CN107272815A - A kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms - Google Patents
A kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms Download PDFInfo
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Abstract
Unimodal MPPT methods of photovoltaic for existing step factor given in advance are it cannot be guaranteed that adaptability under a variety of service conditions；And can cause the problem of hardware cost is higher containing complicated mathematical operation in the variable step MPPT methods of the existing adaptive step factor.The present invention proposes a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms, and this method adaptively asks for the step factor of MPPT controls using SQP Trust Region Algorithms, and arithmetic is only included in method, reduces the complexity and hardware cost of method.The present invention can be achieved at low cost the unimodal MPPT controls of photovoltaic, and with preferable dynamic response capability, improve the efficiency of photovoltaic generation.
Description
Technical field
The present invention relates to solar energy power generating field, more particularly to a kind of photovoltaic list based on SQP Trust Region Algorithms
Peak MPPT methods.
Background technology
MPPT is the common method for improving photovoltaic generating system operational efficiency.Under the conditions of local shades, photovoltaic array
Multi-peak characteristic may be presented in P-U curves, and according to whether the ability with reply multi-peak problem, photovoltaic MPPT methods are divided into
Unimodal MPPT methods and the class of multimodal MPPT methods two.Although unimodal MPPT methods may be absorbed in office under the conditions of local shades
Portion's power maximum point and fail.It is of great significance however, unimodal MPPT methods still have：On the one hand, multimodal MPPT side
Method still needs to rely on the local tracking that unimodal MPPT methods realize maximum power point after searching near global maximum power point；
On the other hand, the development of micro- inverse technology and modularization multi-level converter technology have also been enlarged the applicable model of unimodal MPPT methods
Enclose.
Conventional perturbation observation method and conductance increment method track maximum power point using fixed step size, but this kind of method is present
The contradiction that step-length is chosen：Big step-length is chosen, preferable dynamic response capability can be obtained, but steady-state operation efficiency can be reduced；Choose
Small step-length, can improve steady-state operation efficiency, but its dynamic response capability can decline.To solve this contradiction, mainly there is fuzzy patrol
Collect control and two kinds of variable step schemes of gradient method.Due to fuzzy logic control need carry out fuzzy set, membership function shape and
The design links such as control rule table, design difficulty is larger, therefore presently relevant research focuses primarily upon the change step based on gradient method
Long MPPT methods.
In gradient method, the step-length of next step is multiplied by a step-length to the derivative of voltage by current point of operation power output
The factor is obtained, and wherein step factor is an important parameter：Step factor cross conference cause reference voltage convergence very it is slow not even
Convergence, and step factor is too small, can reduce dynamic response capability.Choose step factor method be mainly previously set and
Adaptive two kinds.The method of step factor is previously set using relatively simple, but because the P-U curves of photovoltaic module can be with
The change of service condition and change, therefore the step factor being previously set is difficult to ensure that and can obtained at various operating conditions
Preferable performance；And adaptive step factor method contains the complicated mathematical operation such as antitrigonometric function, hardware may be increased
Cost.
The content of the invention
The present invention proposes a kind of unimodal MPPT methods of photovoltaic for being based on SQP (SQP) Trust Region Algorithm, with
Solve it is existing the variable step MPPT methods of step factor are previously set it cannot be guaranteed that adaptability under a variety of service conditions, and
The variable step MPPT methods of the existing adaptive step factor contain complex mathematical operation again, can increase hardware cost
Problem.
Technical scheme is as follows：
A kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms, comprise the following steps：
S1：Initialize the voltage U of photovoltaic array componentref (0)With Trust Region Radius r(0), measurement initial output power P(0)
With power output P(0)To voltage Uref (0)First derivative g(0)；
S2：Initialize power output P(0)To voltage Uref (0)Second dervative B(0), by described power output to electricity
The first derivative g of pressure(0), described power output P(0)To voltage Uref (0)Second dervative B(0)With described Trust Region Radius r(0)Calculate described initial voltage and sound out step-length d(0), while initializing iterations mark k=1；
S3：Update described voltage Uref, presently described power output P is measured again；
S4：Judge whether described power output P meets end condition, if described power output P, which is met, terminates bar
Part, then maintain presently described voltage Uref, method termination；If described power output is unsatisfactory for end condition, S5 is carried out；
S5：Calculation Estimation function ρ；
S6：Pass through the value ρ and preset value ε of comparative evaluation function2Make the following judgment, wherein ε2For positive number, if ρ(k)≥
ε2, then current power output P is measured to voltage UrefFirst derivative g, and update described power output P to voltage Uref
Second dervative B；If ρ(k)＜ ε2, then described voltage U is maderefEqual to the voltage U described in before renewalref, power output P etc.
In the power output P before renewal, power output P to voltage UrefFirst derivative g be equal to update before power output P to voltage
UrefFirst derivative g and power output P to voltage UrefSecond dervative B be equal to update before power output P to voltage Uref
Second dervative B；
S7：Described Trust Region Radius r is updated by described evaluation function value ρ；
S8：By power output P to voltage UrefFirst derivative g, second dervative B and new Trust Region Radius r count again
Voltage step size d is calculated, S3 is returned and is iterated.
The present invention adaptively asks for the step factor of MPPT controls by using SQP Trust Region Algorithms, constantly passes through
Adjust voltage step size d so that voltage causes power constantly " tracking " maximum power value, finally when both differences to being continually changing
Away from less than some preset value ε1When, flow terminates.
In a kind of preferred scheme, the voltage U of photovoltaic array component is initialized in described S1ref (0)=40V, initially
Change Trust Region Radius r(0)Power output P is initialized in=3V and described S2 to voltage UrefSecond dervative B(0)=-1.
In a kind of preferred scheme, the voltage described in described S2 and S8 is soundd out step-length d and carried out by below equation
Ask for：
In formula, r(k)Represent the Trust Region Radius in kth time iteration, g(k)Represent that power output is to voltage in kth time iteration
First derivative, B(k)Represent second dervative of the power output to voltage in kth time iteration.
In a kind of preferred scheme, described voltage U is updated in described S3refAsked for by below equation：
In formula,Represent the voltage of kth time iteration.
In a kind of preferred scheme, the end condition described in described S4 is represented by below equation：
|P(k)-P(k-1)| ＜ ε1
In formula, P(k)Represent the power output in kth time iteration, ε1Span be 0.01W≤ε1≤1W。
In a kind of preferred scheme, the evaluation function described in described S5 is asked for by below equation：
In formula, ρ(k)Represent the evaluation result in kth time iteration.
In a kind of preferred scheme, the ε in described S62=0.25.
In a kind of preferred scheme, power output P is to voltage U in described S6refSecond dervative B pass through following public affairs
Formula is asked for：
In formula, B(k+1)Represent second dervative of the power output to voltage in+1 iteration of kth.
In a kind of preferred scheme, the voltage U described in described S6refEqual to the voltage U described in before renewalref、
Power output P is equal to power output P, power output P before updating to voltage UrefFirst derivative g be equal to update before output
Power P is to voltage UrefFirst derivative g and power output P to voltage UrefSecond dervative B be equal to update before power output
P is to voltage UrefSecond dervative B asked for by below equation：
In formula,Represent the voltage before updating, P(k-1)Represent the power output before updating, g(k-1)Represent before updating
Power output is to the first derivative of voltage, B(k-1)Second dervative of the power output before updating to voltage is represented,Represent to work as
Preceding voltage, P(k)Represent current power output, g(k)Represent the first derivative of current power vs. voltage, B(k)Represent current
Power vs. voltage second dervative.
In a kind of preferred scheme, the Trust Region Radius r described in described S7 is asked for by below equation：
In formula, rmaxFor the Trust Region Radius upper limit, rmax=3V, ε3=0.75.
Compared with prior art, the beneficial effect of technical solution of the present invention is：
The present invention only includes simple arithmetic, reduces the complexity of method in addition to multilevel iudge；Utilize SQP
Trust Region Algorithm adaptively asks for step factor, it is ensured that can obtain preferable performance at various operating conditions.
Brief description of the drawings
Fig. 1 is embodiment method flow diagram.
Fig. 2 is this case embodiment 1 and comparative example 2, the voltage waveform of comparative example 3 under standard test condition.Figure
3 be this case embodiment 1 and comparative example 2, the power waveform of comparative example 3 under standard test condition.
Embodiment
Accompanying drawing being given for example only property explanation, it is impossible to be interpreted as the limitation to this patent；
To those skilled in the art, it is to be appreciated that some known features and its explanation, which may be omitted, in accompanying drawing
's.
Technical scheme is described further with reference to the accompanying drawings and examples.
The method of embodiment 1 performs step schematic diagram as shown in Figure 1, comprises the following steps：
A kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms, comprise the following steps：
S1：Initialize the voltage U of photovoltaic array componentref (0)=40V and Trust Region Radius r(0)=3V, measurement is initial defeated
Go out power P(0)With power output P(0)To voltage Uref (0)First derivative g(0)；
S2：Initialize power output P(0)To voltage Uref (0)Second dervative B(0)=-1, by power output to voltage
First derivative g(0), power output P(0)To voltage Uref (0)Second dervative B(0)With Trust Region Radius r(0)Calculate initial voltage
Sound out step-length d(0), while initializing iterations mark k=1；
S3：Update voltage Uref, present output power P is measured again；
S4：Judge whether power output P meets | P(k)-P(k-1)| ＜ ε1If power output P is met | P(k)-P(k-1)| ＜
ε1, then current voltage U is maintainedref, method termination；If power output P is unsatisfactory for | P(k)-P(k-1)| ＜ ε1, then S5 is carried out, wherein,
ε1=0.01W.
S5：Calculation Estimation function ρ, ρ are asked for by below equation：
In formula, ρ(k)Represent the evaluation result in kth time iteration.
S6：Pass through the value ρ and preset value ε of comparative evaluation function2Make the following judgment, wherein ε2=0.25, if ρ (k)≥
ε2, then current power output P is measured to voltage UrefFirst derivative g, and update power output P to voltage UrefSecond order
Derivative B；If ρ(k)＜ ε2, then voltage U is maderefEqual to the voltage U before renewalref, power output P be equal to update before power output
P, power output P are to voltage UrefFirst derivative g be equal to update before power output P to voltage UrefFirst derivative g and defeated
Go out power P to voltage UrefSecond dervative B be equal to update before power output P to voltage UrefSecond dervative B；
S7：Trust Region Radius r is updated by evaluation function value ρ；
S8：By power output P to voltage UrefFirst derivative g, second dervative B and new Trust Region Radius r count again
Voltage step size d is calculated, S3 is returned and is iterated.
Voltage is soundd out step-length d and asked for by below equation in S2 and S8：
In formula, r(k)Represent the Trust Region Radius in kth time iteration, g(k)Represent that power output is to voltage in kth time iteration
First derivative, B(k)Represent second dervative of the power output to voltage in kth time iteration.
Voltage U is updated in S3refAsked for by below equation：
In formula,Represent the voltage of kth time iteration.
If ρ in S6(k)≥ε2, power output P is to voltage UrefSecond dervative B asked for by below equation：
In formula, B(k)Represent second dervative of the power output to voltage in kth time iteration.
If ρ in S6(k)<ε2, voltage UrefEqual to the voltage U before renewalref, power output P be equal to update before output work
Rate P, power output P are to voltage UrefFirst derivative g be equal to update before power output P to voltage UrefFirst derivative g and
Power output P is to voltage UrefSecond dervative B be equal to update before power output P to voltage UrefSecond dervative B by with
Lower formula is asked for：
In formula,Represent the voltage before updating, P(k-1)Represent the power output before updating, g(k-1)Represent before updating
Power output is to the first derivative of voltage, B(k-1)Second dervative of the power output before updating to voltage is represented,Represent to work as
Preceding voltage, P(k)Represent current power output, g(k)Represent the first derivative of current power vs. voltage, B(k)Represent current
Power vs. voltage second dervative.
Trust Region Radius r is asked for by below equation in S7：
In formula, rmaxFor the Trust Region Radius upper limit, rmax=3V, ε3=0.75.
Embodiment 2：Using the variable step MPPT methods based on the angle of contingence.
Embodiment 3：Using the variable step MPPT methods of the fixed step size factor.
Embodiment 1~3 uses model SPR-300E-WHT-D photovoltaic module.The model photovoltaic module is in temperature 25
DEG C and intensity of illumination is respectively 200W/m2、400W/m2And 1000W/m2When maximum power point voltage be respectively 52.51V,
53.71V and 54.70V, peak power output is respectively 57.68W, 117.97W and 300.06W；In intensity of illumination 1000W/m2,
Maximum power point voltage when temperature is respectively 35 DEG C and 45 DEG C is 52.86V and 51.02V, and peak power output is respectively
291.21W and 282.23W.In normal operation, it is believed that the output voltage of photovoltaic module is equal to the reference electricity that MPPT is controlled
Pressure.The sampling period that the MPPT of embodiment 1~3 is controlled is 0.01s, and MPPT starts when being controlled in t=0.01s, the time be 0s extremely
0.9s；Initial voltage is 40V, and the step-length upper limit is 3V, and upper voltage limit and lower limit are respectively 20V and 60V, the end of three kinds of embodiments
Only condition is
|P(k)-P(k-1)| ＜ 0.01W
In standard test condition (25 DEG C, 1000W/m2) under gained reference voltage waveform and power output waveform difference
As shown in Figures 2 and 3.
The test result of embodiment 1~3 is shown in Table lattice 1 at five kinds of temperature and the service condition of illumination.In form 1, the time is
Refer to since being controlled MPPT to the time that end condition is consumed is reached, voltage and power output refer to respectively reaches end condition
When photovoltaic module output voltage and power output, energy refers to start to photovoltaic gust when reaching end condition since MPPT controls
Arrange the difference of the electric energy total amount of the electric energy total amount maximum that can be exported and reality output.
Embodiment test result is analyzed, can obtain and such as draw a conclusion：
Contrast the test result of same embodiment at different conditions, it is known that the inventive method (embodiment 1) is adapted to not
With intensity of illumination and temperature conditionss.
Contrast under identical service condition the not test result of be the same as Example, it is known that the inventive method (embodiment 1) is in each fortune
Dynamic response capability under the conditions of row is superior to other method (embodiment 2 and embodiment 3), reaches the time of maximum power point
Most short, the energy loss caused is smaller, it is possible to increase the efficiency of photovoltaic generation.The term of position relationship is only used described in accompanying drawing
In exemplary illustration, it is impossible to be interpreted as the limitation to this patent；
Obviously, the above embodiment of the present invention is only intended to clearly illustrate example of the present invention, and is not pair
The restriction of embodiments of the present invention.For those of ordinary skill in the field, may be used also on the basis of the above description
To make other changes in different forms.There is no necessity and possibility to exhaust all the enbodiments.It is all this
Any modifications, equivalent substitutions and improvements made within the spirit and principle of invention etc., should be included in the claims in the present invention
Protection domain within.
The embodiment test result of form 1
Claims (10)
1. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms, it is characterised in that comprise the following steps：
S1：Initialize the voltage of photovoltaic array componentWith Trust Region Radius r(0), measurement initial output power P(0)And output
Power P(0)To voltageFirst derivative g(0)；
S2：Initialize power output P(0)To voltageSecond dervative B(0), by described power output to the one of voltage
Order derivative g(0), described power output P(0)To voltage Uref (0)Second dervative B(0)With described Trust Region Radius r(0)Calculate
Described initial voltage sounds out step-length d(0), while initializing iterations mark k=1；
S3：Update described photovoltaic array component voltage Uref, presently described power output P is measured again；
S4：Judge whether described power output P meets end condition, if described power output P meets end condition, tie up
Hold presently described voltage Uref, method termination；If described power output is unsatisfactory for end condition, S5 is carried out；
S5：Calculation Estimation function ρ；
S6：Pass through comparative evaluation function ρ and preset value ε2Make the following judgment, wherein ε2For positive number.If ρ(k)≥ε2, then measurement ought
Preceding power output P is to voltage UrefFirst derivative g, and update described power output P to voltage UrefSecond dervative B；
If ρ(k)＜ ε2, then described voltage U is maderefEqual to the voltage U described in before renewalref, power output P be equal to update before it is defeated
Go out power P, power output P to voltage UrefFirst derivative g be equal to update before power output P to voltage UrefFirst derivative
G and power output P are to voltage UrefSecond dervative B be equal to update before power output P to voltage UrefSecond dervative B；
S7：Described Trust Region Radius r is updated by described evaluation function value ρ；
S8：By power output P to voltage UrefFirst derivative g, second dervative B and new Trust Region Radius r recalculate voltage
Step-length d is soundd out, S3 is returned and is iterated.
2. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 1, it is characterised in that
The voltage of photovoltaic array component is initialized in described S1Initialize Trust Region Radius r(0)=3V and described S2
Middle initialization power output P is to voltage UrefSecond dervative B(0)=-1.
3. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 2, it is characterised in that
Voltage described in described S2 and S8 is soundd out step-length d and asked for by below equation：
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In formula, r(k)Represent the Trust Region Radius in kth time iteration, g(k)Represent that power output is to the one of voltage in kth time iteration
Order derivative, B(k)Represent second dervative of the power output to voltage in kth time iteration.
4. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 3, it is characterised in that
Described voltage U is updated in described S3refAsked for by below equation：
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In formula,Represent the voltage of kth time iteration.
5. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 3 or 4, its feature exists
In the end condition described in described S4 is represented by below equation：
|P(k)-P(k-1)| ＜ ε1
In formula, P(k)Represent the power output in kth time iteration, ε1Span be 0.01W≤ε1≤1W。
6. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 5, it is characterised in that
Evaluation function described in described S5 is asked for by below equation：
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In formula, ρ(k)Represent the evaluation function value in kth time iteration.
7. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 3 or 4 or 6, its feature
It is, the ε in described S62=0.25.
8. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 7, it is characterised in that
In described S6, work as ρ(k)≥ε2When, power output P is to voltage UrefSecond dervative B asked for by below equation：
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<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
</mrow>
<msup>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
</mfrac>
</mrow>
In formula, B(k)Represent second dervative of the power output to voltage in kth time iteration.
9. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 7, it is characterised in that
In described S6, work as ρ(k)<ε2When, described voltage UrefEqual to the voltage U described in before renewalref, power output P be equal to more
Power output P, power output P before new is to voltage UrefFirst derivative g be equal to update before power output P to voltage Uref's
First derivative g and power output P are to voltage UrefSecond dervative B be equal to update before power output P to voltage UrefSecond order
Derivative B, is asked for by below equation：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>U</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>f</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>f</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
In formula,Represent the voltage before updating, P(k-1)Represent the power output before updating, g(k-1)Represent the output work before updating
Rate is to the first derivative of voltage, B(k-1)Second dervative of the power output before updating to voltage is represented,Represent current electricity
Pressure, P(k)Represent current power output, g(k)Represent the first derivative of current power vs. voltage, B(k)Represent current power
To the second dervative of voltage.
10. a kind of unimodal MPPT methods of photovoltaic based on SQP Trust Region Algorithms according to claim 6, it is characterised in that
Trust Region Radius r described in described S7 is asked for by below equation：
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mrow>
<mo>|</mo>
<msup>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
<mo>|</mo>
</mrow>
<mo>/</mo>
<mn>2</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo><</mo>
<msub>
<mi>&epsiv;</mi>
<mn>2</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>&epsiv;</mi>
<mn>2</mn>
</msub>
<mo>&le;</mo>
<msup>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>&le;</mo>
<msub>
<mi>&epsiv;</mi>
<mn>3</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>min</mi>
<mo>{</mo>
<mn>2</mn>
<msup>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
<mo>,</mo>
<msub>
<mi>r</mi>
<mi>max</mi>
</msub>
<mo>}</mo>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>&rho;</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>></mo>
<msub>
<mi>&epsiv;</mi>
<mn>3</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
In formula, rmaxFor the Trust Region Radius upper limit, rmax=3V, ε3=0.75.
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| CN104806450A (en) * | 2015-03-25 | 2015-07-29 | 华北电力大学（保定） | Universal gravitation neural network based wind power system MPPT control method |
| US20150331972A1 (en) * | 2014-05-16 | 2015-11-19 | HST Solar Farms, Inc. | System & methods for solar photovoltaic array engineering |
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| CN104806450A (en) * | 2015-03-25 | 2015-07-29 | 华北电力大学（保定） | Universal gravitation neural network based wind power system MPPT control method |
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