CN103835878A - Maximum power point tracing control method based on neural network optimization starting rotating speed - Google Patents

Abstract

The invention provides a maximum power point tracing control method based on the neural network optimization starting rotating speed. According to the method, on the basis of an existing MPPT control method, a neural network is adopted to obtain the best starting generator rotating speed according to wind speed condition dynamic optimization compensation factors, and then wind energy capture efficiency is further improved. The adopted neural network takes the average wind speed and turbulence intensity as input, and takes the best compensation factor as output. Mass draining data obtained through the traversal algorithm are used for training the neural network, the trained neural network is adopted for obtaining the corresponding best compensation factor through calculation according to the changed wind speed conditions, and then the best compensation factor is used for optimizing the starting generator rotating sped, so that the best MPPT tracking interval is obtained, and the wind energy capture efficiency is further improved. Compared with various traditional MPPT control methods, the effectiveness and superiority of the algorithm are verified.

Description

Based on the maximum power point-tracing control method of the initial rotating speed of Neural Network Optimization

Technical field

The invention belongs to wind power generation field, particularly a kind of maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization.

Background technique

In order to improve the wind energy capture rate lower than rated wind speed interval, speed-variable frequency-constant wind-driven generator group generally adopts MPPT maximum power point tracking (Maximum Power Point Tracking, MPPT) control strategy.Power curve method (also referred to as power signal feedback transmitter or torque curve method) is one of most widely used MPPT controlling method.

Traditional MPPT controls, and particularly power curve method, how based on system stabilization design, and has ignored the dynamic process that blower fan system is followed the tracks of between different steady operation points.But, in the face of the single-machine capacity constantly promoting causes rotary inertia that wind wheel constantly increases and dynamic response performance more slowly thereof, and wind speed is frequently in wave process and very difficult short-term forecast, tradition MPPT controls most time of lower blower fan system in dynamic process, but not operates on steady operation point.Therefore, the wind speed tracking effect of blower fan reality is still to be improved.

For this reason, the L.J.Fingersh in American National renewable energy sources laboratory and P.W.Carlin have proposed to utilize generator electromagnetic torque to help the improvement thinking of blower fan acceleration or deceleration first; On this basis, the people such as Johnson K.E. has proposed to reduce gain of torque (Decreased Torque Gain, DTG) and has controlled.This controlling method has not only improved the acceleration performance of blower fan in the time following the tracks of crescendo fitful wind by reducing electromagnetic torque, more first Application exchange the control thought of the high wind energy capture rate of high wind speed section for to abandon the rotating-speed tracking effect of the low wind speed section of part; Further, consider that DTG controls the constant gain coefficient of employing, the people such as Johnson K.E. design again self adaption torque control, utilize the statistics of adaptive algorithm and history run operating mode, iterative search the online optimum gain coefficient of revising, change with the wind friction velocity responding in iteration cycle time scale.But the crucial difficult problem that this thinking faces is that optimum state and the wind friction velocity of torque adjustment is closely related, but be difficult to find direct quantitative relationship between them.Given this, the people such as Yin Minghui propose the improvement MPPT maximum power point tracking control based on shrinking trace interval, follow the tracks of distance reach the object of improving MPPT tracking effect by shortening.

Above-mentioned research work can be summarized as two Research Thinkings,, by adjusting electromagnetic braking torque and shorten tracking two aspects of distance the dynamic performance and the tracking effect that improve blower fan, has broken through conventional power curve method and has ignored the dynamic narrow limitation of tracking.But, improvement MPPT maximum power point tracking control based on shrinking trace interval does not provide optimum trace interval (being the optimal compensation coefficient), because arranging of trace interval will significantly affect the tracking effect of MPPT, thereby seem very important for the optimization of trace interval.But in prior art, there is no associated description.

Summary of the invention

Technical problem solved by the invention is to provide a kind of maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization, obtains optimum rotating-speed tracking interval further to improve wind energy capture rate by optimizing initial generating rotating speed.

The technical solution that realizes the object of the invention is: a kind of maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization, taking the improvement power curve method based on initial adjustment of rotational speed as basis, adopt neuron network to adjust initial generating rotating speed and realize MPPT maximum power point tracking control, the described improvement power curve method formula used based on initial adjustment of rotational speed is:

$M\stackrel{\·}{\mathrm{\ω}}=T_m(v,\mathrm{\ω})-T_e\left(\mathrm{\ω}\right)$

$T_m(v,\mathrm{\ω})=\frac{0.5\mathrm{\ρ\π}{R}^5{C}_p\left(\mathrm{\λ}\right)}{{\mathrm{\λ}}^3}{\mathrm{\ω}}^2$

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

${\mathrm{\&omega;}}_{\mathrm{bgn}}={\mathrm{\&lambda;}}_{\mathrm{opt}}({\stackrel{\&OverBar;}{v}}_T-\mathrm{\&alpha;})/R$

In above formula, M is rotary inertia, T _mfor the Mechanical Driven torque of wind wheel, T _efor electromagnetic braking torque, v is wind speed, the angular velocity that ω is wind wheel,

for wind wheel angular acceleration, ρ is air density, and R is wind wheel radius, C _pfor power coefficient, λ=ω R/v is tip speed ratio, ω _bgnfor initial generating rotating speed is initial rotating speed, λ _optfor optimum tip-speed ratio, for the mean value of the wind speed sampled value sequence in one-period, α is penalty coefficient, for periodically adjusting initial rotating speed, T _optfor the optimum torque curve of blower fan, be specially:

T _opt(ω)＝K _mω ²

$K_m=\frac{0.5\mathrm{\&rho;\&pi;}{R}^5{C}_p^{\max }}{{\mathrm{\&lambda;}}_{\mathrm{<figure-callout\; id="3"\; label="opt"\; filenames="HDA00003018075200022.png"\; state="\{\{state\}\}">opt</figure-callout>}}^3}$

In above formula

for maximal wind-energy utilization factor;

Wherein initial generating rotational speed omega _bgnthe step that adopts neuron network to adjust is as follows:

Step 1, carry out initialization, to wind speed sample frequency and initial rotating speed revision cycle T _rarrange, wherein wind speed sample frequency is 1～4Hz; Empty T _rcorresponding wind speed sampled value sequence, by ω _bgnbe initialized as the blower fan MPPT maximum power point tracking minimum speed in control stage; Initial rotating speed revision cycle T _rbe preferably 20 minutes.

Step 2, neural network training, utilize ergodic algorithm to obtain multiple mean wind velocity with the optimal compensation factor alpha corresponding under turbulence intensity TI _opt, set it as the training sample of neuron network; When training, with the mean value of wind speed

with the turbulence intensity TI input variable that is neuron network, taking the optimal compensation coefficient as output variable;

Step 3, enter new initial rotating speed revision cycle T _r, with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is read measuring wind speed value, and be saved to wind speed sampled value sequence;

Step 4, judge current initial rotating speed revision cycle T _rwhether complete, if complete, perform step 5; Otherwise, continue with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is saved to wind speed sampled value sequence;

Step 5, ask for the mean value of wind speed sampled value sequence

with turbulence intensity TI;

The mean value of step 6, the wind speed sampled value sequence asked for step 5 , call neuron network and obtain corresponding the optimal compensation factor alpha for input with turbulence intensity TI _opt;

Step 7, the optimal compensation factor alpha obtaining according to step 6 _optto initial rotational speed omega _bgnadjust, be specially:

By ω _bgnbe adjusted into

afterwards with the ω after upgrading _bgnenter new revision cycle T _r, electromagnetic braking torque is still adjusted as follows

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

Empty afterwards wind speed sampled value sequence, and skip to step 3.

Compared with prior art, its remarkable advantage is in the present invention: 1) the present invention adopts the trace interval that Neural Network Optimization MPPT controls, and has further improved wind energy capture rate; 2) the present invention adjusts trace interval according to concrete wind friction velocity, can follow the tracks of better wind speed; 3) the present invention directly estimates optimum initial rotating speed with recent mean wind velocity, is not vulnerable to the impact that wind friction velocity changes; 4) the present invention does not need complicated iterative search procedures, and algorithm is very simple; 5) information that the present invention need to measure is few, and computation burden is light.

Below in conjunction with accompanying drawing, the present invention is described in further detail.

Brief description of the drawings

Fig. 1 is the flow chart of the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization of the present invention.

Fig. 2 is the structure wind series of high degree of fluctuation.

Fig. 3 is maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization of the present invention and the comparison diagram of additive method.

Embodiment

The present invention will based on existing MPPT controlling method propose to adopt neuron network according to wind friction velocity dynamic optimization penalty coefficient to obtain best initial generating rotating speed (being optimal tracking interval), and then further improve wind energy capture rate.

In conjunction with Fig. 1, a kind of maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization of the present invention, taking the improvement power curve method based on initial adjustment of rotational speed as basis, adopt neuron network to adjust initial generating rotating speed and realize MPPT maximum power point tracking control, the described improvement power curve method formula used based on initial adjustment of rotational speed is:

$M\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=T_m(v,\mathrm{\&omega;})-T_e\left(\mathrm{\&omega;}\right)$

$T_m(v,\mathrm{\&omega;})=\frac{0.5\mathrm{\&rho;\&pi;}{R}^5{C}_p\left(\mathrm{\&lambda;}\right)}{{\mathrm{\&lambda;}}^3}{\mathrm{\&omega;}}^2$

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

${\mathrm{\&omega;}}_{\mathrm{bgn}}={\mathrm{\&lambda;}}_{\mathrm{opt}}({\stackrel{\&OverBar;}{v}}_T-\mathrm{\&alpha;})/R$

In above formula, M is rotary inertia, T _mfor the Mechanical Driven torque of wind wheel, T _efor electromagnetic braking torque, v is wind speed, the angular velocity that ω is wind wheel,

for wind wheel angular acceleration, ρ is air density, and R is wind wheel radius, C _pfor power coefficient, λ=ω R/v is tip speed ratio, ω _bgnfor initial generating rotating speed is initial rotating speed, λ _optfor optimum tip-speed ratio,

for the mean value of the wind speed sampled value sequence in one-period, α is penalty coefficient, for periodically adjusting initial rotating speed, T _optfor the optimum torque curve of blower fan, be specially:

T _opt(ω)＝K _mω ²

$K_m=\frac{0.5\mathrm{\&rho;\&pi;}{R}^5{C}_p^{\max }}{{\mathrm{\&lambda;}}_{\mathrm{<figure-callout\; id="3"\; label="opt"\; filenames="HDA00003018075200022.png"\; state="\{\{state\}\}">opt</figure-callout>}}^3}$

In above formula

for maximal wind-energy utilization factor;

Wherein initial generating rotational speed omega _bgnthe step that adopts neuron network to adjust is as follows:

Step 1, carry out initialization, to wind speed sample frequency and initial rotating speed revision cycle T _rarrange, wherein wind speed sample frequency is 1～4Hz; Empty T _rcorresponding wind speed sampled value sequence, by ω _bgnbe initialized as the blower fan MPPT maximum power point tracking minimum speed in control stage; Initial rotating speed revision cycle T _rbe preferably 20 minutes.

Step 2, neural network training, utilize ergodic algorithm to obtain multiple mean wind velocity

with the optimal compensation factor alpha corresponding under turbulence intensity TI _opt, set it as the training sample of neuron network; When training, with the mean value of wind speed

with the turbulence intensity TI input variable that is neuron network, taking the optimal compensation coefficient as output variable;

Step 3, enter new initial rotating speed revision cycle T _r, with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is read measuring wind speed value, and be saved to wind speed sampled value sequence;

Step 4, judge current initial rotating speed revision cycle T _rwhether complete, if complete, perform step 5; Otherwise, continue with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is saved to wind speed sampled value sequence;

Step 5, ask for the mean value of wind speed sampled value sequence with turbulence intensity TI;

The mean value of step 6, the wind speed sampled value sequence asked for step 5

, call neuron network and obtain corresponding the optimal compensation factor alpha for input with turbulence intensity TI _opt;

Step 7, the optimal compensation factor alpha obtaining according to step 6 _optto initial rotational speed omega _bgnadjust, be specially:

By ω _bgnbe adjusted into afterwards with the ω after upgrading _bgnenter new revision cycle T _r, electromagnetic braking torque is still adjusted as follows

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

Empty afterwards wind speed sampled value sequence, and skip to step 3.

Below in conjunction with embodiment, the present invention is done to further detailed description:

By simulation calculation and statistical analysis to simulation wind series, the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization that the present invention is proposed and conventional power curve method, self adaption torque control and the improvement MPPT based on shrinking trace interval control (the improved MPPT control based on reduction of tracking range, control referred to as RTR-MPPT) compare, to verify validity of the present invention and superiority.

1, embodiment's simulation model

1) parameter of simplification blower fan model

The major parameter of blower fan model is set to: fan capacity 1.0MW, rotor diameter 52.67m, rotary inertia 1.1204 × 10 ⁶kgm ².The maximum C of blower fan _pvalue

be 0.4109, optimum tip-speed ratio λ _optbe 8.0.

2) structure of wind series

The construction method of simulation wind series is described in conjunction with Fig. 2.According to autoregressive moving average (ARMA) method, construct 80 endurance and be the wind speed period of 20 minutes.And by wind speed mean value, the above-mentioned wind speed period is carried out to permutation and combination, the acclivity or the rise/fall that construct mean wind velocity replace slope (as shown in Figure 2), to construct the wind friction velocity of high degree of fluctuation.The turbulence intensity of each wind speed period is made as category-A (height) the turbulent flow rank defining in IEC-614000-1 standard.

3) setting of neuron network

The present embodiment adopts the neuron network based on RBF (radical basis function, RBF), and its error performance index is made as 0.1.Error performance index has determined the fitting degree of neuron network to training data, the approximation accuracy by final decision neuron network to the optimal compensation coefficient.

4) the Realization of Simulation of the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization

Adopt neuron network periodically to optimize ω according to step described in summary of the invention _bgncan realize the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization, specific as follows:

Taking the improvement power curve method based on initial adjustment of rotational speed as basis and adopt neuron network to adjust initial generating rotating speed to realize MPPT maximum power point tracking control, adopt neuron network periodically to optimize ω _bgnstep as follows:

Step 1, carry out initialization, to wind speed sample frequency and initial rotating speed revision cycle T _rarrange, wherein wind speed sample frequency is 4Hz; T _rbe 20 minutes, empty T _rcorresponding wind speed sampled value sequence, by ω _bgnbe initialized as the blower fan MPPT maximum power point tracking minimum speed in control stage;

Step 2, neural network training, utilize ergodic algorithm to obtain multiple mean wind velocity

with the optimal compensation factor alpha corresponding under turbulence intensity TI _opt, set it as the training sample of neuron network; When training, with the mean value of wind speed

with the turbulence intensity TI input variable that is neuron network, taking the optimal compensation coefficient as output variable.

Step 3, enter new initial rotating speed revision cycle T _r, with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is read measuring wind speed value, and be saved to wind speed sampled value sequence; Described wind speed sampling period T _wfor 0.25s;

Step 4, judge current initial rotating speed revision cycle T _rwhether complete, if complete, perform step 5; Otherwise, continue with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is saved to wind speed sampled value sequence;

Step 5, ask for the mean value of wind speed sampled value sequence

with turbulence intensity TI;

Step 6, ask for step 5 with TI be that input is called neuron network and obtained corresponding the optimal compensation factor alpha _opt;

Step 7, press

to initial rotational speed omega _bgnadjust, afterwards with the ω after upgrading _bgnenter new revision cycle T _r.Empty wind speed sampled value sequence, and skip to step 3.

2, the comparative analysis of wind energy capture rate

The present invention has constructed 75 groups of emulation experiment examples according to above-mentioned wind speed construction method.For every group of example, apply respectively conventional power curve method, self adaption torque control, RTR-MPPT control and the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization of the present invention, and by the average wind energy utilization P of statistical analysis wind speed period _favgcarry out comparison said method.P _favgbe defined as follows:

$P_{\mathrm{favg}}=\frac{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{P}_{\mathrm{cap}}\left(i\right)}{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{P}_{\mathrm{wy}}\left(i\right)},$

P _wy＝0.5ρπR ²v ³cos ³ψ

Wherein, n is the sampling number in a statistical time range (being iteration cycle); ψ is yaw error angle, and ignoring is herein 0 degree.

Contrast the wind energy capture rate of above-mentioned 4 kinds of methods in conjunction with Fig. 3.The wind series that comprises 80 wind speed periods that above-mentioned 4 kinds of methods is applied to above-mentioned structure, can calculate respectively the P corresponding to every kind of method and this wind series _favgmean value, be designated as

be specially:

${\stackrel{\&OverBar;}{P}}_{\mathrm{favg}}=\underset{i=1}{\overset{80}{\mathrm{\&Sigma;}}}{P}_{\mathrm{favg}}^i/80$

Further, calculate that 75 groups of simulation example obtain

mean value, be designated as

as shown in Figure 3.As seen from Figure 3, the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization that the present invention proposes

(the 4th bar chart) improved 2.03% than conventional power curve method, improved 0.84% than self adaption torque control, controls and improved 0.14% than RTR-MPPT.Verify the superiority of the maximum power point-tracing control method that the present invention is based on the initial rotating speed of Neural Network Optimization.

Claims (3)

1. the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization, it is characterized in that, taking the improvement power curve method based on initial adjustment of rotational speed as basis, adopt neuron network to adjust initial generating rotating speed and realize MPPT maximum power point tracking control, the described improvement power curve method formula used based on initial adjustment of rotational speed is:

$M\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=T_m(v,\mathrm{\&omega;})-T_e\left(\mathrm{\&omega;}\right)$

$T_m(v,\mathrm{\&omega;})=\frac{0.5\mathrm{\&rho;\&pi;}{R}^5{C}_p\left(\mathrm{\&lambda;}\right)}{{\mathrm{\&lambda;}}^3}{\mathrm{\&omega;}}^2$

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

${\mathrm{\&omega;}}_{\mathrm{bgn}}={\mathrm{\&lambda;}}_{\mathrm{opt}}({\stackrel{\&OverBar;}{v}}_T-\mathrm{\&alpha;})/R$

In above formula, M is rotary inertia, T _mfor the Mechanical Driven torque of wind wheel, T _efor electromagnetic braking torque, v is wind speed, the angular velocity that ω is wind wheel,

for wind wheel angular acceleration, ρ is air density, and R is wind wheel radius, C _pfor power coefficient, λ=ω R/v is tip speed ratio, ω _bgnfor initial generating rotating speed is initial rotating speed, λ _optfor optimum tip-speed ratio,

for the mean value of the wind speed sampled value sequence in one-period, α is penalty coefficient, for periodically adjusting initial rotating speed, T _optfor the optimum torque curve of blower fan, be specially:

T _opt(ω)＝K _mω ²

$K_m=\frac{0.5\mathrm{\&rho;\&pi;}{R}^5{C}_p^{\max }}{{\mathrm{\&lambda;}}_{\mathrm{opt}}^3}$

In above formula

for maximal wind-energy utilization factor;

Wherein initial generating rotational speed omega _bgnthe step that adopts neuron network to adjust is as follows:

Step 1, carry out initialization, to wind speed sample frequency and initial rotating speed revision cycle T _rarrange, wherein wind speed sample frequency is 1～4Hz; Empty T _rcorresponding wind speed sampled value sequence, by ω _bgnbe initialized as the blower fan MPPT maximum power point tracking minimum speed in control stage;

Step 2, neural network training, utilize ergodic algorithm to obtain multiple mean wind velocity

with the optimal compensation factor alpha corresponding under turbulence intensity TI _opt, set it as the training sample of neuron network; When training, with the mean value of wind speed

with the turbulence intensity TI input variable that is neuron network, taking the optimal compensation coefficient as output variable;

Step 3, enter new initial rotating speed revision cycle T _r, with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is read measuring wind speed value, and be saved to wind speed sampled value sequence;

Step 4, judge current initial rotating speed revision cycle T _rwhether complete, if complete, perform step 5; Otherwise, continue with wind speed sampling period T _win this cycle T _rin air speed value is sampled and is saved to wind speed sampled value sequence;

Step 5, ask for the mean value of wind speed sampled value sequence

with turbulence intensity TI;

The mean value of step 6, the wind speed sampled value sequence asked for step 5

, call neuron network and obtain corresponding the optimal compensation factor alpha for input with turbulence intensity TI _opt;

Step 7, the optimal compensation factor alpha obtaining according to step 6 _optto initial rotational speed omega _bgnadjust, afterwards with the ω after upgrading _bgnenter new revision cycle T _r, electromagnetic braking torque is still adjusted as follows

$T_e\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{cc}0& \mathrm{\&omega;}<{\mathrm{\&omega;}}_{\mathrm{bgn}}\\ T_{\mathrm{opt}}& \mathrm{\&omega;}>{\mathrm{\&omega;}}_{\mathrm{bgn}}\end{array}\right.$

Empty afterwards wind speed sampled value sequence, and skip to step 3.

2. the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization according to claim 1, is characterized in that, initial rotating speed revision cycle T in step 1 _rit is 20 minutes.

3. the maximum power point-tracing control method based on the initial rotating speed of Neural Network Optimization according to claim 1, is characterized in that, in step 7 to initial generating rotational speed omega _bgnadjust and be specially: by ω _bgnbe adjusted into ${\mathrm{\&omega;}}_{\mathrm{bgn}}={\mathrm{\&lambda;}}_{\mathrm{opt}}({\stackrel{\&OverBar;}{v}}_T-{\mathrm{\&alpha;}}_{\mathrm{opt}})/R.$

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