A MPPT Method based on Improved Fibonacci Search Photovoltaic Array

The P-U curve of photovoltaic arrays (PVAs) has multi-peak characteristics under uneven illumination environments, and the maximum power point tracking (MPPT) strategy for the single peak may fail. An improved Fibonacci search algorithm is proposed to carry out MPPT of photovoltaic arrays under uniform illumination or light mutation. A multiple-interval search algorithm based on a circuit analysis method is presented for different topology arrays and illumination distributions. The series array analysis adopts the current analysis method, the parallel array analysis adopts the voltage analysis method, and the series and parallel array analysis adopts the current and voltage analysis method; then, the output control volume is determined, and the search interval is divided. The search steps in the Fibonacci method and the real-time changes of parameters in the optimization process can be observed by MATLAB simulation. Experimental results show that the algorithm uses less computation time and small area search instead of global search.


INTRODUCTION
Solar power as a renewable energy source is an important way to solve the energy crisis.It is essential to operate the PVA source at maximum power point (MPP) due to its low energy conversion efficiency.PV cells have complex nonlinear output characteristics for the influence of light intensity and temperature, but there is always only one MPP on the output curve.Many MPPT algorithms have been proposed by scholars in order to maximize the conversion of light energy into electrical energy and make the PV cells operate at the MPP.Under a uniform illumination environment (UIE), there is only one peak value on the power-voltage (P-U) curve of photovoltaic arrays.Commonly used MPPT control algorithms in this case include the constant voltage method, perturbation and observation (P&O) method and incremental conductance method [1][2][3][4].PVAs are usually affected by shadows such as trees, billboards and black clouds in applications.The MPPT control method based on sampling data easily falls into local peaks because PV arrays output a multi-peak curve with several peaks in multi-intervals under nonuniform illumination environments (NIEs).Now, a large number of research results for multi-peak MPPT algorithms have been published.These results are mainly divided into algebraic algorithms [5][6][7][8][9][10] and artificial intelligence algorithms [11][12][13][14][15][16].The artificial intelligence algorithms have good adaptability and accuracy independent of the specific mathematical model of the PV array.However, they require a large amount of operation data and experience accumulation developed by complex circuits.Moreover, there are random power oscillations in the MPPT process, which brings great disturbance to the load or power grid.The algebraic algorithm has a small calculation, clear MPPT search trajectory and simple circuit, but the accuracy is closely related to the mathematical model of the PV array.The error of MPPT under NIE still exists.
To overcome the limitations of the above algorithms, an improved Fibonacci search algorithm is adopted in this paper.First, the algorithm ascertains array topology, determines the type of light by measuring the data, and pretreats the characteristic curve of the system.The range of the global MPP is narrowed directly, and the MPPT can be realized quickly under UIE.A search interval is redefined to avoid misjudgment when an abrupt illumination is detected.Dividing the multiple search intervals according to the level of illumination and the array structure under NIE, the local peaks are obtained by the improved Fibonacci algorithm in each interval.The maximum power of the whole output is obtained by comparing several local extreme values.

THE MATHEMATICAL MODEL OF PV ARRAY
A PV module comprises several PV batteries in series, and a PV array comprises PV modules connected in series, parallel or series parallel.

A Double-Diode Mathematical Model for PV Battery
A PV battery model based on double-diode [17] is shown in Fig. 1.
where: I PV -photo current; R s , R p -battery equivalent series and parallel resistance.
The expression of output I and V is as follows:

Mathematical Model of PV Module
The number of series battery N s ( ) ( ) where: open circuit voltage V ocn and short circuit current I scn under standard conditions; T s (25.°C) -standard temperature, ΔT = T − T s ; Illumination G and standard illumination D s (1000 W/m 2 ).

The Mathematical Model of Array
The mathematical model in series array (SA) composed of N sa PV modules is as follows: The mathematical model in parallel array (PA) composed of N pa PV modules: The mathematical model in series parallel of N sa and N pa PV modules:

FIBONACCI SEARCH ALGORITHM 3.1 Search Principle
If the integer sequence {F n } (n=0,1, ...) meets the following conditions,{F n } is called a Fibonacci sequence.1 F n is the nth Fibonacci number and F n−1 /F n is Fibonacci fraction.When an interval is shortened with the nth exploration point, the length shortenings are (2) Select the initial exploration points t 1 and t 2 to satisfy t 1 < t 2 on the interval [a, b]: (3) Calculate the functions f(t 1 ) and f(t 2 ), as shown in Figs.

2(a) and (b).
If f(t 1 ) > f(t 2 ), eliminate the interval (t 2 , b) then move the search interval to the left, setting a 1 = a and b 1 = t 2 on the interval (a, t 2 ), the new test points are t 12 = t 1 and If f(t 1 ) < f(t 2 ), eliminate the interval (a, t 1 ) then move the search interval to the left, setting a 1 = t 1 and b 1 = b on the interval (a, t 2 ), the new test points are t 11 = t 2 and The values of the Fibonacci sequence are The algorithm is an accurate one-dimensional search method of variable step size with a high computational efficiency which uses symmetry to construct the step length.The target function is calculated only once in the search process so that the computational efficiency is high.There is no need to search the entire output voltage range of the PV array, which improves the response speed of the MPPT and reduces the power loss for disturbance.Based on a precise mathematical model, the algorithm can ensure the accuracy of the search results and is suitable for the hardware structure of the traditional MPPT.

The Improved Fibonacci Search Method
The improved Fibonacci method is described as follows: (1) Increase the end conditions and reduce the search numbers The value of the function f(t) is very sensitive to the independent variable t, and small changes of t will make f(t) have a larger error in the vicinity of the MPP.In general, the smaller the tolerance Delta δ is, the more accurate the results are and the longer search times are.Taking |t1 − t 2 | ≤ δ and |f(t 1 ) -f(t 2 )| ≤ δ as the terminating conditions can significantly reduce the search times under the same δ.
(2) The final value is determined by the bisection search method In order to further improve the accuracy of the results, the single bisection search method is used to determine the final f(t) according to the N−1 search results, which can reduce the error caused by the concussion near the MPP.
(3) Sectioning search on multi-intervals The multi-intervals are divided according to the topology and illumination distribution of photovoltaic arrays and each interval has only one local extreme point.Then, the improved Fibonacci method is used in each subinterval to MPPT, and the global MPP is obtained.It can eliminate the faulty judgment theoretically and can be applied to any topological photovoltaic array under nonuniform illumination.(4) When the illumination changes, the parameters are initialized again to determine the new search interval according to the measured illumination variation value.

IMPLEMENTATION OF THE MPPT ALGORITHM
The entire interval of the PV array under a UIE and the subinterval under the NIE meet the single peak characteristics of output.The variable t can express the output voltage or current according to the topology of the array, and the f(t) is the output power of the array.The control flow chart of the MPPT with improved Fibonacci is shown in Figs. 3 and 4. First, determine the topology of the array and judge the illumination environment of the array, then judge whether the illumination mutation occurs according to the change of illumination and output power, which decide whether to redefine the search interval to cope with the uncertainty of the external environment.When searching for k = 5, t 1 = 26.2289V, t 2 = 199.8918V, f(t 1 ) = 199.8862W, f(t 2 ) = 199.8918W, the final result t is found by the single bisection method.That is, t = (t 2 + t 1 )/2 =26.2604V, f(t) = 199.9018W. The difference ΔP m between f(t) and the standard value P m is 0.0982 W. The error ΔV m between t and V m is 0.0504 V.The tracking time t is 0.3 s.
When the illumination changes, such as from G = 1000 W/m 2 to 600 W/m 2 , the power also changes.For example, when setting the ΔP = 0.2, when the power change value is greater than the ΔP, the illumination should be remeasured, and the search interval is defined.The search process is described in Tab. 2, and the power output curve is shown in Fig. 6  The search results are: t = (t 2 + t 1 )/2 =26.5639V, f(t)=119.6276W. Ideally, the maximum power of the module P m is 119.6699W and V m is 26.5554V under 600 W/m 2 , ΔP m is 0.0423 W. The tracking time is adjusted to 0.4 s.Thus, it can be seen that the Fibonacci search can track quickly and accurately to the MPP as the illumination changes.From Tab. 1, it is not advisable to expand the search range in the traditional method, when the direction of the two searches is the same.Because even light does not change, the search interval is not alternating directions each step instead it is moving two or three times in one direction, which causes miscarriage of Justice.In practical applications, the change values of detection illumination and power are used to define the interval, so that the search speed is faster and the result is more accurate.

NIE
According to the flow chart in Fig. 4, the MPPT of the arrays with different topological structures is discussed respectively under NIE.

SA
Taking two modules in series as an example, the parameters are as follows: G1 = 1000 W/m 2 , G 2 = 600 W/m 2 , I cs1 = 8.2279A,I cs2 = 4.9367 A. The current analysis method is adopted as the current of the series module is equal.The output voltages are the sum of the output voltage of two modules, which can be calculated by the formula (4).In SA, the illumination level determines the number of local extreme points, which can be predicted by [0.8•I cs , I cs ], and the location of the global MPP is related to the illumination distribution and module number [18].Therefore, the MPPT is realized by controlling the output current.The power curve has two local power extremes under two illumination levels.When the output current is on the interval [0, I cs1 ], the two modules supply the power simultaneously, and there is a local power point P1.When the current is on the interval [I cs1 , I cs2 ], due to the bypass diode, only G 2 supplies the power, and there is another local point P2.Therefore, there are two search intervals, such as those in Tabs. 3 and 4, the search curve is shown in Fig. 7.In the detection point distribution shown in Fig. 8, the red and blue dots represent points t 1 and t 2 respectively.As shown in Fig. 8, in the two illumination conditions, the SA output presents two peaks, and the global MPP is 267.3118W. As seen in Fig. 7, the first local MPP is achieved at 0.3 s and the second at 0.5 s, which completes the entire search process in 0.6 s with a fast search speed Technical Gazette 26, 1(2019), 163-170 and stable output power.Similarly, M ranges of [0.8•I cs1 , I cs1 ], [0.8•I cs2 , I cs2 ], …, [0.8•I csM , I csM ] are divided when illumination level values are arranged from big to small under the M illumination levels in SA.Then, the global MPP can be derived from M local extremum points.It is obviously difficult for the traditional Fibonacci algorithm to find the global MPP even if the search interval is enlarged in the case of multi-level illumination and multiple local extremums points.The improved Fibonacci search can overcome the limitations of traditional algorithms and can be used for MPPT with any level of illumination.

PA
Taking two modules in parallel as an example, the parameters are as follows: G 1 = 1000 W/m 2 , G 2 = 600 W/m 2 , V oc1 = 32.9V, V oc2 = 32.2V.The output voltage of the parallel modules is equal, and the voltage analysis method is adopted.The output current is the sum of the two modules calculated by Eq. ( 5).Therefore, the MPPT is realized by controlling the output voltage.If the output voltage of each module for the parallel array is equal, the output power is superimposed on the output power of each module.If not equal, the output voltage and power will be superimposed by the maximum output voltage and power of the parallel modules.The output curve has a single peak characteristic, which is close to the UIE, and the level of illumination determines the composition number of the line segment [17].Because the open circuit voltage is mainly related to the temperature, it is less affected by the illumination and the equation of Voc = V oc1 is adopted.The initial parameters for the search interval are: a = 0.75•V oc1 = 24.6750V, b = 0.85•V oc1 = 29.6100V, δ = 0.01.The search process and the curve are shown in Tab. 5 and Fig. 9.
The MPP is: t = 26.3315V, f(t) = 328.2675W/m 2 .The ΔP is 0.0092 W compared with the theoretical value P = 328.2767W. The output voltage of PA under two illumination levels has a turning point due to the series blocking diode, which causes the curve to be composed of two parts.The output curve is composed of M sections for a parallel array with an arbitrary illumination level of M, but the single peak characteristic does not change, which makes MPPT implementation easier, the tracking accuracy higher and the speed faster.The output power of the array is calculated using formula (6), that is P = V(I 1 +I 2 ).Only the G 1 module works in branch I, and two modules works in branch II, so the output voltage in branch I is greater than branch II in the interval [I cs2 , I cs1 ].Only branch II outputs power since the blocking diode is reversed cut-off.The corresponding voltage V is calculated from the current value I of the detection points by taking the current as the search interval.The improved Fibonacci algorithm is used to carry out local MPPT for SPA on the interval [0, I cs2 ] and [I cs2 , I cs1 ], then the MPP can be obtained.The voltage value is calculated in a large current range to ensure that the search voltage interval is completely covered and the extreme points can be determined rapidly.The initial interval for search is: a = 2.0000 A, b = 4.9460 A in the current interval [0, I cs2 ].The voltages are calculated by the current detection points.The search process is shown in Tab. 6.  Local MPP P1 is: t = 4.8925 A, f(V) = 669.8263W, which is compared with the theoretical value P 1 = 69.8743W, the ΔP is 0.048 W.
When the current is in [I cs2 , I cs1 ], the initial search interval is: a = I cs2 = 4.9460 A, b = I cs1 =8.2289 A. The search process is shown in Tab.7 and the output P-V curve is shown in Figs. 10 and 11.
Local MPP P2 is: t = 7.2972 A, f(V) = 438.2817W, which is compared with the theoretical value 438.2911W and the ΔP is 0.094 W.
As shown in Figs. 10 and 11, the output of the SPA shows the current characteristics in the current range, and there are 2 local extreme points under two illumination levels, which are located in two current ranges.Red and blue dots represent detection points t1 and t 2 respectively.The output curve shows parallel characteristics in the voltage range which is composed of three curves in three voltage intervals [17].In M×N SPA, first the current intervals are determined according to the level of illumination, the voltage interval is determined according to the parallel branch.Then, the values of the voltage detection points are calculated by the current detection points, and finally the power is calculated.To track the whole search interval, the whole current interval is taken to ensure that the local extreme points have corresponding convergent voltage values.

CONCLUSION
The MPPT algorithm based on an improved Fibonacci search is proposed based on the accurate mathematical model of PV arrays and increased measurement of environmental parameters.The parameters are used to judge the change and distribution of light in order to provide a basis for MPPT.The MPP is tracked quickly and accurately by dividing the multiple tracking intervals and determining the control voltage or current, and the system is stable at the MPP under the condition of NIV.The algorithm has good anti-interference ability.It can track the global MPP for any environment and PV array topology and improve the output efficiency of PV arrays.

Figure 1
Figure 1 The equivalent circuit of double-diode model for the PV battery and F 1 /F 2 respectively.If the function f(t) is monotonous on the interval I = [a, b] and there exists a unique number P ∈ I such that f(t) is increasing on [a, P] and decreasing on [P, b].The Fibonacci search step is as follows: (1) Select the initial data and determine the search interval [a, b], then calculate the search number n according to the search precision value, δ > 0.

2
Move to the left (b) Move to the left Figure Fibonacci search diagram

Figure 3
Figure 3 MPPT flow chart of the Fibonacci search under UIE

I0 4 .
128×10 −10 A Photocurrent IPV 8.22 A Series resistance Rs 0.335 Ω Parallel resistance Rp 155.48 Ω The simulation uses a single module in order to compare with the module standard output P m and V m .Parameters from Tab. 1 are brought into the formula (3) and the simulation experiment is carried out by the m file of MATLAB according to the flow chart of Fig. 3.The output voltage is controlled and the initial voltage interval parameters are set as follows: G = 1000 W/m 2 , T = 25 °C, a = 0.75•V ocn = 24.675V, b = 0.85•V ocn =27.965V, δ = 0.01.The search process is shown in Tab. 2. The local output curve of the tracking results is shown in Fig. 5.

Figure 5 A
Figure 5 A local magnification chart of P-t for P=1000 W/m 2 . The new interval parameters are: δ = 0.01, a = 0.75•V oc2 = 24.15V, b = 0.85•V oc2 = 27.37 V. V oc2 is the open circuit voltage of the illumination of 600 W/m 2 and V oc2 is 32.19 V.

Figure 7 Figure 8
Figure 7The output power of SA

Figure 9
Figure 9 Output f(t)-t of parallel module

Figure 10 P
Figure 10 P-V output curve of SPA

Table 1
Electrical parameters for improved double-diode model

Table 2
Search data records under UIE
In the second interval, a = 0.8•I cs2 = 3.9494 A, b = I

Table 4
The search data record of the current in the interval [I cs1 , I cs2 ]

Table 5
The search data record of PA

Table 6
The SPA search data record in the interval [0, I cs2 ]

Table 7
SPA searching data record in the interval [I cs2 , I cs1 ]