Gaussian Process for Learning Solar Panel Maximum Power Point Characteristics as Functions of

Environmental Conditions

Nalika N. B. Ulapane*, Sunil G. Abeyratne†, *Centre for Autonomous Systems, University of Technology Sydney, Australia

†Department of Electrical & Electronic Engineering, University of Peradeniya, Sri Lanka

Abstract-This paper proposes a method to learn the variation of solar panel Maximum Power Point (MPP) parameters as functions of environmental conditions using Gaussian Process (GP) based machine learning. As a result of using GP, functions are learned along with the additional information of their uncertainty margins. The paper discusses about learning three functions specifically, where each of them take the two environmental variables 'solar irradiance' and 'cell temperature' as arguments and map these environmental variables to the corresponding MPP parameters, namely, the maximum power, the MPP voltage and the MPP current. Learned functions presented in the paper have been trained for a commercially available solar panel using MPP data generated using a previously published solar panel simulator. The learned function for maximum power has been validated by comparing the function outputs (GP results) against the manufacturer specified power values. A discussion about how learning such functions can help in advancing MPP Tracking (MPPT) is also provoked while highlighting the impact machine learning can make to the field of photovoltaics.

Keywords-Gaussian Process, Machine Learning, MPPT, Solar panel simulation

I. INTRODUCTION

References [1-3] have shown the possibility of simulating and emulating the behavior of commercial solar panels using limited data obtained from solar panel manufacturers. A Matlab application which can simulate commercial solar panels under varying environmental conditions is proposed and validated in [2]. Such applications are efficient and economical because they enable studying the environmental dependence of solar panel characteristics without experimental setups. This work thereby uses the simulating application proposed in [2] to generate MPP data of a commercial solar panel under a wide range of environmental conditions. A Gaussian Process (GP) based method of learning functions which map the environmental conditions to MPP data is then described in the paper. The learned function for maximum power is validated using manufacturer specified MPP data. Finally, the way in which learning such functions can help in the advancement of MPPT is also discussed.

Demonstrating the feasibility of learning three functions which produce three MPP parameters as outputs is the main focus of this paper. The MPP parameters of interest are the

maximum power (Pmax) in W, the panel voltage at maximum power (Vmp) in V and the panel current at maximum power (Imp) in A. Solar irradiance (I) in W/m2 and cell temperature (T) in oC, which are the two dominant environmental conditions which effect solar panel outputs [1-3], are used as the input arguments for the functions. The three functions of interest can thus be expressed as shown in (1), (2) and (3). The terms 1ε ,

2ε and 3ε in those equations denote small differences between the function outputs and experimental values. Those differences can be characterized by uncertainty margins when using GP to learn the functions. Pmax = f1(I,T) + 1ε (1) Vmp = f2(I,T) + 2ε (2) Imp = f3(I,T) + 3ε (3)

The paper unfolds from here onwards under three main sections titled "Methodology," "Results and Validation" and "Discussion." Details about the simulator used for data generation and the GP based method of learning functions are included in the "Methodology" section. The "Results and Validation" section includes graphical representations of the learned functions and the validation of the Pmax function. Following the results, the subsequent "Discussion" section explains the advantages of solar panel simulation, learning functions of MPP parameters and how these functions can help in the advancement of MPPT.

II. METHODOLOGY

The work in this paper has been carried out in relation to the SUNPOWER E19/240 commercial solar panel [4]. Since data was generated for this work using a solar panel simulator, the first subsection under the methodology is allocated for describing the simulator. The second subsection addresses the GP based method of learning functions.

A. Data Generation using a Solar Panel Simulator As mentioned in the introduction, the Matlab based solar

panel simulator published in [2] was used for data generation. It has the capability of facilitating user interaction as well as automated data generation given the requirement. An image of the user interface of this simulator is shown in Fig. 1.

1756978-1-4799-4315-9/14/$31.00 c©2014 IEEE

Fig. 1. Matlab GUI of the solar panel simulator used for this work (adapted from [2]).

The simulator takes seven manufacturer specified values associated with the solar panel as inputs and users have the option of providing them either via the user interface shown in Fig. 1 or to include them in the code itself. These seven values are: open circuit voltage, short circuit current, voltage at maximum power and current at maximum power which correspond to Standard Test Conditions (STC, i.e: 1000 W/m2 Irradiance and 25 °C cell temperature), and the temperature coefficient of open circuit voltage, temperature coefficient of short circuit current and number of solar cells in the solar panel, which are constants. After obtaining these values, the application estimates parameters of the single diode solar cell model which represents the whole solar panel as one unit. Reference [1] addresses this parameter estimation process in detail. Users then have the capability of providing desired environmental conditions (solar irradiance and cell temperature). When the environmental conditions are given, the application proceeds to generate the volt-ampere characteristic curve and the power profile of the solar panel corresponding to the given environmental condition. Subsequently, Pmax, Vmp and Imp are tracked, displayed and stored for the benefit of the user. It was this feature of tracking the MPP which enabled generating MPP data for the work in this paper. References [1] and [2] give an in depth explanation on the full functionality of the simulator application.

For this work, the application was automated to generate MPP data of the SUNPOWER E19/240 solar panel by iterating the cell temperature from 0 °C to 75 °C in steps of 1 °C and the irradiance from 1 W/m2 to 1000 W/m2 in steps of 1 W/m2 within each temperature iteration. This resulted in 76,000 MPP data points per single execution. Since having several sets of noisy data is important for GP [5], the data generation procedure was repeated several times by varying the voltage

iteration step in the power profile. The data generation procedure is graphically elaborated in Fig. 2. Fig. 2. Procedure for generating multiple sets of training data using the simulator.

B. GP based method of learning functions For the convenience of explanation, the following notation

will be used from here onwards. X = [x1, x2, . . . , xm]T = Matrix of training features Y = [y1, y2, . . . , ym]T = Vector of training labels X* = [x*

1, x*2, . . . , x*

n]T = Matrix of testing features Y* = [y*1, y*2, . . . , y*

n]T = Vector of testing labels In the training feature matrix, xi takes the form xi = [Ii, Ti]T

where I and T denote irradiance and cell temperature values respectively while the subscript i is any integer from 1 to m. The variable yi takes values of either Pmax, Vmp or Imp depending on the function which has to be learned and the corresponding training feature values. Variables x*

i and y*i are similar in form

to xi and yi respectively, but contain values that are not identical to those in the training data set.

The training data sets denoted by [X, Y] were used to learn the three functions with uncertainties shown in (1), (2) and (3). Since the learning procedure is the same for the three functions, it will be described from here onwards using the notation y = f(x)+ ε , which is general to all three functions.

The requirement was to train a GP model G = {X, Y, } where is the matrix containing hyper-parameters. The squared exponential covariance function shown in (4) was used for the purpose after evaluating many other commonly used ones.

Start

Stop

Input required number of training data sets (num), Input initial voltage iteration step (Vi = 0.001 V)

Simulate and track MPP values for the accuracy of Vi

num = num - 1

num = 0 ?

Vi = Vi / 10

Yes

No

2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA) 1757

( ) ( )−−= 22

2

21exp, jiji xxxxk (4)

The GP model was trained by learning the hyper-parameters values = ( , , n) through minimizing the negative log marginal likelihood shown in (5) with respect to hyper-parameters.

( ) ( )2log2

log21

21,log 1T mYYXYp +Σ+Σ=− − (5)

( ) IXXK n2, +=Σ (6)

The term K(X, X) in (6) is the m x m covariance matrix whose elements are given by ki,j = k(xi, xj) and I is the corresponding identity matrix. Some Gaussian noise was assumed for the training data with variance n

2 and this variance was also estimated through optimization. When the hyper-parameters are optimized, it has been proved that the Gaussian regression function output y* can be expressed as the conditional Gaussian distributions N( *, *Σ ) with the mean and covariance expressed as in (7) and (8) [5].

( ) ( ){ } yIXXKXXK12

n** ,,

−+= (7)

( )( ) ( ){ } ),(,,

,*12

n*

2n

***

XXKIXXKXXK

IXXK−

+−

+=Σ (8)

Thus, the solution for the addressed problem which yields function values Pmax

*, Vmp* and Imp

* for testing inputs [I*, T*], happens to be a normal distribution with the mean and covariance given by (7) and (8) respectively. Means of these distributions are the interpreted function outputs and the covariance can be considered as uncertainties. Reference [5] provides detailed discussion on GP.

III. RESULTS AND VALIDATION

Fig. 3 to 8 included in this section show the plots of the three learned functions along with their uncertainties. Each plot was generated using 7,503,241 testing data points obtained by iterating cell temperature from 0 to 75 °C and irradiance from 1 to 1000 W/m2 in steps of 0.1 in their respective units.

Results in Fig. 3, 5 and 7 are very similar to those of [6], but are much higher in resolution of irradiance and cell temperature. However, [6] is a reasonable reference for interested readers to find similar results to those of this paper since publications containing such results are rare at present. Fig. 4, 6 and 8 exhibit that the uncertainties of the learned functions are reasonably low, which is a desirable outcome. TABLE I emphasizes this fact even more by providing mean values of the uncertainties. From the figures and TABLE I, it can be concluded that the uncertainties are desirably low in comparison to the magnitudes of the function outputs, but not too low to indicate over fitting.

Fig. 3. Plot of the learned Pmax (W) function expressed in (1), for the SUNPOWER E19/240 solar panel.

Fig. 4. The 99% confidence uncertainty (3 ) of the learned Pmax function for the SUNPOWER E19/240 solar panel.

Fig. 5. Plot of the learned Vmp (V) function expressed in (2), for the SUNPOWER E19/240 solar panel.

1758 2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA)

Fig. 6. The 99% confidence uncertainty (3 ) of the learned Vmp function for the SUNPOWER E19/240 solar panel.

Fig. 7. Plot of the learned Imp (A) function expressed in (3), for the SUNPOWER E19/240 solar panel.

Fig. 8. The 99% confidence uncertainty (3 ) of the learned Imp function for the SUNPOWER E19/240 solar panel.

TABLE I MEANS OF UNCERTAINTIES OF THE LEARNED FUNCTIONS FOR THE E19/240

SOLAR PANEL Function Mean of Uncertainty

Pmax 0.0122 W Vmp 0.00348 V Imp 0.00615 A

Validation of Pmax function was performed using the manufacturer specified temperature coefficient of Pmax provided in [4]. The validation was based on temperature variation of maximum power at 1000 W/m2 irradiance and the results are shown in Fig. 9. Percentage differences of the GP results with respect to values calculated from manufacture's temperature coefficient are shown in Fig. 10. Calculation of those percentage errors was done using (10).

%100

PE ×

−=

Powerer'sManufacturPowerer'sManufacturGP Power

(10)

It is evident in Fig. 10 that all percentage errors are less than 0.14 %. The mean percentage error too resulted in being as low as 0.0495 %. Overall implications of these results are conveyed in the "Discussion" section.

Fig. 9. Comparison between GP results and manufacturer specified values of Pmax (W) for different temperatures at 1000 W/m2 irradiance for the SUNPOWER E19/240 solar panel.

Fig. 10. Percentage error values between GP results and manufacturer specified values for Pmax (W) shown in Fig. 9.

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IV. DISCUSSION

This paper stresses the fact that solar panel MPP parameters can be learned as functions of environmental conditions using machine learning. Software simulation and GP have been used for the purpose and that highlights the applicability of Artificial Intelligence (AI) in the field of photovoltaics. In the present era where many state of the art communication, control, power transmission and storage techniques are partnering with photovoltaics to develop advanced smart grids [7], this paper gives an insight about the role machine learning and AI can play in the field. Therefore, it is important to mention that there is room for more AI researchers to experiment techniques such as machine learning in partnership with photovoltaics, particularly because it has the potential to lead to significant advancements. Besides, the reviewed literature reveal that the application of machine learning in photovoltaics and MPPT has not been significantly exploited yet. Therefore, work similar to that of this paper are rare at present.

This paper also emphasizes how important solar panel simulation can be and how it can be used for practical applications. Simulations are always useful in efficient mass data generation for purposes like machine learning. For instance, if the work in this paper which includes 76,000 data points in one training data set, was done with purely experimental setups, it would have been very costly in terms of time, infrastructure and labor. Contrarily, because simulation was used for this work, the mass data generation and function learning process could be done efficiently with minimal requirement of hardware setups.

The other aspects to be discussed about this work are the implications of the obtained results and how useful theses results can be for further advancement of MPPT. These aspects are discussed under the following two subsections.

A. Implications of Results It is evident that Fig. 4, 6 and 8 show appreciably low 3

uncertainty margins which are acceptable with respect to magnitudes of function outputs, but not too low to an extent where they indicate over fitting. The mean uncertainties shown in TABLE 1 can be considered as evidence supporting the above statement. Uncertainty values of such nature are desired in any GP application [5]. What it implies is that the learned functions predict results with good confidence. Since Fig. 9 and Fig. 10 exhibit low differences between Pmax (W) function outputs and manufacturer specified power values, the accuracy of predictions can be considered high as well. Thus, low errors and low uncertainties indicate that the learned functions predict results with good accuracy and confidence.

As per Fig. 4, 6 and 8, the uncertainties seem to be relatively high for certain irradiances. However, those values are not too high to be alarming and they do not make mean uncertainties in TABLE 1 too high. Two factors can be easily identified as

causes for this phenomenon. The first cause could be the training data not describing the regions with high uncertainty well enough. If that is the case, this occurrence can be avoided by simply increasing the amount of training data for those regions. The second and the most likely cause could be the initial conditions chosen for optimization for learning hyper-parameters not being the best values. Results in this paper were obtained using = ( , , n) = (0.5, 0.5, 0.1) as initial conditions. Before obtaining the published results, the optimization was run using the initial conditions = ( , , n) = (0.1, 0.1, 0.1). Uncertainties obtained for that case were not acceptable. Thus, it is likely that the initial conditions play a crucial role for this GP application. In instances like this, the best initial conditions should be fine tuned. Understandably that process could sometimes be tedious. Therefore, such fine tuning was not performed in this work since decent uncertainties and sufficient accuracies were obtained using the initial conditions = ( , , n) = (0.5, 0.5, 0.1). However, if the uncertainties of the three learned functions are required to be pushed to their best possible values, such fine tuning may be necessary. If conventional approaches do not produce desirably good results, it is possible to extend GP training algorithms with hybrid covariance functions [5]. Such complex approaches were not used since they seemed unnecessary for this work, however, hybrid covariance functions too maybe helpful in further improving the results of the proposed method.

B. How Learning MPP Functions Can Help in the Advancement of MPPT

Though sophisticated and fast MPPT techniques based on different AI approaches such as artificial neural networks, fuzzy logic and genetic algorithms have been proposed and reviewed in publications like [6] and [8], the use of probabilistic machine learning algorithms such as GP is rare. Reference [6] is one of the few publications containing results similar those of this paper and it discusses about a genetic identification function for MPP, which is an approach different from probabilistic machine learning. Therefore, it can be considered as important to elaborate how learning MPP parameters as functions of environmental conditions which effect them is important.

The fact to be noted is that the function learning process is a one off activity for a given solar panel or a photovoltaic system. Once the functions are learned, it is only a matter of storing the function outputs along with the corresponding inputs in memory to be recalled when necessary. The role of GP based machine learning is to learn those functions with the best possible accuracy and produce results with acceptable uncertainty. Thus, as highlighted in this paper, since it is possible to learn the required Pmax, Vmp and Imp values for desired irradiance and cell temperature ranges with desired resolution, the memory based MPPT technique in Fig. 11 can be proposed.

1760 2014 IEEE 9th Conference on Industrial Electronics and Applications (ICIEA)

Fig. 11. Proposed MPPT technique which operates by retrieving learned function values from memory.

Though there are numerous MPPT techniques available at present [8], this memory based technique is proposed since it has the potential to be faster than existing techniques because it has the capability to recall MPP values from a memory and directly adjusts the duty ratio of the switch mode MPPT device to achieve the MPP depending on the sensed irradiance and temperature. The speed of this memory based MPPT technique was not quantified in this work since that was beyond the main objective of this paper. It should therefore be emphasize to researchers that there is room for future work along this line.

It can be said that such a memory based method would be perfect in an ideal environment where the whole solar panel or the whole system is not subjected to factors like partial shading [9] and uneven temperature distribution. To perfect this method in the case of real environments, considering partial shading and uneven temperature distribution would be necessary. Since machine learning is used, it has the capability to learn functions and extend this method by including localized cell temperatures and irradiances as inputs instead of single values which are generalized to the whole panel. There is room for interested researchers to look into that aspect as well. The proposed method thus has the potential of being an accurate, powerful and fast MPPT technique with some further improvements.

A critical factor that would interest readers is the cost of a memory based MPPT technique. It is highly likely that the

implementation of a memory based method along with improvements could be expensive than standard MPPT devices since it could employ multiple memories for data storage and sensors to track environmental conditions. However, such methods might also have the capability to pay the additional costs back by means of more efficient and accurate MPPT. Since the approach in this paper is relatively new to the field of photovoltaics, it is considered too early to comment on factors such cost and payback period and that too creates room for interested researchers. This paper can thus be concluded by emphasizing the applicability of machine learning to learn MPP parameters as functions of environmental conditions, importance of solar panel simulation, potential uses of learning MPP functions and the available space for future work.

REFERENCES [1] N. N. B. Ulapane, C. H. Dhanapala, S. M. Wickramasinghe, S. G.

Abeyratne, N. Rathnayake, P. J. Binduhewa, "Extraction of parameters for simulating photovoltaic panels," Proc. of the 6th IEEE International Conf. on Industrial & Information Systems (ICIIS), pp. 539-544, August 2011, Sri Lanka.

[2] N. Ulapane, S. Abeyratne, P. Binduhewa, C. Dhanapala, S. Wickramasinghe, N. Rathnayake, "A simple software application for simulating commercially available solar panels," International J. of Soft Computing and Software Engineering (JSCSE), vol. 2, no. 5, pp. 48-68, May 2012.

[3] A. M. N. N. B. Ulapane, W. C. H. Dhanapala, W. M. S. M. Wickramasinghe, S. G. Abeyratne, K. R. M. N. Rathnayake, P. J. Binduhewa, "Solar panel emulator based on a switch mode power supply," Annual Trans. of the Institution of Engineers Sri Lanka, vol.1 part. B, pp. 508-515, October 2011.

[4] SUNPOWER E19/240 Solar Panel Datasheet, http://us.sunpowercorp.com, Last Visit: 17/07/2013.

[5] Carl E. Rasmussen, Christopher K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. Online: http://www.gaussianprocess.org/gpml/, Last Visit: 05/08/2013.

[6] Adel El Shahat, "Maximum power point genetic identification function for photovoltaic system," International J. of Research and Reviews in Applied Sciences (IJRRAS), vol. 3, no. 3, pp. 264-273, June 2010.

[7] Janaka Ekanayake, Kithsiri Liyanage, Jianzhong Wu, Akihiko Yokoyama, Nick Jenkins, "Smart Grid: Technology and Applications," ISBN 1119968682, Wiley, 2012.

[8] Trishan Esram, Patrick L. Chapman, "Comparison of Photovoltaic Array Maximum Power Point Tracking Techniques," IEEE Trans. on Energy Conversion, vol. 22, no. 2, pp. 439-449, June 2007.

[9] Hiren Patel, Vivek Agarwal, "MATLAB-Based Modeling to Study the Effects of Partial Shading on PV Array Characteristics," IEEE Trans. on Energy Conversion, vol. 23, no. 1, pp. 302-310, March 2008.

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