CN104242946A - Signal reconstruction method of photovoltaic array state monitoring network - Google Patents

Abstract

The invention discloses a signal reconstruction method of a photovoltaic array state monitoring network. Sensor nodes capable of collecting the electric quantity and environmental parameter information of a photovoltaic power generation system are allocated in the photovoltaic array state monitoring network, and a wireless monitoring network is formed in a self-organization mode when the sensor nodes are initialized. The signal reconstruction method is suitable for the photovoltaic array state monitoring network based on the compressed sensing principle, the reconstruction method for monitoring network sparse signals is provided, the nodes in the photovoltaic array state monitoring network are responsible for collecting information, the information is compressed into the sparse signals to be transmitted to other nodes, the sparse signals need to be reconstructed before the nodes in the monitoring network process the obtained sparse signals, and the information transmission reliability of the monitoring network is guaranteed with the signal reconstruction method.

Description

A kind of photovoltaic array status monitoring network signal reconstructing method

Technical field

The present invention relates to a kind of photovoltaic array status monitoring network signal reconstructing method, namely based on the signal reconfiguring method of the photovoltaic array status monitoring network of wireless monitor network technology, be applicable to the sparse signal by noise jamming reconstructing monitoring network transmission.

Background technology

Being incorporated into the power networks of photovoltaic generating system is the key character of modern power network, and for the random intervals feature of photovoltaic generation, the wireless monitor network technology with flexible monitoring advantage can be its safe and stable operation and provides safeguard.Photovoltaic array status monitoring network is made up of sensor node, and the height of the communication energy consumption of node is related to the working life of node, finally has influence on the reliability of network.The communication pressure reducing node can reduce the energy consumption of node, therefore, node before transmitting the signal, need that compression is carried out to it and obtain sparse signal, the sparse signal of node-node transmission is subject to noise jamming, how to utilize the sparse signal reconfiguring primary signal by noise jamming, i.e. sparse signal reconfiguring problem, major issue to be solved when being photovoltaic array status monitoring network actual motion.And the stable signal reconstruction algorithm still lacked at present for photovoltaic array status monitoring network, sparse signal reconfiguring algorithm can ensure the Effec-tive Function of monitoring network, for signal reconstruction provides new approaches.

Summary of the invention

Goal of the invention: for photovoltaic array status monitoring network signal reconstruction, the invention provides a kind of sparse signal reconfiguring method theoretical based on compressed sensing (Compressed Sensing, CS).When monitoring network is transmitted gathered information, the signal that node obtains is the sparse signal by noise jamming, and the reconstructing method that the present invention proposes can reconstruct the sparse signal be disturbed, and contributes to the reliability improving monitoring network transfer of data.

Technical scheme: a kind of photovoltaic array status monitoring network signal reconstructing method, in photovoltaic array status monitoring network, disposes the sensor node of temperature, intensity of illumination, voltage, electric current and the power information that can gather photovoltaic generating system.Each node adopts Ad hoc mode to form wireless monitor network, and for reducing the communication pressure of monitoring network, monitoring network adopts compressive sensing theory to compress the information that each node gathers.Compressing the signal obtained is sparse signal, and sparse signal, when transmitting, can be subject to the interference of noise, and for ensureing the reliability of monitoring network information transmission, monitoring network needs rationally effective sparse signal reconfiguring method.

The present invention proposes photovoltaic array status monitoring network signal reconstructing method: based on the sparse reconstruct of gradient projection (Gradient Projection Sparse Reconstruction, GPSR) signal reconfiguring method, by obtaining the sparse signal by noise jamming of monitoring network transmission, the raw information that reconstruct monitoring network gathers.Improved the operation efficiency of the sparse restructing algorithm of gradient projection by continuous regular factor simultaneously.

Beneficial effect: photovoltaic array status monitoring network signal reconstructing method provided by the invention, be applicable to reconstruct the sparse signal by noise jamming in monitoring network, by reducing the impact of noise and increasing the degree of rarefication of sparse signal, make reconstruct mean square error (the Mean Square Error of sparse signal, MSE) reach the effect close to zero, and algorithm has the high advantage of operational efficiency.

Accompanying drawing explanation

Fig. 1 is the compressed sensing of monitoring network signal;

Fig. 2 is the reconstruct MSE of GPSR;

Fig. 3 is the reconstruct MSE of GPSR-BB;

Fig. 4 is the sparse signal reconfiguring of GPSR-BB.

Embodiment

Below in conjunction with specific embodiment, illustrate the present invention further, these embodiments should be understood only be not used in for illustration of the present invention and limit the scope of the invention, after having read the present invention, the amendment of those skilled in the art to the various equivalent form of value of the present invention has all fallen within the application's claims limited range.

One, the compressed sensing model of photovoltaic array status monitoring network

1, compressed sensing model

The sensor node of the information such as energy collecting temperature, intensity of illumination, voltage, electric current, power is disposed in photovoltaic array status monitoring network.The feature of compressive sensing theory directly compresses data message when node collection signal.

Suppose to include N number of sensor node in WSN, then the perceptual signal vector of node collection is:

In formula: orthogonal basis be generally the known matrix that analytic signal structure obtains, get wavelet transformation orthogonal basis; X represents the one dimension primary signal vector that length is N; Behalf x exists on projection signal's vector, comprise the strict sparse signal vector of K (K<<N) individual nonzero element.

The sampled signal vector of node can be expressed as:

In formula: θ ∈ R ^{m × N}(M<<N) represent calculation matrix, the present invention gets the random matrix that θ is Gaussian distributed; Θ represents CS matrix.

Formula (2) representative is based on the process of the intelligence sample of compressive sensing theory, and y is the result that node sample obtains.From formula (2), high dimensional signal x passes through calculation matrix θ projectable to low dimensional signal y.

The raw sensed signal x of node is passed through orthogonal basis project on sparse signal vector s is one of important step of Signal Compression perception.Another important step is reconstruct sparse signal vector, is the process that known sampled signal vector y and calculation matrix θ solves sparse signal vector s, can according to formula (1) reconstruct primary signal vector x after obtaining s.The compressed sensing process of signal as shown in Figure 1.

As seen from Figure 1, the sampled signal of node is for containing noisy signal.When containing measurement noises n, sampled signal vector can be expressed as:

y＝Θs+n (3)

In formula: n is additive white Gaussian noise.

2, the quadratic programming problem of belt restraining

The underdetermined system of equations that formula (3) is formed for M the N number of unknown number of equation, have infinite solution in theory, therefore, the solution of s is difficult to determine, solve by finding the most sparse signal meeting M observation vector in y, namely signal s is following l ₀the solution of minimum optimization problem:

$\hat{s}=\arg \min {\left|\right|s\left|\right|}_0$ Make y=Θ s (4)

But the minimum optimization problem solving formula (4) is a NP-hard problem, according to l ₀(0 norm) and l ₁the equivalence of (1 norm) optimization problem when solving sparse solution, Lagrange multiplier is used to formula (4), so, Noise signal reconstruction problem can be expressed as quadratic programming (Bound Constrained Quadratic Programming, the BCQP) problem of belt restraining:

$\hat{s}=\frac{1}{2}\underset{s}{\min }\{{\left|\right|y-\mathrm{\&Theta;s}\left|\right|}_2^2+\mathrm{\&tau;}{\left|\right|s\left|\right|}_1\}---\left(5\right)$

In formula: τ >0 represents regular factor, || s|| ₁represent l ₁norm, represent l ₂norm (2 norm).

In conjunction with iterations i, signal s may be split into positive part and negative part, namely has:

s _i＝u _i-v _i，u _i≥0，-v _i≤0 (6)

In formula: u _i=(s _i) ₊, v _i=(-s _i) ₊, i=1,2 ..., N, wherein, form () ₊represent max{0, s}, representative is got on the occasion of operation.

Further, the l of s ₁norm can be expressed as:

${\left|\right|s\left|\right|}_1=1_N^{T}u+1_N^{T}v---\left(7\right)$

In formula: sequence in u, v difference representative formula (6).

Formula (6), (7) are substituted into formula (5), can obtain:

$\hat{s}=\frac{1}{2}\underset{s}{\min }\{{\left|\right|y-\mathrm{\&Theta;}(u-v)\left|\right|}_2^2+\mathrm{\&tau;}(1_N^{T}u+1_N^{T}v)\}---\left(8\right)$

Finally, formula (8) is write as the form of BCQP and is:

${\min }_{z}F\left(z\right)=b^{T}z+\frac{1}{2}{z}^T\mathrm{Bz}---\left(9\right)$

Make z >=0

In formula: $z=\left[\begin{array}{c}u\\ v\end{array}\right],$ b _t＝Θ ^Ty， $b={\mathrm{\&tau;}1}_{2N}+\left[\begin{array}{c}{-b}_t\\ b_t\end{array}\right]$ With $B=\left[\begin{array}{cc}{\mathrm{\&Theta;}}^T\mathrm{\&Theta;}& -{\mathrm{\&Theta;}}^T\mathrm{\&Theta;}\\ -{\mathrm{\&Theta;}}^T\mathrm{\&Theta;}& {\mathrm{\&Theta;}}^T\mathrm{\&Theta;}\end{array}\right].$

Stepping α be on the occasion of, obtain following Grad renewal equation formula:

$z^{(k+1)}=z^{\left(k\right)}-\mathrm{\&alpha;}\&dtri;F\left(z^{\left(k\right)}\right)---\left(10\right)$

In formula: α > 0, for objective function F (z) is in the gradient in kth moment, k represents moment value, $\&dtri;F\left(z\right)=b+\mathrm{Bz}.$

Two, the restructing algorithm of photovoltaic generating system signal

According to the difference of the above account form of stepping factor-alpha, the sparse restructing algorithm of gradient projection can be divided into the sparse restructing algorithm of basic gradient projection and the sparse restructing algorithm of Barzilai-Borwein gradient projection.Under ensureing the condition that algorithm runs in high accuracy, the operational efficiency of raising algorithm, the present invention will improve for these two kinds of algorithms.

1, the basic sparse restructing algorithm of gradient projection

It is line search method that the stepping α of the sparse restructing algorithm of basic gradient projection upgrades what adopt, first defines new direction vector g ⁽ⁱ⁾:

So, each stepping upgraded is expected to meet following equation:

α ₀＝arg min _αF(z ⁽ⁱ⁾-αg ⁽ⁱ⁾) (12)

The closed solution of formula (12) is:

${\mathrm{\&alpha;}}_0=\frac{{\left(g^{\left(i\right)}\right)}^T{g}^{\left(i\right)}}{\left(g^{\left(i\right)}\right)B{g}^{\left(i\right)}}---\left(13\right)$

In formula: for preventing stepping α ₀not too large or too little, get α ₀∈ [α _min, α _max], 0< α _min< α _max.

The basic step of GPSR algorithm is as follows:

1) initial value z is chosen ⁽⁰⁾, β ∈ (0,1), μ ∈ (0,0.5) are according to formula (12) calculated step α ₀, α ₀=mid (α _min, α ₀, α _max), iterations i=0.

2) successively from sequence α ₀, β α ₀, β ²α ₀... middle selection α ⁽ⁱ⁾value, until meet the following conditions:

$\begin{array}{c}F\left(z^{(i+1)}\right)\&le;F\left(z^{\left(i\right)}\right)-\mathrm{\&mu;}\&dtri;F{\left(z^{\left(i\right)}\right)}^T(z^{\left(i\right)}-z^{(i+1)})\\ z^{(i+1)}=(z^{\left(i\right)}-{\mathrm{\&alpha;}}^{\left(i\right)}\&dtri;F\left(z^{\left(i\right)}\right))\end{array}---\left(14\right)$

3) i=i+1, if the inequality meeting formula (15), so, according to z ⁽ⁱ⁺¹⁾export s ⁽ⁱ⁺¹⁾; Otherwise, perform step 1).

$\left|\right|\hat{s}-{(\hat{s}-\mathrm{\&gamma;}\&dtri;F\left(s\right))}_{+}\left|\right|\&le;\mathrm{tolE}---\left(15\right)$

In formula: tolE is the arithmetic number of very little (close to 0), γ be greater than 0 constant.

2, the sparse restructing algorithm of Barzilai-Borwein gradient projection

There is the large defect of operand in basic GPSR, the GPSR based on Barzilai-Borwein mode can calculated step α fast, improves the operational efficiency of algorithm.

Dark for sea matrix is approximated as follows:

In formula: I is unit matrix, value all get and be greater than 1.

So there is following approximated equation:

The basic step of GPSR-BB algorithm is as follows:

1) initial value z is chosen ⁽⁰⁾, α ₀∈ [α _min, α _max], calculated step: ${\mathrm{\&chi;}}^{\left(i\right)}\&ap;{(z^{\left(i\right)}-{\mathrm{\&alpha;}}^{\left(i\right)}\&dtri;F(\left(z^{\left(i\right)}\right))}_{+}-z^{\left(i\right)}.$

2) line search mode is taked to make F (z ⁽ⁱ⁾+ λ ⁽ⁱ⁾χ ⁽ⁱ⁾) minimize, and upgrade estimated value: z ⁽ⁱ⁺¹⁾=z ⁽ⁱ⁾+ λ ⁽ⁱ⁾χ ⁽ⁱ⁾, wherein, λ ⁽ⁱ⁾∈ [0,1].

3) α is upgraded:

In formula: ε ⁽ⁱ⁾=(χ ⁽ⁱ⁾) ^tb χ ⁽ⁱ⁾.

4) i=i+1, if estimated value z ⁽ⁱ⁺¹⁾meet the inequality of formula (15), so, export z ⁽ⁱ⁺¹⁾and stop iteration; Otherwise, perform step 1).

3, continuity regular factor restructing algorithm

Obviously, the minimum value optimization of formula (5) is divided into two aspects, l ₂the effect of norm is restraint speckle ability, l ₁the effect of norm is the sparse capability strengthening algorithm, and regular factor τ can compromise to these two aspects, and this is the process of a dynamic optimization, and therefore, τ does not get definite value usually.The problem that must solve when selecting suitable τ value to be and to reconstruct the sparse signal containing noise.Therefore, the present invention uses for reference the regular factor update mode in document:

τ＝0.005×max(abs(Θ ^Ty)) (19)

From formula (19), regular factor τ is relevant with sampled signal vector y with CS matrix.

Based on formula (19), the present invention proposes the concept of continuous regular factor, is intended to the operation efficiency improving sparse signal reconfiguring algorithm, continuous regular factor τ _lcomputing formula is as follows:

τ _l＝0.8×max_τ/τ

(20)

max_τ＝max(abs(Θ ^Ty))

From formula (20), in algorithm running, τ _lthe trend tapered off, being convenient to algorithm can the algorithm stop condition that represents in formula (15) of rapid advance.

Three, simulation analysis

The assembly that the distributed photovoltaic power generation system of the present invention's research closes PC14 by model by sky is formed, sensor node is when sampled signal, the impact of the noises such as such as shot noise is had in circuit element, available additive white Gaussian noise (Additive White Gaussian Noise, AWGN) removes these noises of matching.Signal to noise ratio in emulation gets SNR=20dB, for voltage parameter SNR=20Log (Singal _amplitude/ Noise _amplitude).

1, GPSR algorithm

Algorithm that the present invention carries can reconstructed distribution formula photovoltaic generating system sparse signal, is the validity of verification algorithm when practical application and using value.Simulation analysis by the distributed photovoltaic power generation system for a certain 250W in Changzhou, the voltage sparse signal of the sparse signal that need reconstruct to be the sampling period on November 10th, 2013 be 5s.The evaluation criterion of photovoltaic generating system sparse signal reconfiguring is signal reconstruction mean square error, is defined as: two norms of the absolute error of reconstruction signal and primary signal.MSE value is less, and the quality reconstruction of signal is better.

The present invention emulates the major parameter got: N=4096, M=1024, K=160, tolE=10 ^-5, α _min=10 ^-30, α _max=10 ³⁰.

Run GPSR algorithm reconstruct voltage sparse signal, final MSE result as shown in Figure 2,3.As seen from Figure 2, continuous regular factor GPSR is fewer than the iterations of GPSR, operation efficiency is high; As seen from Figure 3, continuous regular factor GPSR-BB is fewer than GPSR-BB iterations, operation efficiency is high.

As can be seen from Fig. 2,3, the iterations of GPSR-BB algorithm is more than the iterations of GPSR algorithm, but (MSE end value is 10 all to reach requirement at signal reconstruction MSE ^-3the order of magnitude is close to 0) when, by comparing the operational effect (see table 1) of four kinds of signal reconstruction algorithms, can reach a conclusion: although the iterations of GPSR-BB algorithm is more than GPSR, the time less run, therefore, the former algorithm operational efficiency is higher; No matter be GPSR algorithm or GPSR-BB algorithm, what the present invention introduced improves the operational efficiency of algorithm based on improving one's methods of continuous regular factor.

Table 1 signal reconstruction algorithm operational effect

2, the reconstruct of monitoring network sparse signal

Because GPSR-BB algorithm has the high advantage of operation efficiency, therefore hereafter adopt the method to be reconstructed analysis to sparse signal, signal reconstruction effect as shown in Figure 4.Node sample signal in Fig. 4 is for containing noisy signal, and as seen from the figure, when GPSR-BB algorithm is reconstructed original sparse signal, reconstruct mean square deviation MSE value reaches 10 ^-3the order of magnitude, close to 0, therefore can think that initial sparse signal reconstructs with obtaining integrality.

Claims (7)

1. a photovoltaic array status monitoring network signal reconstructing method, it is characterized in that: in photovoltaic array status monitoring network, dispose the temperature that can to gather photovoltaic generating system, intensity of illumination, voltage, the sensor node of electric current and power information, wireless monitor network is formed in an ad-hoc fashion during each node initializing, monitoring network adopts the compression method of compressive sensing theory, the signal that node-node transmission in monitoring network is compressed into is to other node, before the signal that node processing in monitoring network obtains, need to reconstruct this signal, what adopt is the sparse reconstructing method of gradient projection, adopt the sparse restructing algorithm of basic gradient projection and the sparse restructing algorithm of Barzilai-Borwein gradient projection respectively, improved the operation efficiency of the sparse restructing algorithm of gradient projection by continuous regular factor simultaneously.

2. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 1, is characterized in that: directly compress gathered data message during sensor node collection signal in monitoring network; Suppose to include N number of sensor node in monitoring network, then the perceptual signal vector of node collection is:

In formula: orthogonal basis be generally the known matrix that analytic signal structure obtains, get wavelet transformation orthogonal basis; X represents the one dimension primary signal vector that length is N; Behalf x exists on projection signal's vector, comprise the strict sparse signal vector of K (K<<N) individual nonzero element;

The sampled signal vector of node can be expressed as:

In formula: θ ∈ R ^{m × N}(M<<N) represent calculation matrix, get the random matrix that θ is Gaussian distributed; Θ represents CS matrix;

High dimensional signal passes through calculation matrix projectable to low dimensional signal; In formula (2), the raw sensed signal x of node is passed through orthogonal basis project on sparse signal vector s is one of important step of Signal Compression perception, another important step is the reconstruct to sparse signal vector, be the process that known sampled signal vector y and calculation matrix θ solves sparse signal vector s, can according to formula (1) reconstruct primary signal vector x after obtaining s.

3. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 2, is characterized in that: the interference that can be subject to noise when node obtains the sparse signal of other node-node transmission; When containing measurement noises n, sampled signal vector can be expressed as:

y＝Θs+n (3)

In formula: n is additive white Gaussian noise.

4. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 3, is characterized in that: the reconstruction by the sparse signal of noise jamming can be converted into quadratic programming problem; The underdetermined system of equations that formula (3) is formed for M the N number of unknown number of equation, have infinite solution in theory, therefore, the solution of s is difficult to determine, solve by finding the most sparse signal meeting M observation vector in y, namely signal s is following l ₀the solution of minimum optimization problem:

make y=Θ s (4)

The minimum optimization problem of solution formula (4) is a NP-hard problem, according to l ₀(0 norm) and l ₁the equivalence of (1 norm) optimization problem when solving sparse solution, use Lagrange multiplier to formula (4), so, Noise signal reconstruction problem can be expressed as the quadratic programming problem of belt restraining:

In formula: τ >0 represents regular factor, || s|| ₁represent l ₁norm, represent l ₂norm;

Signal s may be split into positive part and negative part, namely has:

s _i＝u _i-v _i，u _i≥0，v _i≥0 (6)

In formula: u _i=(s _i) ₊, v _i=(-s _i) ₊, i=1,2 ..., N, wherein, () ₊=max{0, s}, representative is got on the occasion of operation;

Further, the l of s ₁norm can be expressed as:

In formula:

Formula (6), (7) are substituted into formula (5), can obtain:

Finally, formula (8) is write as the form of BCQP and is:

Make z >=0

In formula: b _t=Θ ^ty, with

Need to select suitable stepping α and Grad ▽ F (z), following renewal equation formula can be obtained:

z ^(k+1)＝z ^(k)-α▽F(z ^(k)) (10)

In formula: α > 0, ▽ F (z) is for objective function F (z) is in the gradient in kth moment, and k represents moment value, ▽ F (z)=b+Bz.

5. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 1, is characterized in that: what the stepping of the sparse reconstructing method of gradient projection adopted substantially is line search method;

First new direction vector g is defined ⁽ⁱ⁾:

So, each stepping upgraded is expected to meet following equation:

α ₀＝arg min _αF(z ⁽ⁱ⁾-αg ⁽ⁱ⁾) (12)

The closed solution of formula (12) is:

In formula: for preventing stepping α ₀not too large or too little, get α ₀∈ [α _min, α _max], 0< α _min< α _max;

The basic step of GPSR algorithm is as follows:

1) initial value z is chosen ⁽⁰⁾, β ∈ (0,1), μ ∈ (0,0.5) are according to formula (12) calculated step α ₀, α ₀=mid (α _min, α ₀, α _max), iterations i=0;

2) successively from sequence α ₀, β α ₀, β ²α ₀... middle selection α ⁽ⁱ⁾value, until meet the following conditions:

3) i=i+1, if the inequality meeting formula (15), so, according to z ⁽ⁱ⁺¹⁾export s ⁽ⁱ⁺¹⁾; Otherwise, perform step 1);

In formula: tolE is the arithmetic number close to 0, γ be greater than 0 constant.

6. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 1, it is characterized in that: the sparse restructing algorithm of the gradient projection based on Barzilai-Borwein mode (GPSR-BB) can calculated step fast, improves the operational efficiency of algorithm;

Dark for sea matrix is approximated as follows:

In formula: I is unit matrix, select the condition meeting lowest mean square, its value is greater than 1;

So there is following approximated equation:

The basic step of GPSR-BB algorithm is as follows:

1) initial value z is chosen ⁽⁰⁾, α ₀∈ [α _min, α _max], calculated step:

2) line search mode is taked to make F (z ⁽ⁱ⁾+ λ ⁽ⁱ⁾χ ⁽ⁱ⁾) minimize, and upgrade estimated value: z ⁽ⁱ⁺¹⁾=z ⁽ⁱ⁾+ λ ⁽ⁱ⁾χ ⁽ⁱ⁾, wherein, λ ⁽ⁱ⁾∈ [0,1];

3) α is upgraded:

In formula: ε ⁽ⁱ⁾=(χ ⁽ⁱ⁾) ^tb χ ⁽ⁱ⁾;

4) i=i+1, if estimated value z ⁽ⁱ⁺¹⁾meet the inequality of formula (15), so, export z ⁽ⁱ⁺¹⁾and stop iteration, otherwise, perform step 1).

7. photovoltaic array status monitoring network signal reconstructing method as claimed in claim 1, is characterized in that: the sparse restructing algorithm of gradient projection needs to adopt continuous regular factor to improve operation efficiency;

Obviously, the minimum value optimization of formula (5) is divided into two aspects, l ₂the effect of norm is restraint speckle ability, l ₁the effect of norm is the sparse capability strengthening algorithm, and regular factor τ can compromise to these two aspects, and this is the process of a dynamic optimization, and therefore, τ does not get definite value usually; The problem that must solve when selecting suitable τ value to be and to reconstruct the sparse signal containing noise; Regular factor update mode can be:

τ＝0.005×max(abs(Θ ^Ty)) (19)

From formula (19), regular factor τ is relevant with sampled signal vector y with CS matrix;

Based on formula (19), the present invention proposes the concept of continuous regular factor, is intended to the operation efficiency improving sparse signal reconfiguring algorithm, continuous regular factor τ _lcomputing formula is as follows:

τ _l＝0.8×max_τ/τ

(20)

max_τ＝max(abs(Θ ^Ty))

From formula (20), in algorithm running, the trend that tl tapers off, being convenient to algorithm can the algorithm stop condition that represents in formula (15) of rapid advance.