realized in a printed board hybrid technology. The simulated and measured results obtained by the two different methods are compared in Section V. Finally, Section VI contains concluding remarks. The equivalent electrical circuit of the complex-impedance transformer is shown in Fig. 1. Its total electrical length is θ, and it consists of three transmission-line sections of equal characteristic impedance Zc, and different electrical lengths θ1, θ2, and θ3. The transmission line is loaded by two tunable capacitors Cv1 and Cv2. Assuming that the electrical length θi (i = 1, 2, or 3) of each section is small as compared to the electrical wavelength (λ°=2π), a section can be replaced by its lumped-element equivalent circuit, as shown in Fig. 2 (b). The characteristic impedance Zc and phase velocity vϕ of the transmission line are defined thus:

II. PRINCIPLE

c L Z C = and 1 v LC ϕ = , (1)
where, L is the inductance and C the capacitance per unit length of the transmission line.

From the lumped equivalent circuit of a section (see Fig. 2 (b)), Zci and vϕι can also be expressed thus: ' '

i ci i vi L Z C C = + and '( ' ) i i i i vi l v L C C ϕ = + , (2)
where, '

i i L Ll = and ' i i C Cl =
. The quantities, Li' is the equivalent inductance and Ci' the equivalent capacitance of a section of physical length li :

2 eff i i r c l f θ π ε = , (3)
where, c is the light celerity and eff r ε the effective dielectric constant.

Fig. 3. Simplified equivalent electrical circuit of the complex impedance transformer.

Each section of the impedance transformer is equivalent to a transmission line with a tunable characteristic impedance Zci and phase velocity vϕi, (see Fig. 2 (c)). The maximum tunability of Zci and vϕi was obtained when Ci' << Cvi. This condition was satisfied when Zc was large, leading to the simplified equivalent electrical circuit of Fig. 2 (d). If Zc is large and if the total electrical length θ is small as compared to the wavelength, the equivalent circuit of the

' 2 50 jL ω ' 2 50 jL ω Cv1min Cv1max ZL2 Cv1min Cv1max YLC1 1 50 v j C ω Cv1min Cv2max Cv1max Cv2max Cv1min Cv2min Cv1max Cv2min YLC2 2 50 v j C ω ' 1 50 jL ω Input port 50 Ω ZL1 (e) Zout Cv1min Cv2max ' 3 50 jL ω Cv1max Cv2max Cv1max Cv2min Cv1min Cv2min Zout Zc, vϕ, θi <<λ° Cvi Zci , vϕi Li' Cvi Li' Cvi Ci' Cv1 50 Ω port ZL1 L1' L2' Cv2 L3' ZL2 Zout ZLC1 ZLC2 Impedance transformer Zc, θ1 Zc, θ 2 Zc, θ3 Cv1 Cv2 θ (a) (b) (c) (d) (a) (b) (c) (d)
complex impedance transformer can be simplified, as shown in Fig. 3. This simplified equivalent electrical circuit will be used to explain the principle of the complex impedance transformer. Each section has now been replaced by its equivalent inductance Li'. The quantities ZL1, ZLC1, ZL2, ZLC2 and Zout are the output impedances as seen at different transverse planes indicated by the dotted lines, when the input port is terminated by 50 Ω. The Smith charts of Fig. 4 show the principle of all the impedance transformations, starting from the 50-Ω input port in (a) to the output impedance Zout in (e). These transformations assume a fixed frequency and no losses. In the impedance chart of Fig. 4 (a), one can see that the inductance L1' transforms the 50 Ω input impedance to ZL1 at the output end of inductor L1'. In the admittance chart of Fig. 4 (b), it can be seen that the variable capacitive admittance ωCv1 transforms ZL1 to a range of admittances YLC1.

The

III. OPTIMIZATION METHOD

The Smith charts given in Section II show that the Zout impedance area depends on the three inductances L1', L2', and L3', the minimum and maximum values Cmin and Cmax of the two capacitors, and the operating frequency f. All these parameters need to be optimized according to the tuner application. In this section we develop and compare two different methods for optimizing an impedance transformer. The first method is based on the display of the Zout area (see Fig. 4 (e)) using the S22 parameter for a 50 Ω load. Examples of synthesized impedances obtained by this method are shown in Subsection A below. The second method calculates the S11 and S21 parameters when the impedance transformer is loaded with complex impedances. The principle of this "matching load" method is given in Subsection B. In Subsection C we compare the two methods with a typical example. The comparison is made both by including and omitting the losses to determine their effect on the second optimization method. In practice, not all the elements constituting the impedance transformer are ideal. In the optimization process, the complete equivalent electrical circuit of a commercial varactor diode (Fig. 5 A. First Approach: the "Synthesized Impedance" Method. The first approach, the "synthesized impedance" method, is based on the display of the S22 parameter when the impedance transformer is loaded by 50 Ω. Many state of the art tuners are just characterized this way, and so insertion loss, versus the load, is not known. Fig. 6 shows the synthesized impedances of the impedance transformer at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz. The thick lines correspond to a fixed minimum or maximum value for the capacitance of one varactor and a complete variation of the other. In some cases, this area is not sufficient to show all possible synthesized impedances. With intermediate values of the two capacitors, all the shaded area shown in Fig. 7 can be covered by this impedance transformer. The synthesized impedance area in Fig. 7 (b) is larger than the area in Fig. 6 (b). In this section, simulation results are shown as in Fig. 7. In Section V, measured results are presented as in Fig. 6.

B. Second Approach: the "Matching Load" Method. For the second method, the impedance transformer was loaded by complex impedance. The setup shown in Fig. 8 corresponds to a typical real working configuration of the

(a) (b) (c) (a) (b) (c) Rs C (V) Ls D Cc V V Output Complex load Input 50Ω Impedance transformer S11 S21
transformer, for example when it is used as a matching network for a transistor. Here, the input return loss S11 and insertion loss S21 of the impedance transformer were investigated. In the following discussion, we refer to this approach as the "matching load" method. The simulations were done by a Mathematica [26] program, developed to automatically calculate the Smith chart coverage for the entire range of varactor capacitances. A flowchart of the program is given in Fig. 9. Fig. 10 shows all the complex loads that were tested by the program. For each load, the cascade ABCD and S matrices of the impedance transformer were calculated, with two criteria

= + + + , (7)
where Re(Zin) and Re(Zout) are the real parts of the input and output impedances, respectively. Fig. 10. The complex loads generated by the program.

Comment: The simplified equivalent circuit of Fig. 3 is used in order to understand and easily visualiz all the impedance transformations on the Smith chart. However, in the simulation process, for the "synthesized impedance" and "matching load" methods, we have compared equivalent electrical circuit of Fig. 1 and Fig. 3 by using ABCD transmission line matrix and ABCD inductance matrix respectively. We have demonstrated that results are identical for small length impedance transformers and similar for longer devices. In this paper all the simulations shown in the figures have carried out with the equivalent electrical circuit of Fig. 1 i.e. with the used of real transmission lines.

C. Comparison Between the Two Methods.

In this subsection, simulation results of the impedance transformer obtained by the two different methods, and described at the beginning of Section III are compared. Results are shown for the 1 GHz center frequency. In Subsubsections (1), (2), and (3), the two methods are compared with the varactor Rs as a parameter. In Sub-subsection (4), we demonstrate that the results obtained when an impedance transformer was optimized using the "synthesized impedance" method, without considering losses, can be quite different from those obtained when the varactor losses were included.

1) Lossless Varactors

In lossless microwave devices, the S-parameter moduli are related by

Fig. 11. Simulated coverage areas according to the "conjugate synthesized impedances (S22 * )" approach (▬) and the "matching loads" approach ( 0.04 S = -dB, Fig. 11 shows the "conjugate synthesized impedances" and the "matching loads" obtained from the two different methods. In this lossless case, the results of the two methods are perfectly superposed.

2) Lossy Varactors

In this sub-subsection, the varactor series resistance Rs was considered. Fig. 12 shows the results obtained from the two different methods when 11 max 20 S = -dB, for several 21 max S values. Because of Rs, the "matching load" method gives no possible loads when 21 min 0.1 S = -dB; for 21 2 S < -dB, the covered areas remain small. In this case, a perfect superposition of the results obtained by the two methods is never obtained. These results show that for lossy varactors, the "conjugate synthesized impedance" method, which is much simpler for experimental characterization, fails to give the correct covered area. Fig. 12 proves the importance of the "matching load" method to know the insertion loss of the device versus this complex load. So, a 50 Ω measurement is not sufficient to characterize the tuner insertion loss.

For further simulations, the criterion 2

3) Influence of the Varactor Series Resistance Rs

In this sub-subsection the complex impedance coverage was investigated by varying the series resistance Rs of the varactor as a parameter. Fig. 13 compares the results obtained from the "synthesized impedance" and "matching load" methods, with The difference of area between the two methods increase with Rs so for a fixed complex load the insertion loss increase with Rs.

Rs =0 Ω Rs =0.5 Ω Rs =1 Ω Rs =1.5 Ω Rs =2 Ω Rs =2.5 Ω
Fig. 13. Simulated 1 GHz coverage areas according to the "conjugate synthesized impedance" (S22 * ) method (▬) and the "matching load" method (•••). Here the varactor series resistance Rs is varied.

4) Application Example

To investigate the impact of choosing the design method, an impedance transformer was designed to cover the largest possible area, while using the "synthesized impedance" method. The same MA4ST-1240 M/A-COM™ varactor (Rs=1.6 Ω) was used and the characteristic impedance of the line was also fixed to 200 Ω. The maximum "conjugate synthesized impedance" area was obtained when θ1=40°, θ2=15°, and θ3=8°. The same transformer was then simulated using the "matching load" method, with the criteria 2 S > -dB were found at 1 GHz (see Fig. 14 (a)), but no loads were found at 1.5 GHz (see Fig. 14 (b)). However, a large area was obtained with the "conjugate synthesized impedance" method. These typical results bring to the fore the importance of the design method. For lossy varactors the "synthesized impedance" method, which is used by several researchers in the field, can be very inaccurate because insertion loss information can not be obtained.

The same impedance transformer (θ1=40°, θ2=15°, θ3=8° and Zc=200 Ω) was simulated by the two methods assuming Rs = 0.5 Ω (see Fig. 15). The covered areas of the "matching load" are very different compared to the results with Rs=1.6 Ω (Fig. 14).

To bring to the fore the importance of the "matching load" method, we can compare Fig. 15 (a) to Fig. 13 (Rs=0.5 Ω) and Fig. 14 (a) to Fig. 13 (Rs=1.5 Ω). These results obtained with the same varactors give totally different covered area. So an important difference between the two different methods of simulation is clearly pointed out. We note that insertion loss is more critical for this topology (θ1=40°, θ2=15°, θ3=8°) than for the first topology (θ1=15°, θ2=15° and θ3=8°).

It is obvious that in the case of lossy varactors, i.e. the reality in most cases, the "matching load" method has to be used in order to calculate and optimize by simulation the different parameters as the electrical length and the characteristic impedance of each transmission line, in order to achieve a maximum covered area in the Smith chart. To demonstrate the principles developed in Sections II and III, a tunable impedance transformer proof of concept was designed for a 1-GHz working frequency. The circuit was optimized with a Mathematica program using the "matching load" method. Commercial varactors (M/A-COM™ type MA4ST-1240) were used. Their Ls = 1.8 nH, Rs = 1.6 Ω, and Cc = 0.11 pF. The C(V) range, extracted from experimental results, was 1.0 -8.6 pF for a bias voltage V range from 12 to 0 V. Coplanar waveguide (CPW) was used, the prototype being fabricated on a Rogers™ RO4003 substrate (εr = 3.36, tan(δ) = 0.0035, dielectric thickness 0.813 mm, and copper thickness 35 µm). The transmission line characteristic impedance was set at 200 Ω, leading to a CPW central conductor width of 250 µm and a gap of 2.8 mm. Two varactors were used in parallel to realize the tunable capacitors. This is necessary for CPW symmetry and to lower effective series resistance.

IV. PROTOTYPE DESIGN

The fabricated proof of concept is shown in Fig. 16. By providing an air gap in the ground plane, separate reverse biases V1 and V2 can be applied to the two pairs of diodes. Surface mounted capacitors were used to ensure ground continuity for the RF signal.

The overall electrical length was 38° (θ1=15°, θ2=15°, θ3=8°), corresponding to ~λ/10. The effective εr was 1.75, leading to l1 = 9.4 mm, l2 = 9.4 mm, and l3 =5 mm.

V. RESULTS

In this section, the simulated and measured results obtained by the two methods are compared.

In Subsection A, the tunable frequency range of the impedance transformer loaded by 50 Ω is shown. In Subsections B and C, the simulated and measured results obtained from the "synthesized impedance" and "matching load" methods are compared.

For the simulations, lossless transmission lines were assumed but all parasitic elements of the diodes were considered. Measurements were made using a Wiltron 360 vector network analyzer (VNA).

A. Tunable Frequency Range for a 50-Ω Load.

An initial measurement using a 50-Ω load was made to extract the tunable bandwidth and to confirm the varactor's equivalent electrical model. Fig. 17 shows the frequency tunability of the complex impedance transformer. Simulated and measured results for the parameters S11 and S21, obtained for the extreme tunable frequencies, are shown. These correspond to the extreme capacitances of the variable capacitors. These results show that the transformer can be continuously tuned from 0.6 to 1.6 GHz, that is, ± 60% around 1 GHz, with 2 S > -dB. Good agreement between simulated and measured results was obtained for the whole tunable frequency range. In the experimental setup shown in Fig. 18, the impedance transformer was inserted between the two ports of the VNA, and a coaxial SOLT (short-open-load-through) calibration procedure was applied between the calibration plans P1 and P2. The S22 parameter was measured at the impedance transformer output. The phase shift φ due to the SMA connector used in the prototype of Fig. 16 is given by 15.55

B. "Synthesized Impedance" Method

f ϕ = - , ( 9
)
where f is the frequency in GHz. This phase shift is taken into account in measurement results.

Fig. 19 compares the simulated and measured results from 0.5 to 2 GHz. The conditions are the same as in Fig. 6, with one varactor bias being fixed at its minimum or maximum value and the other being varied over its full range. The tunable S22 area changes with the working frequency, and maximum coverage was obtained between 1.0 and 1.2 GHz. A good agreement between simulations and measurements was obtained for all frequencies.

The area of impedances covered by our device is comparable to best results obtained in the literature [START_REF] Vaha-Heikkila | A 20-50 GHz RF MEMS single-stub impedance tuner[END_REF][START_REF] Lu | A novel MEMS impedance tuner simultaneously optimized for maximum impedance range and power handling[END_REF] measured in this way, with a 50 Ω load. Hoewer, our device is much more simpler.

C. "Matching Load" Method

With the two criteria 11 20 S < -dB and 21 2 S > -dB, Fig. 20 shows the simulation results obtained when the operating frequency was varied from 0.5 to 2.0 GHz.

Measurements were carried out using an experimental approach similar to that used for the simulations. A mechanical tuner [START_REF]C model[END_REF] was used as a complex load for the tunable impedance transformer under test. The measurement steps for the calibration are detailed in Fig. 21. First, a coaxial SOLT calibration was used to define reference plans P1 and P2 (see Fig. 21 (a)). Then the input impedance of the mechanical tuner was measured, as shown in Fig. 21 (b). Finally, the 11 S and 21 S parameters of the tunable transformer, loaded by the mechanical tuner, were measured (see Fig. 21 (c)). We assume that the insertion loss of the mechanical tuner was negligible for the extraction of the insertion loss 21 S of the impedance transformer. The phase shifts of the SMA connectors are taken into account in measurement results. As this measurement procedure is much more timeconsuming than the "synthesized impedance" method, fewer measured load points were obtained, resulting in a "matching area" that is not so well defined as the simulated area. The agreement between the measured results of Fig. 22 (a,b,c) and the simulated points in Fig. 22 (b,c,d) is good. The measurements show a large "matching area" that is slightly smaller than the simulated area. Conclusions and prospects: We believe that the totality of the results presented here validates our approach to both the design and the measurement methods.

VI. CONCLUSIONS

A principle for designing a compact tunable impedance transformer, based on a single transmission line loaded by only two pairs of varactors, has been proposed. The length of the transformer is only λ/10. A prototype with a 1-GHz center working frequency has been realized using commercial varactor diodes.

A good agreement has been obtained between the simulations and measurements and, as expected, the network provided a large coverage of the Smith chart (real part from 20 Ω to 90 Ω at 0.8 GHz and from 30 Ω to 170 Ω at 1.5 GHz), with a range of tunable working frequencies over ±40 %.

Two different approaches to the design and measurements have been investigated. It is shown that an external tuner is necessary for accurate determination of the Smith chart coverage.

A MMIC prototype, in a 0.35 µm BiCMOS technology, is under development. It is believed that such an impedance transformer can be a good candidate for tunable matching of an amplifier embedded in a reconfigurable front-end. VII. APPENDIX Fig. 24 shows a simplified RC equivalent circuit of a single varactor diode inserted between a source impedance Zin and an output load impedance Zout.

We denote Zsin the input impedance as seen from Zin. In admittance form this is VNA (reference impedance: This corresponds to the impedance determined from the measurement of S11 and S21, as shown in Section III, Part B.

Let us now calculate the output admittance Ysout as seen from Zout, as shown in Fig. 25. This is the impedance that is extracted from the measurement of S22, as in Section III, Part A: At this point, it becomes obvious that Yout and Ysout* are different. This is because Rs is not equal to zero. These equations explain why the two measurement approaches investigated in Section III, Parts A and B, do not lead to the same results when the series resistance of the varactors is considered.

Fig. 1 .

 1 Fig. 1. The equivalent electrical circuit of the impedance transformer.

Fig. 2 .

 2 Fig.2. Equivalent circuits of one section of the impedance transformer.

Fig. 4 .

 4 Fig. 4. Principle of impedance transformations. The impedances (a) ZL1, (b) ZLC1, (c) ZL2, (d) ZLC2, and (e) Zout are synthesized at the transverse planes of the impedance transformer, as shown in Fig. 3.

Fig. 5 .

 5 Fig. 5. Equivalent circuit of the reverse-biased varactor diode.

  ) was considered. M/A-COM™ varactors (MA4ST-1240) with a series inductance 1case capacitance Cc = 0.11 pF, and a tunable capacitance C(V) ranging from 1.5 to 8.6 pF were used. A single varactor was used to realize each tunable capacitor in Fig.1. To compare the two approaches, we specified that the impedance transformer should have the parameters ZC = 200 Ω, θ1 = 15°, θ2 = 15° and θ3 = 8°. Ideal transmission lines were assumed for the simulations.

Fig. 6 .

 6 Fig. 6. Synthesized impedances at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz.

Fig. 7 .

 7 Fig. 7. Total area of the synthesized impedances at (a) 0.5 GHz, (b) 1 GHz, and (c) 1.5 GHz.

Fig. 8 .

 8 Fig. 8. Experimental setup for the measurement of S11 and S21.

Fig. 9 .

 9 Fig. 9. Flowchart for the Mathematica simulation program.

Fig. 12 .

 12 Fig.[START_REF] Papapolymerou | Reconfigurable double-stub tuners using MEMS switches for intelligent RF front-ends[END_REF]. Simulated coverage areas at 1 GHz according to the "conjugate synthesized impedance" (S22 * ) method (▬) and the "matching load" method (•••), when Rs = 1.6 Ω. Results are plotted for |S21|min values from -0.1 dB to -20 dB.

  . For each Smith chart, the areas obtained by the two methods decrease as Rs increases.

  Fig.14. A few loads allowing[START_REF] Lange | A reconfigurable double-stub tuner using MEMS devices[END_REF] 20 S < -dB and[START_REF] Lu | A novel MEMS impedance tuner simultaneously optimized for maximum impedance range and power handling[END_REF] 2 S > -dB were found at 1 GHz (see Fig.14 (a)), but no loads were found at 1.5 GHz (see Fig.14 (b)). However, a large area was obtained with the "conjugate synthesized impedance" method.

Fig. 14 .

 14 Fig.[START_REF] Vaha-Heikkila | A reconfigurable 6-20 GHz RF MEMS impedance tuner[END_REF]. Simulated coverage areas according to the "conjugate synthesized impedance" (S22 * ) method (▬) and the "matching load" method (•••), at (a) 1 GHz, and (b) 1.5 GHz for a 40°-15°-8° impedance transformer, with Rs = 1.6 Ω.

Fig. 15 .

 15 Fig.[START_REF] Watley | Reconfigurable RF impedance tuner for match control in broadband wireless devices[END_REF]. Simulated coverage areas according to the "conjugate synthesized impedances" (S22 * ) method (▬) and the "matching load" method (•••), at (a)

Fig. 16 .

 16 Fig. 16. Photograph of the impedance transformer.

Fig. 17 .

 17 Fig. 17. Simulated ( --) and measured ( -) frequency tuning ranges for a fixed 50-Ω load.

  Fig.17shows the frequency tunability of the complex impedance transformer. Simulated and measured results for the parameters S11 and S21, obtained for the extreme tunable frequencies, are shown. These correspond to the extreme capacitances of the variable capacitors. These results show that the transformer can be continuously tuned from 0.6 to 1.6 GHz, that is, ± 60% around 1 GHz, with[START_REF] Lange | A reconfigurable double-stub tuner using MEMS devices[END_REF] 20 S < -dB and

Fig. 18 .

 18 Fig. 18. Experimental setup for the measurement of S22.

Fig. 19 .

 19 Fig. 19. Simulated S22 values (▬) compared with measured results (•••).

Fig. 20 .

 20 Fig. 20. Simulation results for the "matching loads" method with the criteria 11 20 S < -dB and 21 2 S > -dB, versus frequency.The measured results are shown in Fig.22for three different frequencies: 0.8, 1.0, and 1.5 GHz. The lowest frequency was 0.8 GHz owing to the limited mechanical tuner bandwidth. The Smith charts show all the points for which the criteria[START_REF] Lange | A reconfigurable double-stub tuner using MEMS devices[END_REF] 20 S < -dB and 21 2 S > -dB were satisfied. Each of

Fig. 21 .

 21 Fig. 21. Principle of the setups for the measurement of S11 and S21: (a) calibration planes (b) load impedance measurement (c) measurement of S parameters.

Fig. 22 .

 22 Fig. 22. Measured matched complex loads, under the conditions 11 20 S < -dB and 21 2 S > -dB, at three frequencies: (a) 0.8 GHz (b) 1.0 GHz and (c)

Fig. 23 .

 23 Fig. 23. Measured results for three different complex loads at 1 GHz, with the criteria 11 20 S < -dB and 21 2 S > -dB.

Fig. 23

 23 Fig. 23 shows typical measured results obtained at 1 GHz for three different complex loads.Conclusions and prospects: We believe that the totality of the results presented here validates our approach to both the design and the measurement methods.

Fig. 24 .

 24 Fig. 24. The input impedance Zsin seen from Zin.

Fig. 25 .

 25 Fig. 25. The output impedance Zsout seen from Zout.

Table 1 :

 1 Impedance transformers topology and performance.

	Impedance transformer type	Variable capacitor	Technology	Dimensions	Smith chart coverage
		+ operating frequency
		[3] 27 GHz	4 MIM capacitors + HEMT transistors	Monolithic GaAs	1*1 mm² λ/5*λ/5	Very small Some complex impedances
		[5] 18 GHz	8 pHEMT switches	Monolithic GaAs	2.4*0.8 mm² λ/3*λ/9	Small ≈ 50 complex impedances
	Transmission line	[22] 20 to 50 GHz [20] 12 to 25 GHz [21] 27 to 30 GHz [23]	8 MEMS switches 80 MEMS switches 4 MEMS varactors + 1 MIM capacitor FET switches	Monolithic Glass εr=4,6 Monolithic Quartz εr=3,8 Monolithic Si	2.5*1 mm² λ/2*λ/5 10 mm λ 0.49*0.12 mm² λ/8*λ/30	Medium 256 complex impedances Medium 1954 complex impedances Good Continuous Good
		0.05 to 2 GHz				36 complex impedances
		[19] 0.9 GHz	2 varactors	Hybrid Rogers εr=3	30*30 mm² λ/8*λ/8	Small Continuous
		[18] 2.14 GHz	p-i-n diode	Hybrid RF35 εr=3,5	λ/4	Very small 50, 89 or 200 Ω
	Resonant cell	1 resonant cell [6] 2.4 GHz 2 resonant cells [7] 23 to 25 GHz 1 resonant cell [9]	3 varactors MEMS varactors 12 diodes p-i-n	Hybrid Rogers TMM10i Monolithic Quartz Hybrid	λ/2 3.7*2 mm² λ/2*λ/4	Continuous from 4 to 392 Ω No complex impedances Medium Continuous Medium
		0.39 GHz				4096 complex impedances
		Double-stub [8] 29 to 32 GHz	12 MEMS switches	Monolithic Quartz	3*2 mm² λ/2*λ/3	Small 49 complex impedances
		Double-stub [11]	8 MEMS switches	Mixed		Small
		10 to 20 GHz		Si and Alumina	≈ 3λ*2λ
		Double-stub [12] 10 to 20 GHz	8 MEMS switches	Mixed Si and Alumina	18*11 mm² 1.3λ * 0.8λ	Small to good 256 complex impedances
	Stub	Double-stub [13] 10 GHz Double-stub [16]	6 MEMS switches 4 MEMS switches + 1	Monolithic Si εr=11.7 Monolithic	6*7 mm² λ/2*λ/2	Small 64 complex impedances Small
		8 to 12 GHz	varactor			(transistor matching)
		Triple-stub [14] 6 to 20 GHz	11 MEMS switches	Monolithic Glass εr=4,6	7.3*7.3 mm² λ/2*λ/2	Medium to very good
		Single-stub [10] 20 to 50 GHz	10 MEMS switches	Monolithic Glass εr=4,6	λ * 3λ/2 5.6*3.6 mm²	Good
		Triple-stub [15] 4.5 GHz	5 varactors	Hybrid Duroid εr=6,15	28 mm λ	No measurement of complex impedances

Table 1

 1 resumes the impedance transformers topology and performance. Lot of these examples required more than three varactors, complicating the circuit with numerous bias voltages. Those based on quarter-wave transmission lines result in physically long structures with narrow bandwidths.

  YLC1 output admittance values are part of the circle defined

by ωCv1min and ωCv1max where Cv1min and Cv1max are the minimum and maximum capacitance values. A second L'Cv section is necessary to transform this circular arc into a surface, as shown in the impedance and admittance charts (Figs.

4 (c) and (d)

). The final inductance L3' allows the achievable Zout area on the Smith chart to be rotated. This area corresponds to a value of S22 when the output is terminated by 50 Ω.

  Cv1 and Cv2 that satisfy the two criteria were extracted. Then the "matching load" area was plotted on the Smith chart. Equations for calculating S parameters in the case of a complex load are[START_REF] Frickey | Conversions between S, Z, Y, h, ABCD and T parameters which are valid for complex source and load impedance[END_REF]:

								S	22	* out out AZ AZ -	* in out in out B CZ Z B CZ Z + -	in in DZ DZ +
	11 max S	and 21 min S	applied to the 11 S and 21 S parameters.
	The two capacitor values, * 11 out in out out in out AZ B CZ Z S AZ B CZ Z + --= + + +	DZ DZ	* in in	(4)
	12 S	=	2(	)(Re( ) Re( in in out Z B CZ Z AD BC out AZ -+ + +	out DZ Z	1/ 2 )) in	(5)
	S	21	=	2(Re( ) Re( in out in out out Z Z AZ B CZ Z + +	1/ 2 )) DZ +	in	(6)

His main research interest is the design and realization of tunable and miniaturized RF and millimeter wave devices, transmission lines, filters, phase shifters, power dividers, tuners, in PCB and RFIC technologies.

He is also involved in the development of time-domain techniques for the measurement of passive microwave devices and for soil moisture content.


Abderrahmane Barkat
email: abderrahmane.barkat@l2s.centralesupelec.fr

Bogdan Marinescu
email: bogdan.marinescu@ec-nantes.fr

Cédric Join
email: cedric.join@univ-lorraine.fr

Cédric JOIN † ¶ Michel Fliess
email: michel.fliess@polytechnique.edu

Model-free control for VSC-based HVDC systems

Keywords: High Voltage Direct Current (HVDC), Voltage Source Converter (VSC), model-free control, intelligent controllers

come

I. INTRODUCTION

Modern power grids contain more and more complex dynamic elements, most of the times based on power electronics. This is the case of the HVDC links. They are also active elements which have an impact on the neighborhood of the AC network in which they are inserted. Particularly, HVDCs controls may affect the transient stability of the near zone (see, e.g., [START_REF] Sigrist | A fundamental study on the impact of HVDC lines on transient stability of power systems[END_REF]). This impact was optimized in [START_REF] Arioua | Robust grid-oriented control of high voltage DC links embedded in an AC transmission system[END_REF] and [START_REF] Arioua | Multivariable control with grid objectives of an HVDC link embedded in a large-scale AC grid[END_REF] by using an extended control model. However, as this model captures several grid dynamics in addition to the ones of the HVDC link, its order may be high. This situation is encountered in many other power systems applications. For example, analysis and damping of inter-area modes of a grid needs a model of the whole synchronous zone which results in high-order mathematical objects. Another typical example is the so-called secondary regulations (voltage and frequency) for which a global vision of the system is also necessary.

In order to avoid these difficulties related to the size of the control model, a recently developed methodology called "model-free control" [START_REF] Fliess | Model-free control[END_REF], [START_REF] Fliess | Deux améliorations concurrentes des PID[END_REF] is used here. Indeed, this approach does not need a large-scale precise mathematical model since the system is represented by an ultra-local loworder (first or second order) differential equation. This model is input-output data-driven, i.e., continuously updated. It allows the synthesis of a simple controller of PID type, called intelligent-PID, or iPID. This control strategy has already been successfully applied to quite different electric systems in [START_REF] Join | Vers une commande sans modèle pour aménagements hydroélectriques en cascade[END_REF], [START_REF] Michel | Model-free control of dc/dc converters, 12th IEEE Worksh. Contr. Model. Power Electron[END_REF], [START_REF] Bara | Modelfree load control for high penetration of solar photovoltaic generation[END_REF]. Reference [START_REF] Bara | Toward a model-free feedback control synthesis for treating acute inflammation[END_REF] exhibits a rather complete bibliography of the numerous concrete accomplishments of model-free control in the world. Some of them have been patented. As a matter of fact, model-free control yields important advantages for regulating complex systems including HVDCs such as: great conceptual simplicity, simplified parameterization, fast implementation and great robustness with respect to various disturbances.

In comparison to these previous applications, the case of the HVDC link treated here presents much faster dynamics dues to power electronic converters. The method is thus first tuned for this, especially for the estimation of the derivatives of measured signals. Next, it is tested in comparison with the classic vector control on a detailed VSC-HVDC system connected both sides to infinite buses modeled in Matlab/Simulink (Fig. 1).

The paper is structured as follows. The control problem is formulated in Section II. The basic principles of the modelfree control are briefly recalled in Section III. Implementation and validation scenarios and tests are discussed in Section IV. Robustness is analysed in Section V while Section VI is devoted to concluding remarks.

II. SYSTEM DESCRIPTION

The HVDC transmission link considered in our study in Fig. 1 is the detailed HVDC model provided as example (demonstration case) in SimPower Toolbox [START_REF]VSC-Based HVDC Transmission System (Detailed Model) SimPowerSystems Toolbox, Simulink, Matlab-MathWorks[END_REF]. It consists of a 200 MVA (+/-kV DC) forced commutated VSC interconnection used to transmit power from a 230 kV, 2000 MVA, 50 Hz equivalent AC system to another identical one. These equivalent AC grids are modeled in a classic way by infinite buses (E = 1 and f = 50Hz) and equivalent impedances (L gi , r gi , i ∈ {1, 2}) on each side of the HVDC. A circuit breaker is used to apply a three-phase to ground fault on the inverter AC side. The discrete control system generates the three sinusoidal modulating signals that are the reference value of the bridge phase voltages. The power system and the control system are both discretized for a sample time T s power = 7.40 10 -6 s and T s control = 74.06 10 -6 s respectively (see [START_REF]VSC-Based HVDC Transmission System (Detailed Model) SimPowerSystems Toolbox, Simulink, Matlab-MathWorks[END_REF] for more details on the model).

A. Control structure

The vector control is usually structured into two hierachic inner and outer loops which are also decoupled in time. The outer loops are slower and provide references to be tracked by the inner loops. This structure is used here for each converter as shown in Fig. 2. Convertor S1 controls the transmitted active power P and the reactive power injection Q 1 at the left coupling point PCC1 in Fig. 1 while converter S2 controls the DC voltage V DC2 and reactive power injection Q 2 at the right coupling point PCC2.

B. Reactive Power Control

The Reactive Power Control regulator block in Fig. 2 combines an integral (I) control with a feed-forward control to increase the speed response. To avoid integrator wind-up the following actions are taken: the error is reset to zero, when the measured PCC voltage is less than a constant value (i.e., following a disturbance on the AC); when the regulator output is limited, the limitation error is fed back with the right sign to the integrator input.

C. Active Power Control

The Active Power Control block in Fig. 2 is similar to the Reactive Power Control block. The extra ramping block ramps the power order towards the desired value with an adjusted rate when the control is active. The ramped value is reset to zero when the converter is blocked. [START_REF] Fliess | Model-free control[END_REF] For simplicity's sake let us restrict ourselves to Single-Input Single-Output systems. Instead of trying to write down a complex differential equation, introduce the ultra-local model

III. MODEL-FREE CONTROL

ẏ = F + αu, (1)
where

• u and y are, respectively, the input (control) and output variables, • the derivation order of y is 1, like in most concrete situations, • α ∈ R is chosen by the designer such that αu and ẏ are of the same magnitude. The following explanations on F might be useful:

• F subsumes the knowledge of any model uncertainties and disturbances, • F is estimated via the measures of u and y.

A. Intelligent controllers

The loop is closed by an intelligent proportional controller, or iP,

u = - F -ẏ * + K P e α , (2)
where

• y is the reference trajectory,

• e = y -y is the tracking error,

• K P is the usual tuning gain. Combining equations ( 1) and (2) yields ė + K P e = 0, where F does not appear anymore. Local exponential stability is ensured if K P > 0:

• The gain K P is thus easily tuned.

• Robustness with respect to different types of disturbances and model uncertainties is achieved. Remark 3.1: It is clear [START_REF] Fliess | Model-free control[END_REF] that the gain tuning become much simpler than for classic PIs and PIDs ( [START_REF] Åström | Feedback Systems: An Introduction for Scientists and Engineers[END_REF], [START_REF] O'dwyer | Handbook of PI and PID Controller Tuning Rules[END_REF]).

B. Estimation of F

For implementation purpose, F is defined by

F = ẏ -αu(t -h)
where u(t -h) is the control delayed by a small non-zero amount h: it is the short time window needed for computations.

For the estimation of ẏ algebraic numerical differentiation techniques ( [START_REF] Fliess | Non-linear estimation is easy[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]) are employed. These derivatives are replaced by integrals thanks to operational calculus [START_REF] Erdélyi | Operational Calculus and Generalized Functions[END_REF]. Consider a signal x(t) and its Taylor expansion at t = 0

x(t) = +∞ i=1 x (i) (0) t i i! .
A first-order truncation gives:

x(t) = a 0 + a 1 t.
In order to estimate a 1 , i.e., the first-order derivative, write it in the operational domain

X(s) = a 0 s + a 1 s 2 .
In order to get rid of a 0 , multiply both sides by s and take the derivatives with respect to s:

-s -2 a 1 = s dX(s) ds + X(s).
Attenuate the corrupting noise by multiplying both sides by a negative power of s.

1 -s -3 a 1 = s -1 dX(s) ds + s -2 X(s).
Remember [START_REF] Erdélyi | Operational Calculus and Generalized Functions[END_REF] that d ds corresponds in the time domain to the multiplication by -t. It yields the following estimator ã1 of a 1 :

ã1 = 3! T 3 T 0 (T -2τ )y(τ )dτ,
where the length T of the time window may be quite small. Remark 3.2: Estimation of F may be directly derived [START_REF] Fliess | Model-free control[END_REF] without the need of estimating the derivative of y in Eq. (1).

IV. CONTROL IMPLEMENTATION AND VALIDATION

The model-free control is implemented and tested in two stages:

1) Only the active power control loop is implemented at converter S1 with an iP in the hierarchical control structure presented in Fig. 2. The other control loops (for Q 1 , Q 2 and V DC2 ) are not changed and thus maintained in the vector control approach as in the origin HVDC example of [START_REF]VSC-Based HVDC Transmission System (Detailed Model) SimPowerSystems Toolbox, Simulink, Matlab-MathWorks[END_REF].

2) The whole control of converter S1 is synthesized with a multivariable (for P and Q 1 control) model-free control. Remark 4.1: The multivariable extension of Section III is straightforward (see [START_REF] Lafont | A model-free control strategy for an experimental greenhouse with an application to fault accommodation[END_REF], [START_REF] Menhour | An efficient model-free setting for longitudinal and lateral vehicle control. Validation through the interconnected pro-SiVIC/RTMaps prototyping platform[END_REF], and [START_REF] Fliess | Deux améliorations concurrentes des PID[END_REF]).

A. Test scenarios

Basically the simulation runs from 0 to 4s. At t = 0s the reference of the active power is subjected to a slope ramp of 0.43. At t = 1.5s, a -0.1pu step is applied to the reference of active power (decrease from 1pu to 0.9pu). Steps are also applied to the reference of reactive power Q 2 (from 0 to -0.1pu) at t = 2.0s. To quantify interactions with the other controls, a step on the reference of the DC voltage of converter S2 (decrease from 1pu to 0.95pu) is also considered 1 Negative powers of s correspond to integrals which are low-pass filters. at t = 2.5s. Noise of 10 -4 pu is systematically added to measures.

The initial load-flow (steady-state) of the system corresponds to zero active power transit (P = 0), and zero reactive power injection (Q 1 = 0, Q 2 = 0).

B. Active power control

1) Classic controller: Fig. 3 contains results with the classic vector control presented in Fig. 2. As mentioned in Section II, the P loop consist in an intergral control. From the scenarios mentioned above, only the ramp starting at t = 0s and the step of amplitude -0.1 pu at t = 1.5 are applied to the reference of active power at this stage. 2) Intelligent controller: An iP is now synthesized for the active power control loop. Indeed, with this choice, the modelfree control acts like a pure integrator through the coefficient K P in equation ( 2). The latter is chosen arbitrarily in such a way as to ensure desirable performance. The second setting parameter (α), it is chosen to adapt the control to the same order of magnitude as the first derivative of the measurement. The resulting control is given by

u(k) = - F - . P * + 0.002e 5000
where k denotes the k-th sampling time and we select

F = . P -5000u(k -1)
for estimating the active power derivative . P . To implement the latter derivative, we used the algebraic numerical differentiation of Section III with a number of samples n = 50 and a sample time T e = 5.10 -3 s. It gives a window width of T = 0.25s. Figure 4 below represents the response of the resulting model-free control active power loop. Notice first that the tuning of parameters above provides responses at the same dynamics as the vector control loop (response to the step mentioned above in about 100ms). Performances are comparable for both controls. The noise level is also satisfactory in the model-free control loop.

C. Multivariable control of the active power and reactive power

We study here the control of the two powers (active and reactive) simultaneously as we did before; first with the classic vector controller, next using the iP controller.

1) Classic vector controller: In addition to the ramp and the step on the active power reference used in the previous paragraph, a step of -0.1pu is now also applied on the reference of reactive power of converter S1 (from 0 to -0.1pu) at t = 2.0s to see the response of this loop also as well as the interaction with the P loop (Fig. 5 and6). 1 and y 1 be the active power reference and active power measurement respectively and y * 2 and y 2 the reactive power reference and reactive power measurement respectively. u 1 and u 2 represent the active power control and reactive power control respectively. Model (1) is now written in the multivariable form

. P . Q = F 1 F 2 + α 1 0 0 α 2 u 1 u 2 . (3)
The decoupling property, i.e., the diagonal character of the matrix α in Eq. ( 3), simplifies greatly the control synthesis. Computer simulations below confirm this setting. 2 Notice that u 1 et u 2 are defined below for α 1 = α 2 = 5000, K P1 = 0.002 and K P2 = 0.0006

u 1 (k) u 2 (k) = -1 5000 F1 F2 - . P * . Q * + e 1 0 0 e 2 0.002 0.0006 . (4)
As defined above

F1 F2 = . P . Q -5000 u 1 (k -1) u 2 (k -1) . ( 5
)
For the estimation of the two derivatives,

. y 1 and . y 2 , the same algebraic numerical differentiation scheme as before has been employed, with the same parameters: number of samples n = 50 and a sample time T e = 5 10 -3 s. It yields a window width of T = 0.25s.

Responses in Fig. 7 and8 show a good level of performance, i.e., a very low interaction between the two control loops. It is comparable with what has been achieved in the vector control setting. This is due to the fact that, even with this decoupled form of α, information about the coupled dynamics is captured in the estimation of F 1 and F 2 . Notice also that the vector control integrates a scheme of compensation of these interactions (see, e.g., [START_REF] Leonhard | Control of Electrical Drives[END_REF]). The closed-loop provides also a good filtering of noise as in the case of the P loop alone.

V. ROBUSTNESS TESTS

This section investigates the behavior of the controls in non nominal conditions.

A. Grid variations

First, grid variations like, e.g., line tripping, topology modifications, load/generation evolution are considered. On the simple benchmark in Fig. 1, this is implemented as variations of the equivalent impedance of the AC grid. L g1 and r g1 are varied (maintaining their ratio) from the operating point considered before for which L g1 = 62.23 × 10 -3 in Figures 9 and11 for the case of the classic vector controller and in Figures 10 and 12 using an iP. Notice that the controllers have not been recalibrated to take into account the grid variation. For average variations (L g1 = 150 × 10 -3 ) the responses are comparable but for large variations, the new controller has better robustness. As a matter of fact, L g1 = 200 × 10 -3 corresponds to the stability limit of the classic vector control.

B. Responses to large disturbances

The behavior of the controls is now investigated in case of large grid disturbances which strongly excite nonlinear dynamics. Instead of the step on the reference of reactive power Q of station S1, a 150ms three-phase fault was applied at the PCC of station S1. Fig. 13 and 14 show that it is well rejected with both controllers.

VI. CONCLUSIONS

The model-free control has been used to control a VSC HVDC transmission system. It has been shown that the estimation of the derivatives of the output measures can be performed with good performances in the difficult context of the application: fast dynamics due to power electronics and noisy grid environment.

An iP is sufficient to ensure the same level of performances as the classic vector control and a great robustness with respect to the various disturbances. The advantage comes from its simple implementation which reduces to a simple proportional gain setting by trial and error simulations. This is a major advantage in case of devices with complex dynamics like power electronics modules or grid (secondary) regulations which involve several dynamics elements of the power systems and for which it is difficult to obtain both accurate and small-dimension control models. Indeed, in these cases, it is interesting to promote a model-free solution. A next step was done towards implementation of multiinput/multi-output intelligent control loops for one converter first and, next, for the whole HVDC link (the two converters and the DC part) and its validation in a large-scale grid context. This means to evaluate the impact of this new control on the dynamics of the AC neighbour zone of the HVDC. Thanks to Remark 3.2, it is expected that the model-free control provides better robustness against grid variations than the classic vector control.

After the full setting and validation on the HVDC case, this approach could be extended to the control of several grid devices based on power electronics like, e.g., renewable generators, energy storage, etc as an easy mean to take into account the grid environnement of the device.

Figure 2 .

 2 Figure 2. Vector-control of VSC-based HVDC.

Figure 3 .

 3 Figure 3. Active power response with classic vector controller.

Figure 4 .

 4 Figure 4. Active power response with an iP.

Figure 5 .

 5 Figure 5. Active power response with a classic vector controller 2) Intelligent controller: Let y *1 and y 1 be the active power reference and active power measurement respectively and y * 2 and y 2 the reactive power reference and reactive power measurement respectively. u 1 and u 2 represent the active power control and reactive power control respectively. Model (1) is now written in the multivariable form

Figure 6 .

 6 Figure 6. Reactive power response with a classic vector controller

Figure 7 .Figure 8 .

 78 Figure 7. Active power response with a multivariable iP

Figure 9 . 3 )Figure 10 . 3 )Figure 11 .

 9310311 Figure 9. Active power response with classic vector controller (L g1 = 150 × 10 -3 )

Figure 12 .Figure 13 .Figure 14 .

 121314 Figure 12. Reactive power response with a multivariable iP (L g1 = 200 × 10 -3 )

Until today, this decoupling property is valid in any concrete situation[START_REF] Lafont | A model-free control strategy for an experimental greenhouse with an application to fault accommodation[END_REF],[START_REF] Menhour | An efficient model-free setting for longitudinal and lateral vehicle control. Validation through the interconnected pro-SiVIC/RTMaps prototyping platform[END_REF].


Ruqi Zhou

Yuepeng Zhou

The Research of Attribute Granular Computing Model in Cognitive and Decision-making

Keywords: Cognitive and Decision-making, Attribute Granular Computing, Qualitative Mapping, Fuzzy Set, Petri Net

The cognitive activities of human beings are complicate and diversified. So far, there hasn't been a universal cognitive model. Each cognitive model generally only represents cognitive features in one or some aspects.Therefore, this paper aims to, based on the granular computing theory and principles, and with attributes and change laws as the main objects of the cognitive course, proposes a cognitive model which not only can describe the thing through attributes to represent the change law, but also can simulate the cognitive decision-making course with respect to the attribute change of the thing. Attribute granular computing ,based on qualitative mapping,can simulate the cognitive functions of human brain, such as granulation, organization and causation. Petri net has asynchronous, concurrent and uncertainty characteristics, which is similar to the characteristics of some cognitive activities in human thinking process. Petri net is extended based on the basic concept and logic calculation rules of attribute granular computing in this paper.Some basic elements of a cognitive system, such as knowledge representation, reasoning, learning and memory mode are initially showed in the extended Petri net. The results show that this method can reflect the cognitive process of uncertainty identification and decision-making in a certain extent.

Introduction

Neurophysiological studies indicate that human brain has a dual structure consisting of senses and consciousness, which are two independent brain functions respectively 1 processed in two different parts in human brain [START_REF] Rajendra | Neuroanatomical Organization of Perceptual Memory:An fMRI Study of Picture Priming[J][END_REF]. According to the structure and physiological functions of the human brain, paper [START_REF]The fifth generation computer and its cognitive logic methodology[END_REF] propose a physiology-based cognitive logic theory which holds that the human brain is of a dual structure for senses and consciousness; the sensory memory gets information from the external world, and the consciousness memory gets information from the sensory system;the perception is the conjugation of the sensory information and the consciousness information; and the same transformation exists between the sensory information and the consciousness information. Paper [START_REF] Feng | A Mathematical model of perception schema generation and recognition based on attributive in tegration[END_REF] point out that the sense organs only respond to sensory attributes which are sensitive to the sense organs. The information collected by the human brain is the collection of sensory attributes, and the state, movement and change course of a thing is reflected through attributes and the change course of such attributes. The processing of the sensory attributes and their change laws is one of the core problems of the information processing in human brain. The cognitive course of human beings starts from these attributes received through sensory, and an attribute is essentially an information granule.

Paper [START_REF] Zadeh | Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic[END_REF] summarized the human's cognitive ability into: granulation, organization and causation. The concept of the information granule given by paper [START_REF] Feng | Attribute network computing based on qualitative mapping and its applications in pattern recognition[END_REF] can be interpreted with an attribute and its qualitative mapping operator, so that the granular computing can be used to simulate a part of the cognitive functions of the human brain.

Decision making is actually a cognitive process.In the decision-making process, people always make decision evaluation systematically from different perspectives in order to obtain more reliable decision results.Qian [START_REF] Qian | MGRS: a multigranulation rough set[END_REF] proposed the multigranulation rough set theory, and proposed multigranulation decision rules which is consistent with human cognition.Song [START_REF] Song | A two-grade approach to ranking interval data[END_REF] determined the weights of the attributes for the decision objectives through mutual information and conditional entropy, and obtain the efficient sorting results.Yao [START_REF] Yao | The superiority of three-way decisions in probabilistic rough set models[END_REF] put forward three-way decision model, which is a common method of human cognitive decision-making. Based on the semantic model of Bayesian decision, a practical and effective decision-making rough set model was given [START_REF] Yao | Three-way decisions with probabilistic rough sets[END_REF][START_REF] Sun | Multigranulation fuzzy rough set over two universes and its application to decision making[END_REF].Threeway decision has been developed and applied in other areas [START_REF] Liu | Three-way investment decisions with decision theoretic rough sets[END_REF][START_REF] Nauman | A three-way decision making approach to malware analysis using probabilistic rough sets[END_REF].

Feng [START_REF] Feng | Attribute network computing based on qualitative mapping and its applications in pattern recognition[END_REF][START_REF] Feng | Qualitative mapping,linear transformation of qualitative criterion and artificial neuron[END_REF][START_REF] Feng | Conversion degree functions induced by qualitative mapping and artificial neurons[END_REF] put forward a comprehensive decision and evaluation model of attribute coordinate.It is a multi-index factors and multi-attribute decision-making comprehensive evaluation model based on the qualitative mapping.The model is good at simulating the process of psychological cognition of decision maker and gives decision. Its characteristic is that its evaluation method is very close to the brain cognitive thinking mode, which can reflect the cognitive psychological preferences and their change curves of the decision maker.Based on this, this paper describes the brain cognition and decision-making process using the model of attribute granulation.

However, due to the lack of a formal mechanism for the reasoning process of attribute granule, Petri net is used to establish a cognitive decision-making model in this paper.The Petri net is a formalization method with good structure, and is characterized by concurrency, asynchrony, uncertainty and the like. It is similar to certain cognition activity in the thinking course of the human brain. If the places and transitions in the Petri net are simulated with attribute granule and qualitative mapping, the Petri net will initially show the characteristics of some basic elements necessary to a cognition system in terms of knowledge representation, knowledge reasoning, decision-making,learning mode, memory mode, etc.This is a new trying.

Qualitative Mapping method in Cognition

Psychologists believe that human's senses reflect simple attributes of a thing while human's consciousness integrates such simple attributes into comprehensive attributes of the thing. Let a(x) be sensory attributes of a thing x. The value of a(x) generally can be divided into quantitative attribute value d a(x) and qualitative attribute value P a(x) . When x has qualitative attribute value P a(x) , it is said that the x has the "properties" referred in P a(x) . If P a(x) is the property by which the thing x can be distinguished from other things y, the P a(x) is called one property or characteristic of thing x [START_REF] Feng | A Mathematical model of sensation neuron detection based on attributive abstraction and integration[END_REF].

Let d(x) be a quantitative value of the property p(x), and [ i  i  be a certain qualitative criterion domain of p(x). [START_REF] Feng | Granular transformation of qualitative criterion,orthogonality of qualitative mapping system and pattern recognition[END_REF].

If d(x) [ i  i , [ i  i  is a neighborhood of d(x)
If [ i  i  and [ j  j  are the qualitative criterions respectively for property p i (x) and property p j (x),we can perform the following logical operations on [ i  i  and [ j  j .

(

)Logic not:~[ i  i  ,~ [ j  j . 1
(2)Logic and:

[ i  i   [ j  j .The intersection [ i  i ∩[ j  j 
,which can be considered as the qualitative criterions for the conjunction property q(x)= p i (x)∧p j (x).

Property p i (x) and property

p j (x) are homogeneous if a criterion [ ij  ij [ i  i ∩[ j  j  exists and makes a(x) [ ij  ij  b(x) [ ij  ij 
for any quantitative value a(x) of p i (x) and any quantitative value b(x) of p j (x). Or otherwise, p i (x) and p j (x) are heterogeneous.

(3)Logic or:

[ i  i   [ j  j .The intersection [ i  i ∪[ j  j 
can be considered as the qualitative criterions for the disjunction property r(x)= p i (x)∨p j (x).

(4)Logical implication:

[ i  i   [ j  j .
Let ={[ i  i iI} be the cluster of all qualitative criterions of a proposition p. The mapping : is called as the criterion mapping of proposition p [START_REF] Sun | Multigranulation fuzzy rough set over two universes and its application to decision making[END_REF].

If [ j  j  exists for any [ i  i , then  ([ i  i )=[ j  j   is called as the qualitaitve criterion transformation from [ i  i  to [ j  j ,and noted formally as [ i  i   [ j  j .
Obviously, homogeneous relations are equivalent based on the criterion mapping, while heterogeneous relations under the criterion mapping can be transformed when certain conditions are satisfied.

Philosophically, the law of quantitative change to qualitative change is represented as the transformation of quantitative properties to qualitative properties. Literatures [12] provide a mathematic model of this cognitive law:

Definition 1[3] Let ) ( ) ( 1 o a o a i n i  
be the integrated attributes of n factor attributes a i (o), i=1, …, n, and x=(x 1 , ..., x n ) be the quantitative value of a(o), where:

x i X i R is the quantitative property value of a i (o), pi(o)Po is a certain attriibute of a i (o), ={ [ i  i | [ i  i  is the qualitative criterion of pi(o)}, and hypercube [,]=[ 1 , 1 ]...[ n , n ] is the qualitative criterion of integrated attribute ) ( ) ( 1 o p o p i n i  
. The mapping :X→{0,

) ( ]) , [ ( ]) , [ , ( ) ( 1 ? 1 i o p n i i i i n i x x x i             
Where,

      ] , [ 0 ] , [ 1 ) ( ) ( i i i i i i i o p x x x i     
is the real value of the property proposition p i (o).

The numerical domain d a(x) of attribute a(x) of the thing x and the criterion domain cluster ={[ i  i iI} form a criterion topological space T(d a(x) ,Γ). Therefore, the definition 1 actually is a cognitive mode of the granular computing done in the criterion topological space.

Attribute Granules in Cognitive and decision-making

A granule is not only a cluster or a collection of entities, but also the abstraction of such cluster or collection [START_REF] Zhong | Granular data imputation: A framework of Granular Computing[END_REF], [START_REF] Wolski | Rough Granular Computing in Modal Settings: Generalised Approximation Spaces[END_REF]. If all attributes of a thing is considered as a whole, a certain attribute can also be considered as a granule. The qualitative mapping, which creates maps from an attribute to a property, is an intuitive attribute computing but also an intuitive granular computing. Attributes are closely related to the qualitative criterion, or in other words, attributes are closely related to the qualitative mapping. When we say a thing x has the property p(x) referred in P a(x) (the quantity property of an attribute a(x)), what we really mean is the thing has the property p(x) under certain qualitative criterion. Therefore, an attribute with a bigger structure intuitively composed of attributes which contain qualitative criterion or qualitative mapping is called an attribute granule.

An attribute granule in the attribute topological space can be expressed as a 2-tuple:

(P, ) =(p i (x), [ i  i ), or (P, ) = (d a(x) , [ i  i )
Where, a(x) is a sensory attribute of a thing x, p i (x) is the property of a(x), [ i  i  is a qualitative criterion domain, d a(x) is a quantitative attribute value of a(x), and a granule (p i (x), [ i  i ) is referred to as  i (p i ) for short. Formally, an attribute granule sometimes can also be equivalent to a first-order predicate formula.

Definition 2 Let  (p i ) and  (p j ) be two attribute granules. The computing formula of  (p i ) and  (p j ) with respect to logic connector can be defined as follows:

①   (p i ) =(  p i ,  )
, where,  is the complementary operation of the qualitative criterion of ;

② i (pi)∧ j (pj) =( pi(x) pj(x),[ i  i ∩[ j  j ,(i)=j). ③i (pi)∨ j (pj) =( pi(x) pj(x),[ i  i ∪[ j  j ,(i)=j). ④i (pi)  j (pj) =( pi(x) pj(x),[ i  i [ j  j ,(i)=j).
Where,  is attribute disjunction,  is attribute conjunction,  is attribute integration or reasoning, and  is qualitative criterion transformation. ②  is a mapping and P:U→U; ③ For any iI, :U×i→[0,1] is a degree function;

④ i is the qualitative criterion domain.

In the interpretation D, each qualitative criterion transformation formula P corresponds to an element v(P) in [0,1], which is called the true value of qualitative criterion transformation formula P in interpretation D, where:

① If P is the logical formula of the attribute granule (i

(p i )=(p i (x), [ i  i ), then: v(pi (x),[ii) =       ] , [ ] , [ 1 i i i i x iff x iff     ② If P is the logical formula of the attribute granule ~G, then v(~G)=( v(G))= v( p i (x), -[ i  i ); ③ If P is the logical formula of the attribute granule G∨H, then v(G∨H)= v(p i (x)p j (x), [ i  i  ∪[ j  j );
④ If P is the logical formula of the attribute granule G∧H, then v(G∧H)= v( p i (x)p j (x), [ i  i ∩[ j  j );

⑤ If P is the logical formula of the attribute granule G→H, then v(G→H)= v(p i (x)p j (x),[ j  j =([ i  i ))
Where,  is attribute reasoning. Definition 5 G and H in the logical formula of the attribute granule are called equivalence. If the assignment of any interpretation D is v(G)= v(H), then G=H. Definition 6  i (p i ) → i (p i ) is an interference in the logic of the attribute granule, which means that there is a mapping

 ij :  to make  ij ([ i  i )=[ j  j  if (p i (x)p j (x)).

Cognitive decision-making and fuzzy attribute granule Definition 7[5] A mapping : X→[-1, 1]i is a function reprenting the transformation degree of quality property set

p i ( i ). If i(x)[-1, 1]i for (x,N( i ,  i ))X, then: (x, I, i)=|x-i|i=i(x)
Where, The N(i, i) is the qualitative criterion domain, I is radius of the domain, I is core of the domain, i(x) is diversity factor of |x -i| and i as to the mathematics nature.

Zadeh studied the size of divided class or granule and defined the information granularity as a proposition: value of x belongs to the fuzzy subset GU by membership λ. The x is a variable of U and the value of x is an entity of U, represented by g = x is G is λ, and noted formally as:

g = {u∈U: value of x (v(x) = u, v is assignment symbol on U) is calculated based on membership of λ in the fuzzy set G U. }.
Obviously, in the formulation above, 0≤λ≤1. Based on fuzzy set theory, λ is a fuzzy membership function. Based on logic, the λ is the fuzzy truth value or probability of the proposition.

It is easy to figure out by comparison that: under the qualitative mapping  with transformation program function, a information granule

(p i (x), [ i   i  or (d a(x) , [ i   i ) in
Definition 1 shows that 'the fact attribute a(x) having the property p i (x) is calculated based on the membership of the attribute value d a(x) in the criterion [ i   i '. They are the same in essence.

Definition 8 Let the criterion domain N i (o i , r i ) be the core of a fuzzy set A, namely

N i (o i , r i )=A 1 = {x│ A (x)=1 }. The  A (x) is membership of xA.  cut set of A is A={x |  A (x)≥},, , consisting of element x of membership not less than .
 is the  criterion transformation of the fuzzy set A whose core is i o i , r i  if it meets following equation:

 i  [ i ,  i ]o i  r i o k , r k  Definition 9
Let A and B be two fuzzy sets with criterion domains [ i ,  i ] and [ k ,  k ] as cores respectively.  is the criterion transformation from

[ i ,  i ] to [ k ,  k ] if it meets following equation: [ i ,  i ]o i  r i o k , r k 
The following theorem can be obtained immediately:

Based on the definition 10,criterion transformation  as the composite transformation of  and criterion transformation if it meets conditions below:  • i·i =o,r The mapping : X→[-1, 1] is the transformation degree function for p i (x) to reflect its nature characteristics; if

(x, N( i ,  i ))X, i (x)[-1,1], which makes (x,  i ,  i )=|x- i | i = i (x)
Where,  i (x) is the variability between |x- i | and  i .

Set up cognition formal model with attribute granule

The transition nodes in Petri net can be viewed as a qualitative mapping, and the quantity properties, quality properties and qualitative criterion domain mapping of the attributes of a thing care places in the Petri net. Therefore, if the place nodes are extended into an information granule, the transition nodes will naturally be the granular transformation mapping. The definition of ordinary Petri net is extended below, and is considered as a cognitive system model.

Definition 10

The form of a cognitive system can be defined as a nonuple (10-tuple) Petri net: CSPT= { P,T,F; M 0 ,S,O,N,W,, } Where, P is a finite set of place nodes of attribute granules; T is a finite set of transition nodes of attribute granules, equivalent to a granular transformation; and F is a marked relation on PT, representing the connections of place nodes to transition nodes, the rated input and the input strength calculation function S on connection line, as well as the corresponding connection strength. M 0 is a function defined on P with its value range being [0,K] (K is a bounded real number), indicating the initial marking of the place nodes at the beginning of the operation; N is the output strength function at a qualitative level; O is the operation of m-dimensional weighted qualitative operator;  is the criterion transformation, is qualitative criterion; and W is the attribute weight, a function of .

If the number of input link in any place node is 0, the place node is called as the input node of RS. If the number of the output link n any place node is 0, the place node is called as the output node of CSPT.

CSPT is mainly characterized in that:

(1) The starting threshold of transition nodes of CSPT is the detection threshold of sensory nerves, and the starting course is the transformation of the qualitative mapping degree.

(2) Directed edges in CSPT can be divided into input link and output link. For any node, the link pointing to it is called its input link and that deviating from it is called its output link. The link strength is attached to the input and output links.

(3) The capacity of place node is a bounded real number and the start-up of transition node depends on whether value of the input of quantitative (or qualitative) property of attribute on the input link, connection strength and the degree function of qualitative mapping (which is called input strength) is higher than the starting threshold of the transition node. This is an operable net. The logical relationship between place nodes can be represented as shown in Fig. 1 to Fig. 3. Decision problems described in the qualitative mapping model can be represented by CSPT in a vivid and visual manner. Paper [START_REF] Yingxu | Cognitive Informatics Models of the Brain[END_REF][START_REF] Wang | Memory-Based cognitive modeling for visual information processing[END_REF] proposed a logical structure of memory which holds that the sensory storage in the three-stage processing model for memory information is merely an inputoriented storage. In view of the whole process from input to output of the sensation, consciousness and the qualitative mapping, paper [START_REF] Dong | An Attributive coordinate representation of schematic memory[END_REF] indicate that the sensation is a test response to all simple attribute properties of an object, and that the consciousness is responsible for integrate them to form the overall consciousness of human brain to the object. The pattern recognition of the outer attribute information granules through sensation and consciousness can be represented with the following classification function: The representation of attribute granules in CSPT is more natural, and the whole process of knowledge representation reflects the reasoning course of thinking operation in the human brain.

       or y x
Assume the following thinking operation exists, M(y) is the memory set of a subject y, and s i (y)∈M(y)is detecting neurons (or sensor or the like) of the attribute s i (x) of a thing x. Let P x ={p j (x) | j=1,…,n} be the set of the attribute granules of x, and p(x)=∧p j (x) be the integration of r (r≤n) attributes of x, where, ∧ is integration operator. If r=n and T(x)=∧p j (x) is the integrated (comprehensive) attributes of x, then, the attribute detection on T(x) by sensory neuron s i (y) can be represented as a mapping s i (y)∶P

x →M(y) from P x to M(y), and: i (x,y) ∈M(y) is the sensory image of s i (x) in M(y). the above formula indicates that: if T(x) contains attributes s i (x) which can be detected by s i (y), then s i (y) will decompose and extract s i (x)from T(x), and store the detection image s i (x,y) in M(y). or otherwise, s i (y) will tell the computer that x doesn't have attribute s i (x). The decisionmaking process of the above thinking operation can be represented in CSPT as shown in Fig. 4.

(p i (x), [ i  i )

Conclusion

Sense organs of human being receive all kinds of sensory attribute information and respond accordingly. A thing represents its state, movement and change with its attributes and the attribute change courses. The cognitive process of human beings are closely related such attribute information received. Philosophical opinions hold that attribute not only expresses the quality property which needs to be expressed, but also has quantity property which needs to be limited and regulated. Based on such philosophical opinions in combination with the theories with respect to granular computing and qualitative criterion transformation, , this paper provides the basic mathematical model of the cognition and the definition of Petri net. However, the cognitive process of human beings is very complicated. This paper gives merely the framework of a basic cognitive mode. More study work needs to be done.

  1}Po is called the qualitative mapping (QM) of x=(x 1 , ..., x n ) with n-dimentional hypercube [,] as a ceterion. If a propert ) with [,] and [,] as ceterion exists for any xX, then:

Definition 3 ADefinition 4

 34 formula (i(p i )) is true if and only if the true value of its transformation function (i(p i )) is 1 or above 0.5. The logical formula of an attribute granule can be interpreted semantically as:A quadruple (4-tuple) D:D={U,,,{i, iI }} ① U is a non-empty set called domain of interpretation (DOI);

Fig. 1

 1 Fig. 1 One Cause to Multiple Effects Fig.2 One Cause to Multiple Effects

Fig. 3

 3 Fig.3 Multiple Causes to One Effect

Fig. 4

 4 Fig.4 Decision-making in CSPT

  Ruqi Zhou was born in 1971.He is now a Ph.D. candidate in Computer Science and Technology of Sun Yat-sen University.He is also an assistant professor in Guangdong University of Education.His current research interests include machine learning, Intelligent information Processing,Petri net theory & application,Granular Computing.E-mail address: ruqzhou@163.com(R. Zhou). Yuepeng Zhou was born in 1982.He is now a Ph.D. candidate in Computer Science and Technology of Sun Yat-sen University.His current research areas include Intelligent algorithm,data mining.

  is sensory neuron which can be considered as the mapping of memory set M(x, y) of the thing x from attribute set P

	s y	(	p i	(x),	, i   [	) ] i	s y	Γ	(	p i	)	s y	(	x ,	y	s y )	(	,	) If	s y	(	x	is )	a	simple	factor	of	p i	(	x	)
	Where, (p i (x), [

i  i ) and  (p i ) represent the attribute granule of a thing x, s y x to a subject y. Namely, s y : P

x →M(x,y). is an extraction operation.

Acknowledgments

This work is supported by the National High Technology Research and Development Program of China (863) sub project(2012AA101701-06),the Guangdong Science and Technology Plan Project(2012B010100049),the Characteristic Innovation Project of Guangdong Universities(natural science)(2014KTSCX195),open fund of Fujian Provincial Key Laboratory of Big Data Mining and Applications(Fujian University of Techology,2017).