Mathematical modelling of photovoltaic...

IntroductionSome of our main results

Mathematical modelling

of photovoltaic panels

Fco. Javier Toledo Melero∗

Co-authors: José Manuel Blanes∗∗ and Vicente Galiano∗∗∗

*Institute Center of Operations Research (CIO)∗∗Industrial Electronics Group

∗∗∗Department of Computers Engineering

Miguel Hernández University of Elche

February 2021

F.J. Toledo, J.M. Blanes, V. Galiano Mathematical modelling of photovoltaic panels


Photovoltaic panel. Equivalent electrical circuit.The single-diode model

PHOTOVOLTAIC ENERGY




PHOTOVOLTAIC PANEL

When a voltage V is applied to a photovoltaic panel (PV) panel, itreturns a current I. It is possible to generate a cloud of points(V, I) from the short-circuit point, SCP = (0, ISC) to the opencircuit point OCP = (0, VOC). This cloud of points describes theso-called characteristic I-V curve of the panel.Usually, the panel works in a range of voltages around theso-called Maximum Power Point, MPP = (VMPP, IMPP) which isthe point where the power P = V I attains its maximum.




EQUIVALENT ELECTRICAL CIRCUIT

The characteristic I-V curve tells us about the state of the panelat the moment and current conditions, so, it is important toextract as much as possible information from the curve. Thebest way is model it using an equivalent electrical circuit.




THE SINGLE-DIODE MODEL EQUATION

Depending on the electrical components of the circuit, numberof diodes and resistors in series and in parallel, a differentmodel is obtained, the more components the more complex .It has been checked that, under minimum conditions oftemperature and irradiance, the equivalent electrical modelwith a single diode, a resistance in series and a resistance inparallel, called Single-Diode Model (SDM), correctly describesthe behavior of the solar panel, therefore, it is one of the mostused models in the literature.Associated to the SDM, Kirchoff’s laws give rise to amathematical equation that relates, I, the panel current, with V,the panel voltage

I = Iph − Isat

(exp

(V+ I Rs

nVT

)− 1

)− V+ I Rs

Rsh




RESOLUTION OF THE SINGLE-DIODE MODEL

To solve the single-diode model from a data set

I = Iph − Isat

(exp

(V+ I Rs

nVT

)− 1

)− V+ I Rs

Rsh(1)

is to obtain the parameters equationIph = npIcell

ph panel photocurrent in Amperes

Isat = npIcellsat panel diode saturation current in Amperes

Rs =nsnp

Rcells panel series resistance in Ohms and

Rsh =nsnp

Rcellsh panel shunt resistance in Ohms

n ideality factor

where VT =kq T thermal voltage, where T is the temperature in

Kelvin degrees, k = 1.3806488× 10−23J/K is the Boltzmann’sconstant and, q = 1.60217653× 10−19C is the electron charge.




DIFFERENT TYPES OF DATA SETSFrom experimental measurements: cloud of points sweepingthe curve from the short-circuit to the open-circuit points.

From manufacturer’s datasheet: values of the remarkable I-Vpoints with some additional information.




SOME RESOLUTION TECHNIQUES

Using experimental measurements:

With five voltage-current points one obtains a nonlinear system offive equations with five unknowns

Resolution with numerical methods, for example, theNewton-Raphson method for nonlinear systems ofequations, or variations of this methodSimplify the equations with approximations of someterms, for instance, removing terms which are negligibleunder certain conditionsSimplify the model equation, for instance, giving fixedvalues for some parametersHeuristics and metaheuristics algorithms




With less than five points and their slopes (a selection)

The analytical five-point method: with the three remarkablepoints (SCP, OCP, MPP) and their slopes, together with anapproximation of Rsh

The analytic quasi-explicit four arbitrary point method: withfour arbitrary points and their slopes, the problem isreduced to solve a single-variable equation which must besolved numerically. Slopes comput: α-power functionThe reduced form method: with the three remarkable pointsand their slopes, the problem is reduced from five to twoparameters, then with two well selected extra points, asystem of two variables and two equations must be solvedThe oblique asymptote method: with the SCP and its slope(the I-V curve oblique asymptote is approximated), tehn,with three more points well distributed below the MPP, theparameters are explicitly obtained




With all (or a sufficiently large selection of) the points of the cloud

Curve fitting: find the parameters that provide the modelthat best fits the data points- Regression: ordinary least squares, total least squares- Heuristics and metaheuristics algorithms

Mixed methods: combination of different techniques- Reduced form method (RF): after reducing the problem totwo parameters, do curve fitting with them- Two-step linear least-square method (TSLLS): obliqueasymptote method optimized. For each aproximation ofthe oblique asymptote, search the best selection of threepoints for obtaining the remaining parameters.




Using manufacturer’s datasheet:

With the three remarkable I-V points:

Short-circuit point: SCP = (0, Isc)Open circuit point: OCP = (VOC, 0)Maximum power point: MPP = (VMPP, IMPP)

Together with the slope of the I-V curve at the MPP (it is awell-known fact): I′MPP = −IMPP/VMPP

one obtains a nonlinear system of four equations with fiveunknowns.

Possibilities:

Compute the infinite I-V curves satisfying the previousconditions: first time in a very recent paperAdd a fifth equation (with a physical motivation) to obtaina system of five equations with five unknowns




RESULTS ANALYSIS

All the methods have advantages and dissadvantages fromdifferent points of view. Depending on the needs of the user, itshould be taken into account:

Accuracy of the solution versus the data: for instance, rootmean square error (RMSE)Accuracy of the solution versus the hypotheticalparameters: Goodness of fitFeasibility: mathematical vs physical meaningLocal or global (approximate) solutionsStability of the solutions: small perturbations of the datagives rise to close solutionsComputational time consumptionSimple programming for: aerospace applications, real-timefacility control, mechanisms with microcontrollers, etc.




OTHER RELATED PROBLEMS

Known the model parameters, the following mathematicalquestions also arise in the study of the SDM:

Compute the current values for the voltage values and viceversa, from the implicit model equation. The equationdoes not have analytical solutions, so, numerical methodsmust be used to obtain the approximate solutions.One possibility is to use the well-known Lambert Wfunction, which is indeed a function defined implicitely.Compute the maximum power point, that is, the pointwhere the function P = VI, also given implicitely, attainsits global maximum.Compute the slopes of the I-V curve at some of its points.Study the analytical and geometrical properties of theSDM to guarantee that the model satisfies the expectedphysical properties.



Geometric properties of the SDM and their consequencesAnalytical and quasi-explicit four arbitrary point method

One of our most interesting contributions to the topic ofmodeling the behavior of PV panels has been the theoreticalstudy of the SDM properties.

In this work it is proved that the SDM equation definesimplicitly the current as an infinitely differentiable function ofthe voltage, and vice versa, at any point satisfying the equationand the properties that we show below.




In this work, the following properties of the I-V curvegenerated by the SDM were demonstrated for the first time:

It is strictly decreasing and concaveIt has an oblique asymptote at left. This behaviour isalready strongly noticeable near the SCP.

A key tool of the previous work was to rewrite the SDMequation with the following reparametrization

I = A+ B− EV− BCVDI (2)




The new parameters in (2) are given by:

A =IphRsh

Rsh + Rs, B =

IsatRshRsh + Rs

, E =1

Rsh + Rs

C = exp(

1nsnVt

), and D = exp

(Rs

nsnVt

)The nice geometrical interpretation of the parameterssuggested us the Oblique Asymptote (OA) method based onthe following approximation:

A+ B− EV = ISC + I′SCV

So, it is needed to compute ISC and I′SC from the data. After thisapproximation, one has from (2) that

ln B+V ln C+ I ln D = ln(ISC + I′SCV− I

)Now, with three more points, a linear system of three equationand three unknowns is obtained and the model is solved.




In [Batzelis, 2019] 17 methods found in the literature werecompared. Some extracts of the conclusions can be seen below:




The simplicity of the OA method and the possibility of workingcompletely blind led us to design an algorithm to optimize theOA method.

Just as a curiosity, this article had five reviewers and with two ofthem we reached the third review.It was worth it! The year of publication, this journal had:Impact Factor: 7.503. Position at cathegory INSTRUMENTS &INSTRUMENTATION: 1/61. Position at AUTOMATION &CONTROL SYSTEMS: 2/62.




The proposed algorithm, called Two-Step Linear Least-Squares(TSLLS) Method, consists, as its name indicates, in two steps:

1 Extract the oblique asymptote with the best fit of the pointsnear the SCP. A progressive sweep of points is madestarting from the SCP to the optimal point.

2 Extract the parameters B, C, D of the exponential part ofthe equation wich provides the best fit to the points abovethe OCP. A progressive sweep of points is made startingfrom the OCP to the previous optimal point.

Refinement: A finall step consists on a curve fit starting fromthe five previous parameters.




TSLLS method, with the first two steps, have the sameorder of accuracy of the best documented methods in thefield of parameters extraction, but, with the refinement, itattains the best accuracy documented until now in twoimportant case studies usually used in the literature aswell as in a large-scale I–V curve repository with morethan one million of curves.TSLLS method can work completely blind, so, it has beenpossible to implement it on a Web page that currentlyreceives visits from researchers and agencies from all overthe world: https://pvmodel.umh.es




Using again the oblique asympte property of the SDM I-Vcurve, we have been able to obtain for the first time the infinitepossible I-V curves just using the three remarkable points(SCP, OCP and MPP) that are usually provided in the datasheetof photovoltaic panels.




First, using the extrem slopes at the remarkable I-V points, webounded the feasible region of the I-V curves

The domain of parameter E (the slope of the OA) wasstablished and it was imposed the same hypothesis over theordinate at the origen of the oblique asymptote which was usedin the OA and TSLLS methods: A+ B = ISC.




Then, the parameters B, C and D and the orginal ones Iph, Isat,Rsh, Rs and n, where explicitly given in terms of E.The more difficult part of the paper was to realize and then todemonstrate the monotonic behaviour of all the parameters interms of E.

Once monotonicity was proved, the interval to which each ofthe parameters belongs was stablished.




Likewise, a method to obtain each one of the possible infiniteI-V curves was also described.

We noticed that some methods in the literature, under certainhypothesis (usually not justified), obtain infeasible I-V curves.For instance, a widely cited article with more than 4000citations (Google Scholar), uses a hypothesis (n = 1.3) which isincompatible with large number of PV panels.




Finally, there exist in the literature a variety of extra conditionswhich try to complement the three remarkable pointsinformation in order to determine a unique solution betweenthe possible infinite ones. Some of the used methodologies toextract the unique I-V curve are such unsupported that finishgiven solutions which are infeasible.This work provides a simple methodology that allows tocalculate the corresponding I-V curve when an extra conditionis incorporated to the remarkable points.




In https://pvmodel.umh.es/ has also been implemented thecomputation of the parameters intervals with a sample of thepossible I-V curves with their graphs in the feasible region.

An interesting application of the knowledge of the parametersintervals is its use in metaheuristic algorithms that needstrongly to know a priori the intervals where the solutionbelongs to ensure the convergence of them.




The following paper provides an analytical resolution of theSDM, reducing the problem of solving a nonlinear system offive equations with five unknowns to the resolution of asingle-variable five-degree polinomial equation.

Once the equation is solved and the solution selected, theparameters are obtained explicitly.The AQE method is exact in the sense that no approximationsof the model or the resulting equations are made.




The AQE method requires 4 arbitrary points on the I-V curvewith their corresponding slopes.A real distribution of points of a curve can be seen in thefollowing image where a section of the curve is shown withzoom.

The greatest difficulty of the AQE method lies in finding theslopes of the 4 selected points. Standard mathematicaltechniques, such as approximations of the derivative by meansof the corresponding incremental quotients, do not work at all.




In this paper, it is provided an easy optimization method toobtain, directly from real data measurements, the 4 points andtheir slopes without using any kind of sophisticatedtechniques. The idea is inspired by the calculation of themaximum power point obtained with the maximum of thepower function P = VI.Given a positive real number α, the α-power functionassociated to a PV panel is defined as:

Pα = VαI

With the properties of I as a function of V in the SDM, it issatisfied that Pα has a unique relative (indeed global) maximumon the interval [0, VOC] which is attained at its unique criticalpoint. Observe that

P′α = 0⇔ I′ = −αIV




So, in a cloud of points, if we compute the maximum value ofPα, we automatically obtain a point (Vα, Iα), called maximumα-power point, in which the slope of the I-V curve isI′α = −α Iα

Vα. The case α = 1 provides de MPP.




Illustrations of some tests of the AQE method

Original (real or synthetic) curve: in black.Computed curve by the method: in red.Selected points of original curve: black small balls.




OPEN PROBLEMS

The UMH PVmodel team formed by Dr. José Manuel Blanes,Dr. Vicente Galiano, Dra. Ma Victoria Herranz and myself,together with our Ph.D. students and collaborators of otheruniversities, are currently working in the following openproblems:

Best method which recovers the original parameters of theSDM (with Dr. Batzelis from Imperial College of London)Faster computation of the SDM equation solutionsFunctional analysis of parameters and remarkable pointswith respect to the environmental conditions: temperature,irradiance, ... (with Ph.D. student)Study the behaviour of the SDM under low irradianceconditions (with Ph.D. student)Generalizations of the TSLLS method to providealternative solutions for extrem curves: organic panels




OPEN PROBLEMS

Comparation of methods to obtain SDM solutions notdependent on the resolution methodArtificial intelligence applications to photovoltaicmodelling (with Grade student)

The following open problems have yet to be addressed.

Analytical and geometrical properties of the double-diodemodelParameters stability under perturbations dataThe AQE method optimized

Thank you very muchfor your attention!


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