Estimation of PV Module Parameters from Datasheet Information Using Optimization Techniques Mohamed A. Awadallah
Bala Venkatesh
Centre for Urban Energy Ryerson University Toronto ON, Canada
[email protected]
Department of Electrical and Computer Engineering Ryerson University Toronto ON, Canada
[email protected]
Abstract—The paper presents a technique for parameter estimation of photovoltaic (PV) modules from datasheet information. The manufacturer normally provides open-circuit voltage, short-circuit current, maximum power, and voltage at maximum power under the standard test conditions (STC). The four datasheet values represent the targeted performance of the module. Parameter estimation is formulated as an optimization problem solved by traditional nonlinear programming techniques as well as global search algorithms. The objective is to search a set of parameters that minimizes the error between targeted and computed performance. The methodology is successfully applied to single- and double-diode equivalent circuits of PV modules. Evidently, the need to perform prototype lab testing, for the purpose of parameter estimation, is eliminated. Results show that genetic algorithms (GA) outperform other optimization techniques in obtaining the equivalent circuit parameters of a commercially available PV module. Index Terms—Genetic algorithms, simulated annealing, nonlinear programming, parameter estimation, PV modules.
I.
INTRODUCTION
Solar energy comes next to wind as the fastest developing renewable source. The light energy of sun is converted to DC electricity by photovoltaic (PV) cells. When the rays of sun light hit the semiconductor material of PV cells, the electrons gain energy allowing them to transfer from the valence band to conduction band. A PV cell typically generates about 2 W at 0.5 V; accordingly, cells are connected in series to form a PV module that produces higher voltage. The modules are then connected in different series and parallel configurations forming PV arrays in order to attain the desired level of voltage and current. The operation and control of PV arrays require accurate mathematical modeling, where the estimation of module parameters becomes vital. The equivalent circuit parameters of many electrical systems are usually extracted from the experimental results of prototype laboratory testing. Nevertheless, some publications in the literature consider parameter estimation of PV modules from the manufacturer datasheet information. Analytical
methods are used to express the parameters in terms of known performance of the module under certain conditions [1]-[4]. The performance could be obtained from the module ratings given in the datasheet [1], [2], or from direct measurement [3], [4]. Numerical iterative techniques may be required to extract the parameters from given performance indices [1]; otherwise, direct solution is possible upon some mathematical simplification assumptions [2]. Although most of the literature work considers parameter estimation of the single-diode equivalent circuit of PV cells, some publications are dedicated to the double-diode model as well [4]. Fitting computed performance to measured performance with minimal error is another technique to search a set of parameters that best represent the PV module [5]-[7]. The curve fitting methods vary between least square [5], voltage fitting near open circuit and current fitting near short circuit [6], and auxiliary function fitting [7]. Optimization algorithms are exploited to estimate PV module parameters by minimizing certain objective functions [8]-[13]. In [8], the discrepancy between measured and computed module current is used as an objective function to be minimized by particle swarm optimization (PSO) and genetic algorithms (GA). Statistical and cluster analyses are used with PSO to fit measured performance to that computed through the seven-parameter double-diode model of PV modules [9]. The difference between measured and computed current is minimized via GA to extract PV module parameters [10]. In [11], an objective function based on the rate of change of module current with respect to its voltage is minimized using bacterial foraging (BF), GA, and artificial immune systems (AIS). A similar objective function is minimized by differential evolution (DE) in [12]. Mean square error comparing computed current and that supplied by the manufacturer datasheet is minimized by DE to estimate the single-diode model parameters [13]. This paper presents a methodology for estimating the equivalent circuit parameters of PV modules from datasheet information. The manufacturer-supplied open-circuit voltage, short-circuit current, maximum power, and voltage at
This work was sponsored by the Centre for Urban Energy, Ryerson University, Toronto ON, Canada.
978-1-4799-7800-7/15/$31.00 ©2015 IEEE
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maximum power under standard test conditions (STC) represent the targeted performance. An optimization algorithm searches the parameter set which minimizes the relative absolute error (RAE) between computed and targeted performance. The optimization problem is solved using three nonlinear programing algorithms implemented by the Matlab function ‘fmincon’ as well as the global search techniques of GA and simulated annealing (SA). The method is applied to the single- and double-diode models of a market-available PV module. Results show the effectiveness of proposed method and superiority of GA to other techniques. The need to perform prototype testing in order to estimate the parameters is evidently eliminated. Complicated mathematical derivations and obscuring approximating assumptions are also avoided. The proposed methodology relies on performance characteristics spanning the whole range of operation from open circuit to short circuit. Unlike other literature work, the objective function formulation is straightforward and requires no mathematical derivations. II.
raise the level of output voltage and current, Fig. 1(b), the output current is expressed as
0
= - − -. /
1 3 4 6 2 25
− 17 −
8
95
9
: + ; (4)
where Np and Ns are the number of cells in parallel and series, respectively.
(a)
MODELING OF PV CELLS
A. Single-Diode Model As shown in Fig. 1(a), a PV cell can be modeled with a current source in parallel to a diode, a shunt resistance to account for leakage current, and a series resistance to represent losses related to load current. Accordingly, the cell current is given as
= −
− 1 −
(1)
where Ic is the cell current, A, Iph is the photocurrent, A, Ios is the reverse saturation current of the diode, A, q is the electron charge, C, A is the diode ideality factor, K is Boltzmann constant, J/oK, T is the cell temperature, oK, Vc is the cell voltage, V, Rs is the series resistance, Ohm, and Rsh is the shunt resistance, Ohm. The photocurrent depends on the solar irradiance and cell temperature, and is given as = + ! − !" #
(2)
where λ is the solar irradiance, kW/m2, In is the nominal shortcircuit current at 1000 W/m2 and 25 oC, ki is the short-circuit current temperature coefficient, A/oK, and Tr is the reference temperature, oK. Meanwhile, the reverse saturation current of the diode varies with temperature, and is given as $ &
= "
%$ '( * * ) + ,
%
(3)
where Ior is the reverse saturation current of the diode at reference temperature and irradiance, A, Eg is the band gap energy of the semiconductor material, J/C. When PV cells are connected in series and parallel forming a PV array in order to
(b) Fig. 1. Single-diode model of: (a) PV cell, and (b) PV array.
Equations (1) through (3) describe the performance of a PV cell. The model correlates the output variables of the cell, Vc and Ic, with the independent variables representing the environmental conditions, λ and T, through physical constants and system parameters. The Boltzmann constant (K), electron charge (q), band gap energy of the semiconductor (Eg), and reference temperature (Tr) are all constants. Whereas, series resistance (Rs), shunt resistance (Rsh), diode ideality factor (A), nominal short-circuit current (In), reverse saturation current at reference temperature and irradiance (Ior), and short-circuit current temperature coefficient (ki) are parameters of the PV cell. The numbers of cells in series and parallel are required to compute the performance of a PV array using (4). B. Double-Diode Model In the double-diode model of a PV cell, two diodes are in parallel with the photocurrent source as shown in Fig. 2. The model is known to be more accurate than the single-diode model, especially at low irradiance levels. One diode represents the diffusion current in the p–n junction, whereas the other takes the space-charge recombination effect into account. Both single- and double-diode models are widely
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accepted to represent the behavior of monocrystalline and polycrystalline semiconductor PV cells. The output I-V characteristic equation of the double-diode model becomes =
= − 8 < >*?$
=
− 1@
− A < >B ?$
− 1@ −
: + ; ; (5)
The reverse saturation currents of the two diodes are expressed in terms of their values at reference irradiance and temperature as $ &
'( * * ) + ,
* %
(6)
$ &
'( * * ) + , %
(7)
8 = "8
%$and A = "A B
%$Equations (5), (2), (6), and (7) represent the mathematical model based on the double-diode equivalent circuit. The model parameters are series resistance (Rs), shunt resistance (Rsh), ideality factors for both diodes (A1 and A2), nominal short-circuit current (In), reverse saturation current at reference temperature and irradiance for both diodes (Ior1 and Ior2), and short-circuit current temperature coefficient (ki). It is obvious that the second diode adds two parameters to the model.
Fig. 2. Double-diode model of a PV cell.
Manufacturer datasheets normally supply the open-circuit voltage, short-circuit current, and maximum power point under STC (1000 W/m2 irradiance, 25 oC temperature, and 1.5 air mass). Datasheets also provide temperature coefficients of short-circuit current, open-circuit voltage, and maximum power. III.
PROBLEM STATEMENT
The objective of this research is to estimate the model parameters of PV modules from datasheet information. The module characteristics supplied by the manufacturer at STC are considered the targeted performance. A set of parameters which yields computed performance as close as possible to targeted performance is sought. The targeted performance signifies four indices, i.e., the short-circuit current, opencircuit voltage, maximum power, and voltage at maximum power, all at STC, as supplied by the manufacturer. The obtained parameter set has to minimize the relative absolute error (RAE) that measures the discrepancy between computed and targeted performance. The RAE is expressed as
;CD = ∑JK8
|GHI + GI | GI
(8)
where XCi is the ith computed index and XTi is the ith targeted index. The short-circuit current temperature coefficient (ki) and nominal short-circuit current at STC (In) are usually supplied by the manufacturer through datasheet. Therefore, the parameters to be estimated for the single-diode model are the series resistance (Rs), shunt resistance (Rsh), diode ideality factor (A), and diode reverse saturation current at reference temperature and irradiance (Ior). However, for the doublediode model, six parameters are to be estimated including Rs, Rsh, A1, A2, Ior1, and Ior2. The RAE given in (8) is considered an objective function to be minimized by optimization techniques. The optimization problem is independently solved by the nonlinear programming algorithms coded in the Matlab function ‘fmincon’ as well as the global search routines of GA and SA. The objective function formulation of this work covers a wide operating range and requires no prototype testing or mathematical derivation, unlike other practices [8]-[13]. IV.
OPTIMIZATION TECHNIQUES
A. Nonlinear Programming Algorithms The Matlab function ‘fmincon’ enables the use of four different nonlinear optimization algorithms [14]. The trustregion algorithm is one of the most basic, yet powerful, search techniques of optimization. However, the algorithm requires user-defined derivatives of the objective function, which is not possible for the problem at hand. Accordingly, the trust-region algorithm is not used with the present optimization problem. The active-set algorithm solves the Karush-Kuhn-Tucker (KKT) equations, which are necessary conditions for optimality of constrained problems. The algorithm, therefore, attempts to compute the Lagrange multipliers directly. It assures superlinear convergence by accumulating secondorder information using a quasi-Newton updating procedure. Active-set is, however, a medium-scale algorithm. The sequential quadratic programming (SQP) algorithm is quite similar to active-set; however, some differences exist in favor of SQP. The differences include strict feasibility with respect to bounds, adaptive step size, fast convergence, efficient use of memory, possibility of objective and constraint functions combination, and second-order approximation of constraints. The interior-point algorithm of nonlinear programming solves a sequence of approximate minimization problems. Solution of an approximate problem incorporates either a direct step or a conjugate gradient step using a trust region. The optimization problem of this work is independently solved using the active-set, SQP, and interior-point algorithms as coded by the Matlab function ‘fmincon’. B. Simulated Annealing An optimization method that mimics the physical process of heating, then slowly cooling down, a material to decrease defects and minimize system energy is called simulated annealing [15]. Annealing is the technique of closely controlling the temperature when cooling the material to ensure that it reaches an optimal state. The SA routine works in such a way that a new point is randomly generated in the
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search space every iteration. The distance of the new point from the current point, or the search extent, follows a probability distribution with a scale proportional to temperature. The algorithm accepts any new point that lowers the objective function. However, points that raise the objective function could be accepted with certain probability in order to help the routine escape local minima. The temperature is an SA parameter which affects the distance of a trial point from the current point as well as the probability of accepting a trial point with higher objective function. An annealing schedule is selected to systematically decrease the temperature and reduce the search extent as the algorithm proceeds in order to converge to a minimum. Temperature decreases gradually as the algorithm proceeds; it could be a function of the iteration number. The slower the rate of temperature decrease, the better the chance to find an optimal solution, but the longer the convergence time. The annealing parameter is a proxy for the iteration number. The algorithm can raise temperature by setting the annealing parameter to a lower value than the current iteration. Re-annealing raises the temperature after the algorithm accepts a certain number of new points, and starts the search again at a higher temperature to escape local minima. Unlike other multi-agent optimization techniques, SA is a single-point global search algorithm with lesser computational burden per iteration. Stopping criteria of the routine include stagnation of fitness function, reaching runtime limit, processing certain number of iterations, obtaining a desired value of the objective function, evaluating the objective function for a pre-defined number of times, or a combination of such conditions. C. Genetic Algorithms The GA is a probabilistic random guided search technique inspired by the Darwinian theory of evolution, which employs the “survival of the fittest” concept of natural biology [15]. One distinct feature of GA is that the routine starts searching from a population of points, not a single point, with no need to have information about the derivatives of the objective function. The algorithm codes prospective solutions of the problem as a population of individual chromosomes of different genes. The population is randomly initialized and the individuals are evaluated based on the corresponding values of an objective fitness function. Fit individuals are probabilistically copied to the “mating pool”, while weak individuals likely die as their probability of selection is small due to the poor fitness. The natural genetic processes of crossover and mutation are then imitated in order to mate parents of the current generation, and produce the offspring of the next generation. Based on Darwin’s theory, the population evolves from one generation to the next as the best fitness improves. In spite of the remarkable robustness of GA in finding global optima, the slowness of operation could be a significant obstacle in some applications. However, GA has proven notable effectiveness in many types of optimization problems. Coding of individual solutions has granted GA one more apparent plus, which is its adaptability to certain optimization problems that could not be solved by classical or even some other evolutionary techniques. The sequentially applied genetic operations of selection, crossover, and mutation
attributed GA with the ability to escape local optima, besides the robustness of attaining global ones. V.
RESULTS
The equivalent circuit parameters of a commercial PV module are estimated via the proposed technique. The datasheet information of the Solarex MSX60 PV module is given in Table I. The short-circuit current at STC (In) and short-circuit current temperature coefficient (ki) are provided by the manufacturer. Therefore, the single-diode model lacks four parameters (Rs, Rsh, A, and Ior), whereas the sought parameters of the double-diode model are six (Rs, Rsh, A1, A2, Ior1, and Ior2). The target performance denotes the short-circuit current, open-circuit voltage, maximum power, and voltage at maximum power under STC (Isc, Voc, Pmax, and VMPP, respectively) as supplied by the manufacturer. The nonlinear optimization problem is solved in four- and six-dimension search spaces for the single- and double-diode models, respectively. TABLE I.
DATASHEET INFORMATION OF THE PV MODULE
Short-circuit current at SCT
3.8 A
Open-circuit voltage at STC
21.1 V
Maximum output power at STC
60 W
MPP Voltage at SCT
17.1 V
SC current temperature coefficient (ki)
0.065 %/oC
Number of cells in series
36
The objective function of (8) is independently minimized using three nonlinear programming algorithms and two global search techniques. The obtained parameters of the single- and double-diode models are given in Table II for different optimization techniques. Due to the stochastic nature of SA and GA, both algorithms are run for 10 times on every case, where the best results are reported in Table II. In terms of the objective function value at convergence, GA always yields the best result. Meanwhile, the global search techniques generally reach a less value of the objective function than all nonlinear programming algorithms. In terms of the convergence time, nonlinear programming algorithms are distinctly faster. The quality of the estimated parameters is tested by inserting each set into the corresponding model in order to compute the performance. Computed performance indices are compared with targeted ones in Table III, and the percentage error is also given. It appears that the performance indices computed via the parameters estimated by GA are the closest to targeted ones. In general, the global search techniques of SA and GA converge to better solutions of the present problem than nonlinear programming algorithms. Also, the performance computed by the double-diode model is more accurate than that of the single-diode model. The most challenging part for a parameter set is to attain the targeted maximum power at the targeted voltage. Best result is obtained via single-diode model with parameters estimated by GA. An experimental verification study of these results will be published in a subsequent paper.
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TABLE II.
ESTIMATED PARAMETERS OF THE PV MODULE Single-diode model
Parameter Rs, Ω Rsh, Ω A Ior, A Convergence time, sec Objective Function
Nonlinear programming algorithms Active set SQP Interior point 0.5201 0.7677 0.571 200 200 200 1.1393 1.1952 1.1737 6.53×10–9 1.02×10–8 1.022×10–8 8.96 9.89 11.26 0.0158 0.0559 0.0243
Global search algorithms SA GA 0.2548 0.3379 400.77 455.83 1.2814 1.2732 7.6593×10–8 6.2178×10–8 48.62 48.61 0.0057 0.0011
Double-diode model Rs, Ω Rsh, Ω A1 A2 Ior1, A Ior2, A Convergence time, sec Objective Function
0.761 200 1.1808 2.0002 7.28×10–9 2.32×10–6 9.34 0.0531
0.7677 200 1.1952 2.0002 1.02×10–8 1.02×10–8 11.1 0.0558
TABLE III.
0.4217 199.69 1.1437 1.9982 8.03×10–9 8.97×10–9 8.71 0.0072
0.3905 322.36 1.1615 1.6319 1.21×10–8 5.88×10–8 64.4 0.0067
0.3471 430.86 1.2506 1.9858 4.45×10–8 4.98×10–9 38.38 8.0530×10–4
SOLUTION COMPARISON BASED ON COMPUTED PERFORMANCE OF THE PV MODULE Single-diode model
Performance Index (STC)
Targeted
Isc, A Voc, V Pmax, W VMPP, V
3.8 21.1 60 17.1
Isc, A Voc, V Pmax, W VMPP, V
3.8 21.1 60 17.1
Nonlinear programming algorithms Active set SQP Interior point Error, Error, Error, Computed Computed Computed % % % 3.7901 –0.26 3.7855 –0.38 3.7892 –0.28 21.2455 0.69 21.7939 3.29 21.4003 1.42 59.9999 –0.0002 60.265 0.44 59.9484 –0.09 17.208 0.63 17.352 1.47 17.208 0.63
Global search algorithms SA GA Error, Error, Computed Computed % % 3.7976 –0.06 3.7972 –0.07 20.9933 –0.51 21.1065 0.03 59.9979 –0.004 59.9997 –0.001 17.1 0 17.1 0
Double-diode model 3.7856 21.806 59.677 17.28
–0.38 3.35 –0.54 1.05
3.7855 21.7935 60.2602 17.352
–0.38 3.29 0.43 1.47
3.792 21.1087 59.971 17.172 [2]
VI.
CONCLUSIONS
The paper presents a methodology for estimating PV module parameters based on optimization techniques. The task is formulated as a nonlinear optimization problem that minimizes the discrepancy between computed and targeted performances. Computed performance is obtained independently through single- and double-diode models; whereas, targeted performance represent nominal outputs at STC as reported in the datasheet. The problem is solved using different nonlinear optimization algorithms and global search techniques. The obtained sets of parameters are evaluated by comparing their computed performance to targeted performance. Although nonlinear optimization algorithms require lesser time for convergence, results of global search techniques are clearly more accurate. Best results are usually obtained through GA.
[3]
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