The Output Characteristics of the Photovoltaic Array for Laser Wireless Power Transmission Within Non-Kolmogorov Turbulence considering Pointing Error

This paper examines the output characteristics of the photovoltaic (PV) array in a laser wireless power transmission (LWPT) system considering atmospheric turbulence and pointing error <italic>r</italic> (<italic>r</italic>, <italic>θ</italic>). Firstly, we propose a theoretical model for studying the laser energy of each PV cell in non-Kolmogorov turbulence by introducing power in bucket (PIB). Then, to obtain the output characteristics of the PV array, we build the equivalent model based on the carrier continuity equation and bipolar transport equation. The results demonstrate that the power at the maximum power point (MPP) <italic>P</italic><sub>mpp</sub> and the system efficiency <italic>η</italic><sub>2</sub> descend 50.9% and 51.5%, in the cases of <italic>r</italic> (20 mm, 45°) compared with <italic>r</italic> (0, 0), respectively, when the transmission distance is 1000 m. The variation range (5.1 V to 22.3 V) of the voltage at MPP <italic>V</italic><sub>mpp</sub> with <italic>r</italic> (10 mm, 45°) is distinctly larger than that (21.9 V to 23.2 V) with <italic>r</italic> (0, 0) when the transmission distance varies from 0 m to 2000 m. The findings of this study are valuable for the design of DC-DC converter, maximum power point tracking (MPPT) algorithm and the closed loop control circuit in LWPT system to make it efficient and stable operation.

Abstract-This paper examines the output characteristics of the photovoltaic (PV) array in a laser wireless power transmission (LWPT) system considering atmospheric turbulence and pointing error r (r, θ).Firstly, we propose a theoretical model for studying the laser energy of each PV cell in non-Kolmogorov turbulence by introducing power in bucket (PIB).Then, to obtain the output characteristics of the PV array, we build the equivalent model based on the carrier continuity equation and bipolar transport equation.The results demonstrate that the power at the maximum power point (MPP) P mpp and the system efficiency η 2 descend 50.9% and 51.5%, in the cases of r (20 mm, 45°) compared with r (0, 0), respectively, when the transmission distance is 1000 m.The variation range (5.1 V to 22.3 V) of the voltage at MPP V mpp with r (10 mm, 45°) is distinctly larger than that (21.9 V to 23.2 V) with r (0, 0) when the transmission distance varies from 0 m to 2000 m.The findings of this study are valuable for the design of DC-DC converter, maximum power point tracking (MPPT) algorithm and the closed loop control circuit in LWPT system to make it efficient and stable operation.
Index Terms-Laser wireless power transmission, nonkolmogorov turbulence, output characteristics, photovoltaic array, pointing error.

I. INTRODUCTION
O VER the past decades, laser wireless power transmission (LWPT) technology (as depicted in Fig. 1), has attracted increasingly research interests as a means of supplying energy between space stations, satellites, and unmanned aerial vehicles [1], [2].The output characteristics of the photovoltaic (PV) array, as the input source connects to the DC-DC converter, in the LWPT system have significant impact on its conversion efficiency, maximum power point tracking (MPPT)algorithm, and closed-loop control loop.Therefore, the output characteristics of the PV array, serving as the pivotal issue in LWPT, have received considerable attention [3], [4].
In practice applications, the PV array is composed by multiple PV cells to overcome the limitation of individual PV cells, which typically have insufficient output voltage and current.However, it's worth noting that the output currents of the PV cells within a series module may not match when exposed to unequal laser illumination, resulting in multiple peaks in the P-V curve and causing power loss.Moreover, in a conventional LWPT system, the laser beam may be affected by atmospheric turbulence, leading to spot distortion ad variations in irradiation intensity.Additionally, the pointing error is nonnegligible in tracking and aiming at the PV array installed on the dynamic target.As a consequence, the laser intensity distribution on the PV array renders more irregular, and then the output voltage and power of the PV array fluctuate more drastically, which will compromise the stability and operational security of the LWPT system.And the instability and volatility of the output voltage and power of the PV array will cause the DC-DC converter fail to achieve stable control, effective conversion, and efficient storage of the power from PV array.Thus, investigating the output characteristics and stability of the PV array in LWPT under atmospheric turbulence is crucial.
Recently, a number of studies have been done on the output characteristics of the PV cells [5], [6], [7], [8], [9], [10], [11].He et al studied the efficiency of the GaAs PV cell with respect to wavelength, laser intensity and temperature [5].Zhang evaluated the impacts of laser parameters and the PV cell profile on laser cells efficiencies [6].Meng studied multi-field coupling characteristics of the PV cell under non-uniform laser beam irradiance [7].Li analyzed the performance, in particular, the electrical efficiency of the PV cell under non-uniform and uniform distributions [8].The previous researches on PV cells have focused on output characteristics under different conditions, but has not fully addressed the issue of non-uniform illumination causing mismatch in the PV array.A limited number of researchers  have examined the effects of intensity distribution on the PV array output.Okan compared and analyzed the power, efficiency and mismatch loss of the PV array with different configurations under partial shading conditions [9].To optimize and improve the output power and efficiency of PV array, Zhou proposed an optimal PV array configuration search mechanism for arbitrarily sized PV arrays to enhance the power under Gaussian laser beam condition [10].However, the effects of atmospheric turbulence on the LWPT system have received comparatively little research attention.Meanwhile, the transmission characteristics of laser beams in atmospheric turbulence have been extensively studied [12], [13], [14], [15], with a particular focus on beam wander, intensity fluctuation, and spot distortion.However, the abovementioned studies have not been applied on LWPT systems.
In this paper, we focus on the output characteristics of the PV array for LWPT within non-Kolmogorov turbulence considering pointing error and provide theoretical support for the design of the DC-DC converter, the maximum power point tracking (MPPT) algorithm and the closed loop control circuit in LWPT system, which will further enhance the output power, the laserto-electricity conversion efficiency and the output stability of the PV array.For the propagation along the vertical path, the turbulence indicates non-Kolmogorov characters [16].Firstly, we build the mathematical model of laser propagation by using the Huygens-Fresnel principle.Secondly, we introduce the non-Kolmogorov model, as a more general model than the Kolmogorov model, to describe atmospheric turbulence [17], [18].Thirdly, the I-V characteristic equations are established by using the carrier continuity equation and bipolar transport equation.Finally, we obtain and analyze the output characteristics of the PV array on the propagation path.Our findings provide valuable insights for the optimization of the PV array structure and the laser aperture within the LWPT system, as well as the optimal system parameter range under different transmission distance and turbulence parameters is given.
The remainder of this study is structured as follows: Section II presents the analytical expressions for the average intensity, while Section III examines the numerical results.Finally, Section IV summarizes the conclusions drawn from the study.

II. THEORY AND METHOD
The laser beam propagating through non-Kolmogorov turbulence is illustrated in Fig. 2(a).Fig. 2(c) displays the changes of intensity distribution.Particularly, we present the pointing error, which is caused by the misalignment between the transmitter and receiver.Fig. 2(d) represents the power distribution of each cell considering pointing error, Fig. 2(e) shows the I-V and P-V characteristic curve of the PV array under uniform (Ir 1 ) and non-uniform (Ir 2 , Ir 3 and Ir 4 ) laser irradiance.It can be seen from Fig. 2 that atmospheric turbulence and the pointing error have significant effects on the output characteristics and photoelectric conversion efficiency of the PV array.

A. Intensity Analysis of Gaussian Beam in Atmospheric Turbulence
Fig. 2 indicates that atmospheric turbulence has an impact on the light field during the laser beam propagating through the atmosphere.Accordingly, utilizing the Huygens-Fresnel principle [19], we can express the cross-spectral density function in the source plane.
where w 0 is the waist width of GABs when z = 0, The average intensity of laser beam propagating through non-Kolmogorov turbulence is written as where ρ 1 = (x, y), k = 2π/λ is the wave number with wavelength λ, and < …> m represents the average over the ensemble of statistical realizations of the atmospheric turbulence, which can be expressed as [20] exp where J 0 (•) is the Bessel function of the first kind and order zero, Φ n (κ, α) is the spatial power spectrum of the refractive-index fluctuations in the atmospheric turbulence, κ is the magnitude of spatial wave number [20].Considering the inner-and outerscale effects, the non-Kolmogorov spectrum Φ n (κ, α) can be represented as [13], [14] Φ where A(α) = Γ(α-1) cos(απ/2)/(4π 2 ), κ 0 = 2π/L 0 , and is the outer scale and l 0 is the inner scale, α is the generalized exponent, C 2 n is the generalized structure parameter.By substituting (3) to ( 2), and after straightforward integral calculations [see Appendix], we can obtain where,

B. Calculation of Laser Power
The laser power is determined by the distribution of light intensity at the corresponding area Ω.Subsequently, the expression (p (Ω) (z)) of the laser power received on the designated area Ω can be calculated by means of an appropriate mathematical integral.As suggested by Siegman, the PIB is a measure of laser power focus-ability in the far field.It clearly indicates how much fraction of the total beam power is within a given bucket Ω [21].The PIB is defined as [22] P IB = Ω I(x, y, z)dxdy ∞ I(x, y, z)dxdy Then, we can obtain where, p (total) (z) is the total light power, and we can obtain the total light power p (total) (z) at arbitrary propagation distance in (11).
Comparing ( 9)-( 11), and we can obtain Substituting ( 5) to (12), by using the integral formula in (A-6), we can obtain Assuming the area of the i-th PV cell is denoted as Ω i .Then, considering pointing error, Ω i can be expressed as where, the pointing error is represented as r = (r, θ), 2 represent the left, right, lower, upper boundaries of the region Ω i without pointing error, respectively.
where, Φ(x, y) is the CDF of normal distribution function, which the variance and mean are Then, we can obtain Based on the power distribution and the physical dimension of PV cells, the laser energy on the received plane of each PV cell can be obtained by using (21).

C. Principle and Output Characteristics of PV Cells
The equivalent circuit model of the PV cell is shown in Fig. 3(a).The I-V characteristic equation corresponding to the equivalent circuit model is described by (22).
Here, I and V are the output voltage and current of the PV cell, respectively.R s and R sh are equivalent series and parallel parasitic resistors of PV cells.I d1 , I d2 and I dby are the forward current of the diode D 1 , D 2 , and D by , respectively.n is the diode ideal factor.I ph is the photo-generated current of PV cell, which can be expressed as (23) [24].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(22).According to the law of kirchhoff, the output I-V characteristic of the PV modules can be described as (27) when PV cells are connected in series.
In this study, the universal gallium arsenide photovoltaic cells are used to compose the PV array.The wavelength of the laser is 808nm.The specific parameters and physical constants are shown in Table 1 [24], [25].

D. PV Cells Distribution and Connection Mode in the PV Array
In this section, we present two typical and reliable connections of PV arrays in Fig. 4

A. Laser Energy Distribution on the Surface of the PV Array
The laser power distribution of each PV cell is obtained by (21), which ensures the conservation of the total power of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.laser beam.Fig. 5 illustrates the influence of the five parameters (C 2 n , α, w 0 , r, z) on the power distribution of each photovoltaic cell.In Fig. 5(a-1)-(a-5), as the generalized structure parameter of atmospheric turbulence (C 2 n ) increases, the overall intensity of the laser on all cells decreases, although the energy distribution tends to become more uniform among the individual cells.As shown in Fig. 5(b-1)-(b-5), the variation of the parameter α (ranging from 3 to 3.8) has a negligible effect on the laser energy distribution, while the total laser intensity on all cells reaches its minimum value when α is equal to 3.2.Fig. 5(c-1)-(c-5) illustrate that increasing the initial waist width w 0 leads to an overall increase in the total laser energy, while the two-dimensional distribution of energy remains unaltered.In Fig. 5(d-1)-(d-5), pointing error primarily affects the energy distribution of the laser by causing the peak intensity to shift from the center of the PV array to the surrounding regions.Finally, as presented in Fig. 5(e-1)-(e-5), the total laser energy decreases as the distance of laser transmission increases, leading to a more homogeneous distribution of energy on the PV cells.These results infer that the pointing error, the initial waist width w 0 , propagation distance z, generalized structure parameter C 2 n , and generalized exponent α are crucial factors in determining the power distribution.Then, we carry out the output characteristics of PV array versus the mentioned parameters.

B. Effects of Atmospheric Turbulence on the Output Characteristics of the PV Array
In Section III-A, we analyzed the distribution of laser energy on the PV array under atmospheric turbulence and pointing errors.Using the equivalent circuit model and the topological structure of the PV array we obtain the I-V and P-V characteristic curves.For SP and PS arrays, the distribution of laser energy is symmetrical without pointing errors.From simulation results, we can find that the output characteristic curves of the PV arrays under SP and PS configurations are consistent.Fig. 6 shows that the output characteristics vary C 2 n , and the maximum output power and current have significant differences and attenuates as the turbulence intensity increases.As shown in Fig. 7, the turbulence generalized exponent α has weak influence on PV array output characteristics.In Fig. 8, the initial beam waist width is used to study the relationship between output characteristic curve and the size of the laser spot.The difference is less than 0.01 V in the output characteristic curve when the initial beam waist width is between 10 mm and 20 mm (accounts for 1/4 to 1/2 the side length of the PV array).The output current and power of the PV array are smaller than that of the above-mentioned condition (w 0 = 10, 20 mm) when the initial beam waist width is 5 mm (accounting for 1/8 of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the side length of the PV array and 1/2 of the PV cell).When the laser power is constant, the more concentrated the energy, the greater the irradiation intensity in per unit area.The main reason is that few PV cells in the central area of the PV array receive more laser energy.Therefore, when the strong laser irradiation exceeds the maximum absorption and conversion capacity of PV cells, the remaining laser energy is converted into heat energy, resulting in power loss.The peripheral PV cells receive less intense laser irradiation, result in weaker output power.When the laser transmitting a distance of 1000 m through the atmospheric turbulence, the laser intensity of the initial beam waist width of 10 mm to 20 mm is more uniformly distributed on the PV array.Meanwhile, the laser energy can be fully absorbed and converted into electric energy by the PV cell.Therefore, when the laser power is fixed, within a certain range, the power of the LWPT system will increase with the increase of the beam waist width.However, When the beam waist is wider than the size of the PV array, the laser energy will be lost, and the overall efficiency of the system is reduced.Based on the above analysis, we can obtain that selecting an appropriate initial waist width and mitigating the pointing error represent effective strategies for enhancing the LWPT system performance.
To further analyze the impact of transmission distance and pointing errors on the output characteristics of PV cells considering atmospheric turbulence, the representative parameters are used.From Fig. 9, it can be inferred that atmospheric turbulence exerts a noticeable impact on the I-V characteristic curve within 1000 m.The results also suggest that pointing error significantly affects the output characteristics of the PV array, resulting in the multi-peaks P-V curves and attenuation of the maximum output power.Meanwhile, the voltage and current at the maximum power point will decrease obviously, which may lead to the output voltage of the DC-DC converter failing to meet the application requirements.Therefore, we assume z = 1000 m in the following parts.Fig. 10 illustrates the output  characteristic curves of the PV array versus pointing error r = (r, θ).Results show that when the point error r aligns with the diagonal of the square PV array, the output characteristics of SP and PS arrays are identical.When the pointing error r is symmetrical about the diagonal of the square photovoltaic cell, the output characteristics of SP or PS PV arrays are consistent with themselves but differ from each other.From Fig. 10, it can be seen that at an affirmatory θ (angle of pointing error), the I-V characteristic curve of the PV array presents more steps with the increase of r (distance of pointing error), and the P-V curve has more peaks.The output power of the PV array decreases obviously and the difficulty of MPPT algorithm to search the maximum power point increases, which will significantly reduce the stability and efficiency of the LWPT system.

C. The Maximum Power Point and Efficiencies
According to the numerical results in Section III-B, we can obtain the output voltage (V mpp ) and power (P mpp ) at the maximum power point of the PV array.And the P mpp and V mpp are the input power and voltage of the DC-DC converter in the LWPT system.The decreases of P mpp and V mpp will lead to the decreases of the output power and voltage of the LWPT system.As displayed in Fig. 11, subfigures (a), (b), (c) and (e) depict the same variation trend for P mpp and V mpp with the x-axis parameter, while in (d) and (f), the variation trend is opposite.In Fig. 11(a), the P mpp and V mpp decrease with the increasing C 2 n .That is because the larger the C 2 n is, the larger the laser intensity attenuation is, which makes the beam spreading and leads to the decrease of the laser energy received by the PV array.And the trend of laser intensity versus C 2 n is consistent with that of P mpp and V mpp .In Fig. 11(b), the changes in P mpp and V mpp are due to the change of α and have the similar trends with α.In Fig. 11(c), the P mpp and V mpp initially increase and then decrease with w 0 .This phenomenon can be attributed that where the energy incident on the PV array surpasses its absorption and conversion capabilities, thereby leading to energy dissipation when initial beam waist width has a little size.However, excessive waist width will cause the spot area to exceed the PV array area, resulting in the loss of laser energy.In Fig. 11(d), the P mpp decreases with the increase of r (distance of pointing error).When r = 20 mm, the attenuation percent of P mpp is 50.9% (from 3.3 W to 1.62 W).And the V mpp initially increases and then decreases, where the change interval is 22.26 V to 22.88 V. Fig. 11(e) shows the changes of P mpp and V mpp with transmission distance z when r = 0.For P mpp , the attenuation in 1000 m accounts for 66.7% (from 9.9 W to 3.3 W) of the initial position.The attenuation in 2000 m is 74.8% (from 9.9 W to 2.5 W) of the initial position.This indicates that under the influence of atmospheric turbulence, the change rate of P mpp is larger at short distances than at long distances.The change in V mpp is 1.3 V (from 23.2 V to 21.9 V).In Fig. 11(f), it is observed that the transmission distance has obvious effect on P mpp and V mpp when considering pointing error.P mpp decrease with the transmission distance.The change in V mpp is 17.2 V (from 5.1 V to 22.3 V).Therefore, it is necessary for the DC-DC converter in the LWPT system to have a wide input voltage range and high voltage gain.
In Fig. 12, the η 1 , η 2 and η 3 represent the photoelectric conversion efficiency of the PV array, the total efficiency of the initial laser energy to output power of the PV cell and the efficiency of laser transmission in the atmosphere, respectively.Simulation results indicate that the turbulence parameters and initial beam width, as shown in Fig. 12(a), (b) and (c), have little effect on the photoelectric conversion efficiency of PV array when the transmission distance is 1000 m and the beam is not offset.At this time, according to the efficiency η 2 , it can be inferred that the efficiency of the system will increase initially and decrease afterwards with the beam waist width w 0 .Therefore, when the laser energy is constant, the optimal waist width should be selected based on the actual size of the PV array.From Fig. 12(d), it can be seen that η 1 decreases as the pointing error increases.This is because the pointing error makes the energy distribution on the PV array more uneven, which causes more power loss.The efficiency η 2 descent 51.5% (from 16.5% to 8%) when the pointing error changes from r = (0 mm, 0°) to r = (20 mm, 45°).Comparing Fig. 12(e) with Fig. 12(f), it is obvious that the pointing error changes the variation trend of η 1 versus z. η 1 reaches its maximum when the pointing error is 0 and the propagation distance is 1000 m.And the pointing error attenuates the overall efficiency η 2 of the system.The η 2 and η 3 decreases as the x-axis parameter increases as shown in Fig. 12(a), (d), (e), and (f).The η 2 and η 3 initially decrease and then increase with the α, which opposite with w 0 in Fig. 12(b) and (c).Comparing Fig. 12(d), (e) and (f), we can find that the photoelectric conversion efficiency η 2 is affected by the pointing error and transmission distance.Fig. 13 shows the P mpp , V mpp and efficiency of SP and PS arrays at the maximum power point versus pointing error.One can find that the V mpp of SP and PS arrays are attenuated when the angle θ is 0°and the offset distance r is larger than 15 mm.At this time, the V mpp plunges to 10.1 V and 16.4 V for the SP and PS array when r = 20 mm, θ = 0°.And the V mpp of the PS array are larger than the SP array.The decrease of the V mpp will result in the decrease of the output voltage of the DC-DC converter.The DC-DC converter cannot supply power to the storage battery and load in the LWPT system, when its output voltage less than the voltage of the storage battery.Therefore, the large pointing error may cause the failure of the LWPT system.The differences between V mpp of the PS array and SP array are less than 0.5 V when the offset distance is within 15 mm.And the V mpp of the PS array and SP array are more than 22 V.In addition, P mpp with different offset angle θ have little difference, but which obviously decrease with the offset distance r.Therefore, the offset distance r of the pointing error will significantly decrease the efficiency of the LWPT system.In Fig. 13(b), the η 1 , η 2 and η 3 of the SP array and PS array attenuate as the offset distance r increases.And the difference of the η 1 , η 2 and η 3 of the SP array and PS array at the same pointing error is less than 10%, respectively.The main reason is that the beam spreading at 1000 m makes the changes of laser intensity distribution smaller than that within 1000 meters.

IV. CONCLUSION
In conclusion, this paper presents a study of the output characteristics of the PV array in the LWPT system considering non-Kolmogorov turbulence and pointing error.A theoretical model of the laser intensity and equivalent circuit of the PV array in non-Kolmogorov turbulence is proposed.Further, the output characteristics of the PV array versus the generalized exponent, the generalized structure parameter, atmospheric turbulence, transmission distance, initial beam width and pointing error are studied.The results reveal that the SP array and the PS array have the same output characteristics when the angle of the pointing error is 45°.The P mpp and the system efficiency η 2 descend 50.9% and 51.5%, respectively, when the transmission distance is 1000 m and the pointing error changes from r = (20 mm, 45°) to r = (0, 0).Moreover, the descend values of P mpp are 66.7% and 74.8% at 1000 m and 2000 m, respectively, when the pointing error r = (10 mm, 45°).Meanwhile, the V mpp increases significantly (from 5.1 V to 22.3 V) with the increasing of transmission distance.Furthermore, the results also suggest that the system efficiency of the PV array decreases with the increase of the generalized structure parameter, pointing error and transmission distance while less affected by generalized exponent and initial waist width.The simulation results demonstrate that turbulence and pointing error will cause the output voltage and power of the PV array attenuation.And the P-V characteristic curve of the PV array will present multi-peaks.Which will not only result in the decrease of the output voltage and power or failure of the LWPT system, but also have negative effects on the stability.The findings of this study have significant implications for the optimization of the overall structure of LWPT systems especially the design of PV array and DC-DC converter.

APPENDIX
When κξ|ρ 1 −ρ 2 | << 1, J 0 (κξ|ρ 1 −ρ 2 |) can be approximated to the first two terms in the power series expansion [26], i.e., J 0 (κξ Assuming after some tedious mathematical manipulation, the term I t is expressed as [27]  Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 1 .
Fig. 1.Diagram of laser wireless energy transmission system in atmospheric.

Fig. 3 .
Fig. 3. Equivalent circuit model of the PV cell.
(a) and (b), series-parallel (SP) array and parallel-series (PS) array, respectively.To avoid the issue of multiple peaks in P-V curve mentioned above, we propose an optimal configuration design of PV array, where each PV cell in the series module receives the same irradiation power.The location of each cell in the series module is symmetrically designed with respect to the center point of laser beam considering Gaussian distribution.As shown in Fig. 4(c) and (d), the configurations of PV cells are designed for the two connection modes shown in Fig. 4(a) and (b), respectively.The number M-N in Fig. 4(c) and (d) represents the PV cell of column M and row N in Fig. 4(a) and (b).

Fig. 5 .
Fig. 5. Laser power distribution on each PV cell in the PV array under different parameter.

TABLE I UNIVERSAL
GALLIUM ARSENIDE (ASGA) PHOTOVOLTAIC CELL PARAMETERS AND PHYSICAL CONSTANTS USED IN THIS PAPER