Rethinking solar photovoltaic parameter estimation: global optimality analysis anda simple efficient differential evolution method

Shuhua Gao1, Cheng Xiang1,∗, Yu Ming2, Tan Kuan Tak3, Tong Heng Lee1

Abstract

A large variety of metaheuristics have been proposed for photovoltaic (PV) parameter extraction. Our aim here is notto develop another new metaheuristic method but to investigate two important yet rarely studied issues. We focuson the two most widely used benchmark datasets and try to answer (i) whether the globally optimal parameters havealready been found; and (ii) whether a significantly simpler metaheuristic, in contrast to currently sophisticated ones, canachieve equally good performance. We address the former using a branch and bound algorithm, which certifies the globalminimum rigorously for the single diode model (SDM) and locates a fairly tight upper bound for the double diode model(DDM) on both datasets. These obtained values can serve as useful references for evaluation and design of metaheuristicmethods. However, this algorithm is excessively slow and unsuitable for time-sensitive applications (despite the greatinsights that it yields). Next, extensive examination and comparison reveals that, perhaps surprisingly, a classic andremarkably simple differential evolution (DE) algorithm can consistently achieve the certified global minimum for theSDM and obtain the best-known result for the DDM on both datasets. Moreover, the simple DE algorithm takes onlya fraction of the runtime required by other contemporary metaheuristics and is thus preferable in real-time scenarios.This novel, unusual, and notable finding also indicates that the employment of increasingly complicated metaheuristicsmight be somewhat overkill for regular PV parameter estimation. Finally, we discuss the implications of these resultsand suggest promising directions for future development.

Keywords: Photovoltaic modeling, Parameter identification, Metaheuristic algorithms, Global optimization,Differential evolution

1. Introduction

Solar energy is one of the most important renewable en-ergy resources characterized by its emission-free cleanness,ubiquitous abundance, high sustainability, favorable powerdensity, and almost cost-free exploitation except the ini-tial investment (Navabi et al., 2015; Jordehi, 2016a; Guoet al., 2016). The major application of solar energy isphotovoltaic (PV) power generation through a solar PVsystem composed of many PV cells connected in series orparallel (Li et al., 2019; Li and Xu, 2018). A PV cell isa fundamental unit that converts sunlight to electricity,which is basically a semiconductor diode with a P-N junc-tion exposed to the light (Villalva et al., 2009). Despitethe potential advantages of solar energy, current PV sys-tems still suffer from low efficiency and large generation

∗Corresponding authorEmail addresses: elegaos@nus.edu.sg (Shuhua Gao),

elexc@nus.edu.sg (Cheng Xiang), ming.yu@pa.com.sg (Yu Ming),KuanTak.Tan@SingaporeTech.edu.sg (Tan Kuan Tak),eleleeth@nus.edu.sg (Tong Heng Lee)

1Department of Electrical and Computer Engineering, NationalUniversity of Singapore, 119077 Singapore

2Power Automation Pte Ltd, 438B Alexandra Road, AlexandraTechnoPark, 119968 Singapore

3Engineering Cluster, Singapore Institute of Technology, 10 DoverDrive, 138683 Singapore

fluctuations, whose power primarily depends on the vary-ing environmental factors like irradiation and temperature(Navabi et al., 2015; Jordehi, 2016a). Accurate modelingof PV cells is thus necessary for the design, simulation,power forecasting, and optimal control of solar PV sys-tems (Jordehi, 2016a; Navabi et al., 2015; Gong and Cai,2013; Li et al., 2019; Guo et al., 2016; Yang et al., 2020).

The dominating method to describe solar PV systemsdepends on an analogous electrical circuit model (Villalvaet al., 2009), which, based on the number of diodes, hasbeen further specialized to the single-, double-, or eventhree-diode models (Chenouard and El-Sehiemy, 2020; Liet al., 2019; Yang et al., 2020; Jordehi, 2016a; Chin et al.,2015; Qais et al., 2019; Zaimi et al., 2019). Despite the in-tuitiveness of these circuit models, the main difficulty liesin the accurate determination of unknown parameters in amodel. To address this challenge, an increasing number ofstudies have been conducted in recent years, yielding var-ious algorithms to identify PV model parameters to highprecision (see, e.g., (Chin et al., 2015; Yang et al., 2020;Jordehi, 2016a) for detailed reviews). There are generallytwo kinds of data sources for PV parameter estimation:datasheets provided by manufacturers and experimentallymeasured I-V (current-voltage) curves (Chin et al., 2015;Yousri et al., 2020; Ishaque and Salam, 2011). In contrast

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to the few essential data points restricted to the standardtest condition in the datasheet (Almonacid et al., 2010),an experimental I-V curve generally contains more datapoints measured in any practical operation condition.

Parameter estimation methods are broadly divided intotwo classes, analytical methods and nonlinear optimiza-tion algorithms, which correspond roughly to the two datasources mentioned above (Yang et al., 2020; Li et al., 2019;Yousri et al., 2020). Analytical methods are distinguishedby their simplicity and rapid estimates through extractinginformation at the essential points to formulate a set ofequations to identify unknown parameters (Yousri et al.,2020; Li et al., 2019; Navabi et al., 2015; Tong and Pora,2016; Zaimi et al., 2019). Nonetheless, to allow completeparameter estimation with limited information extractedfrom merely essential points, some possibly restrictive as-sumptions and simplifications have to be made, which maycompromise the accuracy and physical reliability of the re-sults (Li et al., 2019; Yousri et al., 2020). We refer readersto (Chin et al., 2015) and (Cotfas et al., 2013) for a moredetailed review of analytical methods.

The second class of methods usually formulates PVparameter estimation as a nonlinear optimization problemfrom the perspective of I-V curve fitting (Jordehi, 2016b),which has drawn a great deal of research interest in thepast decade. Due to the nonlinearity and nonconvexitypresent in the problem, a large number of nature-inspiredmetaheuristics have thus been employed to tackle this chal-lenge in view of their global search ability (Yang, 2020).The employment of metaheuristics for PV parameter esti-mation is also the focus of our study in this paper.

Most metaheuristics are population-based by exploit-ing a swarm of interacting agents to search the solutionspace efficiently, e.g., the classic genetic algorithm (Dizqahet al., 2014) and particle swarm optimization (Nunes et al.,2018). Since such nature-inspired metaheuristic algorithmsare generally not problem-specific, any metaheuristic opti-mizer may be applied to parameter estimation of PV cellsin principle. As reported in a recent survey (Yang, 2020),there are more than 100 different nature-inspired algo-rithms and variants. It is thus unsurprising that numerousmetaheuristic methods have been employed for PV param-eter estimation. Some recent examples include cat swarmoptimization (Guo et al., 2016), artificial been swarm opti-mization (Askarzadeh and Rezazadeh, 2013), moth-flameoptimizer (Sheng et al., 2019), improved JAYA optimiza-tion (Yu et al., 2017b, 2019), sunflower optimization (Qaiset al., 2019), marine predators algorithm (Yousri et al.,2020), self-adaptive ensemble-based differential evolution(Liang et al., 2020a), multiple learning backtracking search(Yu et al., 2018), teaching-learning-based optimization (Liet al., 2019; Chen et al., 2018), chaotic whale optimization(Oliva et al., 2017), repaired adaptive differential evolu-tion (Gong and Cai, 2013), and variants of these algo-rithms, among many others. Interested readers may referto the latest survey (Yang et al., 2020) for a comprehen-sive review of metaheuristics in PV parameter identifica-

tion. Finally, we note that there also exist studies thattry to hybridize traditional numerical optimization meth-ods with metaheuristic algorithms in the hope of bringingbenefits from both worlds, for example, the combination ofthe Nelder-Mead simplex algorithm with the artificial beecolony algorithm (Chen et al., 2016) and with the mothflame optimization (Zhang et al., 2020).

Despite the great expansion in interest of such meta-heuristics, it should be noted that, to our best knowl-edge, none of them can guarantee the discovery of globaloptima; even though almost all of them have designedspecific mechanisms to help get away from local minima(Yang, 2020) (e.g., the well-known probabilistic acceptanceof worse solutions during iterations in simulated annealing(El-Naggar et al., 2012)). In the literature of PV param-eter estimation, the performance of a metaheuristic algo-rithm is typically justified by examining it empirically onsome benchmark I-V curve datasets, like the widely usedRTC France solar cell data and the Photowatt-PWP201PV module data originally introduced in (Easwarakhan-than et al., 1986). However, it can be observed in re-cent studies that, on the two representative benchmarkdatasets, the difference in performance measured by rootmean square error (RMSE) among many metaheuristicmethods appears negligible. For example, it is only atthe level of 1E-6 that the majority of methods exhibit per-ceptible difference in terms of RMSE using either the sin-gle diode or the double diode model on the RTC Francedataset (see (Li et al., 2019, Table 3), (Gong and Cai,2013, Table 9), (Guo et al., 2016, Table 3), and Table 8in this paper). Thus, one may wonder naturally whethera significant improvement is still possible by introducingmore advanced and typically more complex metaheuris-tics. Also, since a variety of metaheuristics characterizedby different complexity and distinct working principles areall able to get the same RMSE value, a natural query ishow sophisticated a metaheuristic has to be, e.g., with ex-tra parameter adaptation (Liang et al., 2020a) and strate-gic hybridization (Chen et al., 2016), in order to estimatePV parameters effectively. To put it more formally, wehave at least the following two open problems at present:

(i) Have the existing methods arrived at the minimumpossible RMSE in respect of the two most widelyused benchmark datasets, since a bottleneck (i.e., nofurther reduction) seems to emerge for methods pro-posed in the last five years?

(ii) If the above global minimum has already been at-tained by a few sophisticated metaheuristic methodsin the literature, can a simple and computationallycheap metaheuristic get equally good results?

We believe that answers to the above questions arevaluable to both the research community and industrialpractitioners. In one aspect, if the best-known minimumof the objective function that the current literature hasachieved can be shown to be the global minimum (in thesense of ε-optimality (Floudas, 2013)), future researchers

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should not expect to get even better solutions on thisbenchmark by striving for novel and convoluted meta-heuristics. Instead, researchers may put more efforts onother factors of PV parameter identification aside fromaccuracy, e.g., performance robustness, usability, and timeefficiency. In another aspect, a practitioner in the solar in-dustry may get unfortunately overwhelmed by the abun-dance of metaheuristics in the current literature and can-not figure out quickly which method is ideal, if any, fora practical task. Instead of a blind and tedious exami-nation of existing algorithms, it is more meaningful andof practical importance to start with a simple yet promis-ing algorithm without time-consuming parameter tuning.Such a simple, efficient, yet effective algorithm will be rec-ommended in this study, which answers the second ques-tion above and provides valuable insights for forthcomingstudies.

In this study, we investigate the above two problemsusing the two most broadly adopted benchmark datasets(Easwarakhanthan et al., 1986). The main contributionsof this paper are summarized as follows.

• We consider two popular PV cell models, includingthe five-parameter single diode model (SDM) and theseven-parameter double diode model (DDM). An in-terval arithmetic based branch and bound methodis used to get the certified global minimum for theSDM and a useful upper bound of the global min-imum for the DDM in terms of RMSE. These val-ues can serve as useful references for the evaluationand further development of metaheuristics. The ob-tained results reveal that a few current methods havealready obtained the ε-global minimum for the SDM.

• We show, for the first time, that a simple differen-tial evolution (DE) algorithm can adequately attainthe global minimum for the SDM and achieve equallygood accuracy for the DDM compared with a varietyof sophisticated state-of-the-art metaheuristics. Be-sides, the DE algorithm is characterized by high sta-bility and incomparable time efficiency. Overall, theDE algorithm turns out to be a favorable alternativefor PV parameter estimation despite its simplicity.

• All obtained results are analyzed, compared, and val-idated with other contemporary algorithms. Basedon our findings, we recommend the simple DE to so-lar industry practitioners as the first choice in prac-tical applications, especially time-sensitive ones. Be-sides, we provide useful suggestions for future studiesto refresh viewpoints on PV parameter estimationand to refrain from possible over-engineering causedby overcomplicated metaheuristics.

The remainder of this paper is organized as follows.Common PV models are introduced in Section 2. The pa-rameter estimation is formulated as an optimization prob-lem in Section 3. After that, the two optimization methods

𝐼𝑝ℎ 𝐼𝑑 𝐼𝑝

𝑅𝑝

𝑅𝑠 𝐼+

−

𝑉

𝐼𝑝ℎ 𝐼𝑑1 𝐼𝑝

𝑅𝑝

𝑅𝑠 𝐼+

−

𝑉

𝐼𝑑2

Figure 1: Equivalent circuit of the single diode model.

we use in this study are described in Section 4. We thenpresent results obtained on two benchmark datasets forboth models and conduct a detailed comparison in Sec-tion 5. Finally, in Section 6, we give concluding remarkson our findings and suggest possible future work.

2. Mathematical modeling of PV systems

In this section, we introduce the two most widely usedcircuit models for PV systems, namely the single diodemodel (SDM) and the double diode model (DDM). Thephysical rationale behind both models is that an ideal PVcell acts like a semiconductor diode whose P-N junctionis exposed to the light to produce an electrical currentthrough the photovoltaic effect (Villalva et al., 2009).

2.1. Single diode model (SDM)

The SDM is arguably the most popular model in theliterature, especially for parameter estimation, due to theirleast number of parameters and yet provable effectiveness.The electrical circuit corresponding to the SDM is shownin Fig. 1. Specifically, the circuit contains a current sourceIph, which refers to the photocurrent generated by the PVcell, a diode flowing current Id, and two resistors withresistance Rp and Rs, respectively. The output current Iis obviously

I = Iph − Id − Ip. (1)

where Ip is the current through the parallel resistance Rp

(also known as shunt resistance (Jordehi, 2016a; Zhanget al., 2020)). Supposing the output voltage is V , thecurrent Ip is obviously

Ip =V + IRs

Rp. (2)

Next, we can calculate the diode current Id using theShockley equation as follows,

Id = I0

[exp

(q(V + IRs)

nkT

)− 1

], (3)

where I0 is the reverse saturation current of the diode, n isthe diode ideal factor, T is the temperature in Kelvin, andV is the output voltage of the cell. The other terms are justphysical constants: the electron charge q = 1.60217646 ×

3

𝐼𝑝ℎ 𝐼𝑑 𝐼𝑝

𝑅𝑝

𝑅𝑠 𝐼+

−

𝑉

𝐼𝑝ℎ 𝐼𝑑1 𝐼𝑝

𝑅𝑝

𝑅𝑠 𝐼+

−

𝑉

𝐼𝑑2

Figure 2: Equivalent circuit of the double diode model.

10−19 C and the Boltzmann constant k = 1.3806503 ×10−23 J/K.

By inserting Eq. (2) and (3) into Eq. (1), we get thecomplete equation of the SDM:

I = Iph − I0[exp

(q(V + IRs)

nkT

)− 1

]− V + IRs

Rp. (4)

There are five unknown parameters in (4), which arecollected into a vector θS = [Iph, I0, n,Rs, Rp].

2.2. Double diode model (DDM)

Despite the simplicity and usefulness of the above SDM,it does not consider the effect of recombination current lossin the depletion region (Jordehi, 2016a; Chin et al., 2015;Zhang et al., 2020). An additional diode can be introducedinto the circuit to compensate for this specific loss to at-tain higher accuracy. The equivalent circuit of the DDM isillustrated in Fig. 2, in which there are two parallel diodeswith current Id1 and Id2 respectively. One diode plays therole of a rectifier, while the other emulates the recombina-tion effect (Zhang et al., 2020; Yousri et al., 2020).

The only difference between the SDM and the DDMfrom the perspective of mathematical equations is the in-clusion of an additional current term. In analogy to theSDM (4), the DDM is derived straightforwardly as follows:

I =Iph − Id1 − Id2 − Ip

=Iph − I01[exp

(q(V + IRs)

n1kT

)− 1

]−

I02

[exp

(q(V + IRs)

n2kT

)− 1

]− V + IRs

Rp, (5)

where I01 and I02 are the reverse saturation current of thetwo diodes, and n1 and n2 denote the ideality factor of thetwo diodes, respectively. In contrast to the SDM discussedabove, the DDM has seven parameters in total, denotedby θD = [Iph, I01, I02, n1, n2, Rs, Rp].

We note that, though in theory the DDM may bettercapture fine-grained electrical characteristics of PV cellsespecially at low irradiance conditions, the inclusion oftwo additional parameters poses more difficulty for param-eter estimation. In particular, the DDM is seldom used indatasheet-based analytical methods because the accuratedetermination of seven parameters requires at least sevenequations, but at most five equations can be formed with

the manufacturer’s datasheet in general (Chin et al., 2015;Tong and Pora, 2016). To further refine the circuit-basedmodel, another diode may be inserted to form the threediode model (Yousri et al., 2020; Qais et al., 2019), whichhas in total of nine parameters. However, there is no solidevidence justifying the usage of such complex models: theynot only challenge the effective estimation of PV param-eters but also increase the computation time, whose ap-plication in PV system simulations is consequently quitelimited (Jordehi, 2016a; Villalva et al., 2009).

2.3. PV module model

A practical PV module usually contains multiple PVcells that are connected in series or parallel. SupposingNs is the number of series PV cells and Np is the numberfor parallel connection, the overall model of a PV modulebased on the SDM of an individual cell is (Li et al., 2019;Zhang et al., 2020; Yu et al., 2019)

I =IphNp − I0Np

[exp

(q(V/Ns + IRs/Np)

nkT

)− 1

]−

V/Ns + IRs/Np

Rp/Np. (6)

Note, however, that Eq. (6) holds only if all PV cells inthe module are identical, i.e., they share the same parame-ter values. This assumption is unlikely to be true in realitydue to the intrinsic heterogeneity among cells. For this rea-son, it is standard in the literature to fit the SDM (4) toPV module data directly by lumping all cells into a single,functionally equivalent cell (Easwarakhanthan et al., 1986;Yu et al., 2019). Of course, the corresponding parametervalues of a PV module (6) are supposed to be much largerthan those of an actual cell. The extension of the DDM(5) to a PV module is straightforwardly analogous to (6)and omitted here.

3. Problem formulation

We focus on the I-V curve fitting methods in this study,which typically formulate parameter estimation into a non-linear optimization problem. The core principle is to findappropriate parameter values such that the current valuescalculated with either the SDM (4) or the DDM (5) matchthe measurement values closely for a set of current-voltagedata points. To simplify notations, we shorten the SDM(4) and the DDM (5) as the following two equations:

I = fS(V, I;θS) (SDM) (7a)

I = fD(V, I;θD) (DDM) (7b)

Without loss of generality, we continue our discussion to-wards problem formulation using the SDM fS in (7a) (withθS omitted for conciseness) in the following part, whoseprinciple can be transplanted to the DDM seamlessly.

From a traditional perspective of curve fitting, e.g., ourprevious work (Gao et al., 2018), we may view the voltage

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as the input and the current as the output in (7a). How-ever, the peculiarity of PV parameter estimation is thatthe current is related to the voltage only implicitly (Cot-fas et al., 2013), that is, we cannot write down a simpleclosed-form solution I = f−1S (V ) for the model fS to com-pute I given V (Nunes et al., 2018). Consequently, givena voltage value V , the corresponding current value I gen-erally has to be computed iteratively using a numericalroot-finding algorithm like the Newton-Raphson methodto solve the nonlinear equation I − fS(V, I) = 0 (Yousriet al., 2020; Jordehi, 2016b; Nunes et al., 2018; Kler et al.,2019). It deserves to be mentioned that in some recentstudies an analytical solution to f−1S has been obtained us-

ing the Lambert W function (Yousri et al., 2020; Calasanet al., 2020). However, this method only applies to the5-parameter SDM (Calasan et al., 2020), and the compu-tation of the Lambert W function itself is iterative (Ab-dulrazzaq et al., 2020) (see equations (6)-(11) in (Calasanet al., 2020)). Another potential drawback of such itera-tive methods is the increased computational burden, espe-cially in the case of curve fitting based parameter estima-tion, since thousands of function evaluations usually needto be performed. Hence, given a measurement (V m, Im)and a parameter vector θS (not necessarily the groundtruth), the majority of metaheuristic-based studies in theliterature compute the predicted current with the modelapproximately but computationally economically as

I ≈ fS(V m, Im;θS), (8)

and try to reduce the deviation between I and Im by ad-justing θS (see (Chen et al., 2016), (Li et al., 2019), (Nuneset al., 2018), (Yu et al., 2019), and (Liang et al., 2020b)among others). It is evident that, if we can find a parame-

ter vector θS such that the computed I in (8) is sufficiently

close to Im for each measurement, then θS is a good esti-mation of the ground truth θ∗S , which in turn justifies therationality of the approximation in (8).

Two metrics are frequently used in the literature toquantify the difference between a computed current valueI and the true measurement value Im (Yang et al., 2020;Jordehi, 2016a), namely, the mean absolute error (MAE)(Chenouard and El-Sehiemy, 2020; Guo et al., 2016) andthe root mean square error (RMSE) (Liang et al., 2020a;Zhang et al., 2020; Yousri et al., 2020; Gnetchejo et al.,2019). In line with the majority of the literature and forpurposes of comparison, we adopt the RMSE metric in thisstudy as follows:

RMSE =

√√√√ 1

N

N∑i=1

(Ii − Imi )2, (9)

where N is number of measured data points.A well-known trick to slightly promote computational

efficiency of (9) is to avoid the square root calculationand use an equivalent metric called the sum of squarederror (SSE), whose optimal solution is exactly the same

as the one of RMSE. In summary, we get the followingconstrained optimization problem, which is known widelyas nonlinear least squares regression in the literature.

minimize J(θ) =

N∑i=1

(f(V mi , Imi ;θ)− Imi )

2, (10a)

subject to θ ∈ Θ. (10b)

In the above equation (10), f refers to either the SDMfS (4) or the DDM fD (5), and θ is the correspondingparameter vector θS or θD. (V m

i , Imi ) is the i-th datapoint in measurement. Θ denotes the bound constraintsof θ that take physical reality into consideration (see theparameter ranges listed in Table 1 for examples).

4. Optimization methods

To address the open problems we listed in Section 1, weattempt to (i) determine rigorously the global minimum orat least a reasonably tight bound of the global minimumusing numerical optimization algorithms and (ii) demon-strate that an intentionally simple metaheuristic can achievethe global minimum or at least attain the best minimumknown in the literature. To our best knowledge, no existingmetaheuristic algorithms, belonging to stochastic globaloptimization family, can certify the global minimum orquantify the optimality gap for the PV parameter esti-mation problem (10) (Chenouard and El-Sehiemy, 2020;Yang, 2020). To evaluate the optimality rigorously in task(i), we apply a branch and bound (B&B) based determinis-tic global optimization technique. For task (ii), we choosea simple and widely used evolutionary algorithm, differen-tial evolution (DE). We detail the mathematical principlesof both methods in the remainder of this section.

4.1. Deterministic global optimization with an interval arith-metic based branch-and-bound algorithm

The fundamental task of deterministic global optimiza-tion (DGO) is to determine rigorously (i.e., with theo-retical guarantees) the global minimum of an objectivefunction f subject to a set of constraints C (Floudas andGounaris, 2008; Floudas, 2013). DGO is notoriously trickywhen handling nonconvex optimization problems: even tocheck the feasibility of a given problem may be out ofreach (Floudas and Gounaris, 2008; Burer and Letchford,2012). In fact, a general nonconvex optimization problemlike the nonlinear least squares in (10) has been provedto be NP-hard (Floudas, 2013). From a practical point ofview, we are usually more interested in identifying a solu-tion sufficiently close to the true global minimum, calledthe ε-global minimum, which is defined formally below.

Definition 1. (Floudas, 2013) Consider the problem ofminimizing f(x),x ∈ S. A feasible solution x ∈ S is anε-global minimum if f(x) ≥ f(x)− ε, ∀x ∈ S, where ε ≥ 0is a small prescribed tolerance.

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The most popular algorithmic framework of DGO tohandle such general NLP problems is arguably the branch-and-bound (B&B) method and its variants like branch-and-reduce (Burer and Letchford, 2012; Floudas and Gounaris,2008; Floudas, 2013). The B&B framework for global opti-mization is usually attributed to McCormick (McCormick,1976), whose general principle is intuitive. The searchspace is divided recursively into smaller subspaces andforms accordingly a tree structure of subproblems. Onecrucial factor to success lies in the determination of properbounds of each subproblem, usually via convex or concaverelaxation. The consequential pruning of search space isperformed by eliminating subproblems whose lower boundis no better than the best upper bound found so far (Burerand Letchford, 2012; McCormick, 1976). In the worst case,if only a fraction of subproblems can be pruned, the perfor-mance of B&B is close to exhaustive search. Fortunately,interval analysis is often a useful tool for efficient estima-tion of the lower and upper bounds of regions/branches ofthe search space.

The definitive characteristic of interval analysis (alsoknown as interval arithmetic) is to represent a single num-ber x as an interval [a, b] that contains x in order to tracknumerical errors such as rounding errors and measurementerrors. Apart from their numerical reliability, intervalmethods have important applications in rigorous global op-timization and location of all solutions to nonlinear equa-tions (Hansen and Walster, 2003). Common arithmetic op-erators and mathematical functions have been extended tointervals. For example, given any binary operator among(+,−,×,÷), we have

[x1, x2] [x2, y2] = x y|x ∈ [x1, x2], y ∈ [y1, y2].

As mentioned above, one critical operation in B&B forglobal optimization is to reduce efficiently the hierarchicalsubproblems by pruning the search tree; otherwise, B&Bdegenerates to brute-force search that is intractable formost nontrivial problems. Recall that pruning efficiencyrelies significantly on the tightness of estimated bounds(especially the lower bound) of the objective function fora specific subproblem. Since applying a function to an in-terval yields another interval, interval analysis turns out tobe valuable in this particular aspect due to its capacity tocompute rigorous yet sharp bounds for a function f overan interval vector [x] (also known as an interval box, whosecomponents are intervals) using techniques like hull con-sistency, Taylor expansion, and interval Newton method(Hansen and Walster, 2003, Chapter 7). Note that ourproblem (10) has only bound constraints, which is viewedas an unconstrained optimization problem with an initialinterval box defined by the bound constraints in intervalanalysis. Once the function bounds over a subregion (i.e.,a subbox) are available, we can detect and discard an un-promising subregion, whose lower bound is greater thana known minimum value at hand, from the subsequentsearch for a global minimum.

The general B&B framework with interval arithmeticis depicted in Algorithm 1, where [x] denotes an intervalbox and [f([x]), f([x])] denotes the interval value of apply-ing function f to [x]. Each iteration is composed of threemain components: box selection (Line 5), box contracting(Line 6), and box splitting (Line 12). Starting from theinitial box [x]0 corresponding to the bound constraints X,the list L tends to accumulate many boxes over the time.There is no standard way in Line 5 to choose the box [x]from L to be processed next, but a simple heuristic is topick the box that has the smallest lower bound f([x]),which is more likely to contain the global minimizer x∗.The next key step is to contract (i.e., prune) the chosenbox [x]. The purpose of contracting is to delete subboxesinside [x] that cannot contain a globally optimal solution.There are multiple techniques to perform contraction, e.g.,to delete subboxes wherein the gradient of f is nonzero orwherein f is not convex. The rationale of the latter is thatf must be convex in some neighborhood of a global min-imum point x∗ (Hansen and Walster, 2003, Chapter 12).Another important technique to prune [x] is to remove asubbox [x′] ⊆ [x] with f([x′]) ≥ f , where f is the mini-mum value of f we have evaluated so far, which serves asan upper bound of the global minimum value f∗. In gen-eral, such a box pruning operation is conceptualized as acontractor, which is basically an operator that transformsa box to a smaller (included) box by safely removing pointswithout affecting the global minimum. Similarly to con-tracting, there is no single best strategy for box splitting inLine 12. Typical choices include splitting along the widestedge of [x] or over a coordinate in which f varies mostly.After multiple iterations of box contracting and splitting,a small box may be added to the candidate solution listLS as a tentative solution, i.e., one that may contain theglobal minimum x∗. In order to be included in LS , a box[x] must satisfy two conditions that are checked in Line 9:

w([x]) ≤ εx, w(f([x])) ≤ εf , (11)

where w(·) denotes the width of a box defined by its largestdiameter among all components. εx and εf are two toler-ance parameters provided by the user, often known as theprecision. Obviously, smaller values of εx and εf demandmore iterations and prolongs the execution time. After themain loop finishes, we post-process the solution list LS inLine 15 to discard boxes which cannot contain the globalminimum x∗ according to the latest knowledge of f .

At the end of Algorithm 1, we get the list of boxes LS

containing the global minimum x∗ and the (usually verytight) bounds of the global minimum value f∗ ∈ [f, f ].Note that f([x]) − f∗ ≤ 2εf ,∀[x] ∈ LS , is guaranteed(Hansen and Walster, 2003). Recall Definition 1, and wesee that any x inside the remaining boxes in LS becomesan ε-global minimum with ε = 2εf in this case. Thoughthe exact global minimum x∗ and f∗ are still unknown ingeneral, a reasonably tight bound acquired by setting smallεf and εx in Algorithm 1 is usually enough in practice.

6

Algorithm 1 Interval arithmetic based branch-and-bound algorithm for deterministic global optimization

Input: objective function f : Rn → R and bound con-straints X ⊂ Rn, precision parameters εf and εx

Output: bounds of the global minimum [f, f ] and a listof boxes LS that contain all global minima

1: initialize a list L← [x] with [x] corresponding to X2: initialize an empty candidate solution list LS

3: f ←∞ . Upper bound of f∗

4: while L 6= ∅ do5: choose [x] ∈ L and remove [x] from L6: contract [x]7: evaluate f at the center of [x] and get value fc8: Update f by f ← minf , fc9: if [x] satisfies criteria (11) then

10: append [x] to LS

11: else12: split [x] into subboxes and add them to L13: end if14: end while15: remove any box [x] ∈ LS from LS with f([x]) > f16: f ← min[x]∈LS

f([x])

Note that Algorithm 1 only sketches out the basic skele-ton of interval B&B algorithms. There are many practicalissues and improvements to be considered for high-qualityimplementation. We thus resort to a well-developed open-source numerical library called ibexopt4, which is basedon interval analysis and contractor programming, to per-form interval B&B optimization in PV parameter esti-mation. We refer interested readers to the monograph(Hansen and Walster, 2003) and the documentation ofibxopt for more details.

Remark 1. The global optimization solver ibexopt hasbeen employed in a recent study (Chenouard and El-Sehiemy,2020). Nonetheless, the results obtained therein have in-sufficient precision due to the timeout setting of 20000 sand, consequently, cannot yield sufficiently tight optimal-ity bounds. In this study, we use the interval B&B algo-rithm (implemented by ibexopt) only to validate globaloptimality but do not recommend it as a routine optimizerfor PV parameter identification due to its excessively longrunning time. Thus, we are free to allow it more time andset smaller tolerances for promoted precision. We will givemore details in Section 5.2.

4.2. Stochastic global optimization with a simple differen-tial evolution algorithm

As we have discussed above, though an interval B&Balgorithm has the capacity to ascertain the global opti-mum rigorously in theory, it is generally much more com-putationally expensive and runs considerably slower than

4http://www.ibex-lib.org (v2.8)

stochastic metaheuristic algorithms, which can avoid localminima to a certain extent but with no guarantee (Yang,2020; Yang et al., 2020). The ability to find satisfacto-rily high-quality solutions in reasonable computation timemotivates the broad application of stochastic search basedmetaheuristics in PV parameter estimation (Chen et al.,2016; Yang et al., 2020; Nunes et al., 2018). In this sec-tion, we take a remarkably simple evolutionary algorithmon purpose, namely the differential evolution (DE), to ap-proach the parameter estimation problem (10) in order toaddress the second concern listed in Section 1.

DE is a simple, efficient, yet powerful evolutionaryalgorithm (EA) originally proposed by Price and Storn(Price et al., 2006), which has got proved success in a vari-ety of optimization problems arising from diverse domainsof science and engineering (Das and Suganthan, 2011).The exceptional popularity of DE is partly attributed toits simplicity, e.g., the main body of the algorithm takesonly four to five lines of code (see Algorithm 2), as wellas the small number of control parameters, only four inclassical DE. Similar to other EAs, DE follows the Dar-winian principle to evolve a population of solutions (calledvectors in the DE community), and each iteration of DEcomprises three key steps: selection, crossover, and muta-tion. The distinguishing feature of DE is its mutation withdifference vectors (Price et al., 2006; Das and Suganthan,2011). In addition, by making a parent vector competewith a child vector for survival into the next generation,DE’s selection operator enforces elitism and ensures thatthe highest quality vector ever found will never get lostduring long evolution.

Suppose that we want to minimize a scalar functionf : Rn → R with bound constraints X ⊂ Rn. A candi-date solution is represented by a vector x ∈ Rn in DE. Theprocedure of classical (and also the most simple) DE is out-lined in Algorithm 2. The initial population P comprisesNp vectors, and each vector x0

i , i ∈ [1, Np] is generatedrandomly as follows,

x0i,j = bj + rand(0, 1) · (bj − bj), (12)

where x0i,j denotes the j-th component of x0i , j ∈ [1, n],

bj and bj represent the lower and upper bound of the j-thvariable respectively, and rand(0, 1) generates a randomnumber between 0 and 1, which is called anew indepen-dently for each pair of i and j.

The mutation operation in DE is characterized by itsusage of difference vectors. Several mutation strategieshave been developed in the literature (Das and Sugan-than, 2011; Price et al., 2006). Here we adopt the mostcommonly used one called the “DE/rand/1” scheme. Foreach vector in the g-th iteration, an auxiliary vector vgi , i ∈[1, Np], is generated by

vgi = xga + F (xg

b − xgc), a 6= b 6= c 6= i, (13)

where a, b, c ∈ [1, Np] are randomly chosen and F is a con-trol parameter, called the scaling factor, which typically

7

Algorithm 2 Differential evolution with bound con-straints handled by bounce-back

Input: objective function f : Rn → R and bound con-straints X ⊂ Rn, control parameters Np, Cr, F,G

Output: the best vector discovered x and the correspond-ing function value f

1: generate randomly an initial population P 0 ←x0

i , i = 1, 2, · · · , Np with (12)2: for g from 0 to G− 1 do3: for each vector xg

i ∈ P g do4: generate a donor vector vgi by (13) . mutation5: vgi ← bounce-back(vgi ) by (14)6: ug

i ← crossover(vgi ,xgi ) by (15)

7: xg+1i ← select(ug

i ,xgi ) by (16)

8: insert xg+1i into the new population P g+1

9: end for10: end for11: x← the best vector in PG

12: f ← f(x)

lies in the range [0.4, 1] (Das and Suganthan, 2011). Theresultant vgi is known as the donor vector in contrast tothe target vector xg

i .Note, however, that the donor vector vgi produced in

(13) may lie outside the feasible region if the problem hasadditional constraints, like bound constraints X (see Table1 for PV parameter estimation). We adapt a simple strat-egy called bounce-back from (Das and Suganthan, 2011;Price et al., 2006) to handle bound constraints in Line5, which relocates each infeasible component between thebound it exceeds and the corresponding value of the targetvector, as follows.

vgi,j =

bj + rand(0, 1) · (xgi,j − bj) if vgi,j < bjbj − rand(0, 1) · (bj − xgi,j) if vgi,j > bj

(14)

As is common in EAs, a crossover operation is appliedin DE to enhance the diversity of the population. Thefundamental idea is to exchange components between thedonor vector vgi and the target vector xg

i so as to producea new vector ug

i named the trial vector. An intuitive andwidely used crossover scheme is the binomial crossover,which is performed as follows:

ugi,j =

vgi,j if rand(0, 1) ≤ Cr or j = β

xgi,j otherwise(15)

where β ∈ [1, n] is a random integer that is generated anewfor each i. Besides, Cr is the user provided crossover rate.The role of β is to ensure that at least one component ofthe trial vector ug

i comes from vgi (Das and Suganthan,2011).

Finally, differing from other EAs like genetic algorithm,DE imposes elitism by selecting the better one between thetarget vector xg

i and the trial vector ugi as the i-th vector

in the next generation. That is,

xg+1i =

ugi if f(ug

i ) ≤ f(xgi )

xgi otherwise

. (16)

One interesting observation is that ugi replaces xg

i even ifthey share the same objective value, a useful feature thatallows DE to move over flat fitness landscapes for efficientexploration of the search space (Das and Suganthan, 2011).After the selection with (16) in Line 7, the population sizeremains Np. Note that the termination criterion of DEcan be defined in various forms, e.g., when the minimumfunction value obtained so far has not changed for a fewconsecutive generations (early stopping), apart from spec-ifying a fixed number of generations, G, in Algorithm 2.

Remark 2. Many variants of DE have been proposed (seethe latest survey (Bilal et al., 2020) for an extensive re-view), which include advanced mutation and/or crossoveroperators and control parameter adaptation. For example,the repaired adaptive DE (Gong and Cai, 2013) and theself-adaptive ensemble-based differential evolution (SEDE)(Liang et al., 2020a) has been used to extract solar cellparameters. We refer readers to Section 4.1.2 of (Yanget al., 2020) for a review of advanced DE in PV parameteridentification. Algorithm 2 represents the simplest formof classic DE, which is chosen intentionally here to vali-date our hypothesis: the increasingly complicated meta-heuristics pervading the literature may not be necessaryfor pragmatic PV parameter estimation.

5. Results and discussion

In this section, we apply the optimization methodsintroduced in Section 4 to two widely used benchmarkdatasets and try to answer the two questions raised inSection 1 by analyzing and comparing the results.

5.1. Datasets and experimental settings

The experimental data are acquired from (Easwarakhan-than et al., 1986), which serve as the de facto benchmarkin the PV parameter estimation literature (Yousri et al.,2020; Yang et al., 2020). The first dataset contains 26 datapoints measured for an RTC France solar PV cell of 57mm diameter that operates at 33 C and under irradiance1000 W/m2. The second dataset is measured from a solarmodule called Photowatt-PWP201 that contains 36 poly-crystalline silicon cells in series and operates at 45 C tem-perature with an irradiance of 1000 W/m2. Though theoriginal dataset for Photowatt-PWP201 in (Easwarakhan-than et al., 1986, Table II) includes 26 data points, moststudies discard the first data point, e.g., (Gong and Cai,2013; Li et al., 2019; Chen et al., 2016; Liang et al., 2020b;Jordehi, 2016b). We follow this convention in our studyfor fair comparison and compatible analysis. Note that,the methodology developed in this study can be applied

8

Table 1: Parameter search range of SDM and DDM (LB: lowerbound; UB: upper bound).

ParameterRTC France Photowatt-PWP201

LB UB LB UB

Iph (A) 0 1 0 2I0, I01, I02 (µA) 0 1 0 50n, n1, n2 1 2 1 50Rs (Ω) 0 0.5 0 2Rp (Ω) 0 100 0 2000

seamlessly to any other dataset acquired under other con-ditions in practical applications.

Regarding the bound constraints in (10b), we adopt theparameter search range used in the majority of literaturefor fair comparison purposes (see, e.g., (Chenouard andEl-Sehiemy, 2020; Chen et al., 2016; Nunes et al., 2018;Li et al., 2019; Liang et al., 2020b), among others). Theparameter search ranges for both datasets are listed inTable 1. Since the reverse saturation current of a diode isextremely small, the unit µA is adopted to permit similarscales among all parameters for numerical stability.

Algorithm 1 is implemented in the open-source soft-ware ibexopt (v2.8). We code algorithm 2 with the Juliaprogramming language for its flexibility and efficiency. Theresults presented below were obtained on a desktop PCwith a 3.4 GHz Core i7-3770 CPU, 16 GB RAM, and Win-dows 10.

5.2. Global optimality analysis via interval B&B

In this part, we investigate the global minimum of thenonlinear least squares problem (10) for both the SDMand the DDM using the interval B&B method describedin Section 4.1. As mentioned above, lots of metaheuristicshave been employed in the literature to extract parametervalues for solar PV models. For example, the results of69 different methods, including the SDM’s five parametersand the resultant RMSE values, for the RTC France so-lar cell dataset are collected in Table 1 of a latest article(Calasan et al., 2020). It is reasonable that the primaryobjective of most studies is to reduce further the fitting er-ror evaluated conventionally by the RMSE. Nevertheless,none of these 69 methods can certify the the global min-imum or quantify the optimality gap even for the simpleSDM case. From a practical standpoint and to be com-patible with the vast majority of studies, we restrict ourdiscussion of RMSE values to five significant digits. Anotable observation of existing methods, e.g., those sum-marized in (Calasan et al., 2020, Table 1), (Li et al., 2019,Table 3), (Gnetchejo et al., 2019, Table 11), (Yousri et al.,2019, Table 1), and (Yu et al., 2019, Table 4), is that theminimum RMSE attained so far remains 9.8602E-4 whenthe SDM is utilized on the RTC France cell dataset. Inline with the prior observation, a natural speculation iswhether 9.8602E-4 is the global minimum RMSE value inthe SDM case. Similar observations exist regarding the

DDM and the Photowatt-PWP201 dataset. Such a con-jecture motivates our work present in this section.

Before proceeding to global optimization, we note thatseveral studies have reported RMSE values even smallerthan 9.8602E-4 when fitting the SDM to the RTC Francesolar cell, e.g., some of the results collected in (Gnetchejoet al., 2019, Table 2). There are primarily two reasonsfor these unexpectedly small values, either due to acci-dental errors in calculation as pointed out by (Gnetchejoet al., 2019) and (Calasan et al., 2020) or because of anevaluation metric different from the one defined by (8)and (9) (Yousri et al., 2019). Regarding the latter specifi-cally, instead of Eq. (8), some studies decide to calculatethe estimated current more precisely by solving a nonlin-ear equation despite the increased computational cost (seeSection 3), which often contributes to smaller RMSE val-ues (Jordehi, 2016b, Table 1). Nevertheless, such precisebut expensive computation is adopted in only a small por-tion of existing studies, which is thus not considered here.Another subtle yet influential reason for such discrepancyis possibly that the RMSE value is quite sensitive to round-off errors (Jordehi, 2016b), especially to those appearingin the exponential term of (4) and (5). To be as preciseas possible and consistent with the mainstream studies,the values of the two constants q and k refer to exactlythe numbers listed below Eq. (3) (see, e.g., (Chen et al.,2016), (Yu et al., 2019), (Long et al., 2020), and (Lianget al., 2020b)). In our discussion below, we have excludedthose erroneous and incompatible results. Finally, notethat the interval B&B method in Algorithm 1 is a de-terministic one with no control parameters to be tuned(except the timeout and desired precision), which meansmultiple runs with the same inputs will always producethe same result (Chenouard and El-Sehiemy, 2020).

5.2.1. SDM results

We first applied the interval B&B algorithm to solveproblem (10) specialized for the SDM and the RTC FrancePV cell data. The bound constraints, i.e., the initial box inAlgorithm 1, are specified by the search range in Table 1.Following (Chenouard and El-Sehiemy, 2020), we limitedthe maximum runtime of ibexopt to 20000 seconds. Onthe other hand, unlike (Chenouard and El-Sehiemy, 2020),we did not use the default absolute and relative precisionbut set them to smaller values in order to obtain tighterbounds of the global minimum (that is, [f, f ] in Algorithm1). In our setting, the absolute precision was 1E-13, andthe relative precision was 1E-9 (in contrast to the defaultvalues 1E-7 and 1E-3, respectively; see also Remark 3).Note that our objective value in optimization denotes theSSE (10), and the corresponding RMSE is obtained after-wards by (9) once the optimization finishes. The resultsobtained with the SDM and the RTC France dataset arereported in the first column of Table 2. Recall that, un-like metaheuristics, the interval B&B algorithm yields atight interval that contains the global minimum, while thedisplayed values of the five parameters correspond to the

9

Table 2: Optimization results for SDM using interval B&B.The bounds of the global minimum RMSE value are reported, whoseupper bound is attained by the parameter values above it.

Variable RTC France Photowatt-PWP201

Iph (A) 0.760779120136 1.03052020484I0 (µA) 0.322873926858 3.48287904343n 1.48113747635 48.6435574734Rs (Ω) 0.0363792207867 1.20123680201Rp (Ω) 53.7009537057 981.263690780

RMSE[9.860250397955652E-4,9.860250417458982E-4]

[2.425076598320144E-3,2.425076599532477E-3]

Gap 1.950333050615427E-12 1.2123329007351913E-12Time (s) 13547 38924

upper bound of the produced interval. On the other hand,despite its theoretical preciseness, the main drawback ofthe interval B&B algorithm is the exceptionally long exe-cution time, that is, around 4 hours versus a few secondsof most metaheuristics.

We notice from the “RTC France” column of Table 2that the optimality gap between the lower and the upperbound of the RMSE is extremely small. By focusing on fivesignificant digits only, we can certify safely that 9.8602E-4is indeed the global minimum value of RMSE by fittingthe SDM to the RTC France dataset, which is in agree-ment with our observation of the literature. The five pa-rameters’ values are also very close to those acquired withvarious metaheuristics in the literature (see, e.g., (Calasanet al., 2020, Table 1), (Yu et al., 2019, Table 4), (Longet al., 2020, Table 5), and Table 6 in the next section),though these metaheuristic algorithms cannot recognizethe global minimum even if they have actually found it.Such an agreement also confirms the correctness of ourimplementation.

Next, we applied the same methodology to the Photowatt-PWP201 PV module dataset. The major difference liesin the considerably widened parameter search range inTable 1, which may pose a big challenge to the intervalB&B algorithm. The results of fitting the SDM to thePhotowatt-PWP201 dataset are reported also in Table 2.As expected, it took ibexopt much more time to solvethe minimization problem (10) in the Photowatt-PWP201case probably due to the enlarged search space. As far aswe know, the best minimum value of the SDM’s RMSEacquired in the literature for this PV module is 2.4250E-3 (truncated to five significant digits), as summarized in(Gnetchejo et al., 2019, Table 15), (Long et al., 2020, Ta-ble 13), (Li et al., 2019, Table 6), (Chen et al., 2016, Table8), (Yu et al., 2019, Table 3), and (Liang et al., 2020b,Table 7), which shows exact agreement with the globalminimum value’s bounds reported in Table 2. Further-more, the metaheuristics that have succeeded in acquiringthe global minimum all yield parameter values extremelyclose to those in Table 2, such as results presented in (Gongand Cai, 2013), (Chen et al., 2016, Table 8), (Yu et al.,2019, Table 6), (Liang et al., 2020a, Table 7), and (Longet al., 2020, Table 7). Again, such a consensus indicates

the correctness of each other.

Remark 3. The ibexopt toolkit has also been utilizedin a recent study (Chenouard and El-Sehiemy, 2020). Incontrast to our setting, the default precision tolerances areused in (Chenouard and El-Sehiemy, 2020), which leadsto a fairly larger optimality gap roughly equal to 1.95E-6.Consequently, the result of (Chenouard and El-Sehiemy,2020) appears insufficient to certify the global minimumvalue 9.8602E-4 in the RTC France SDM case. More in-terestingly, it is surprising that the extra running timerequired for the higher precision in our setting was essen-tially negligible, less than one second. Besides, the priorstudy (Chenouard and El-Sehiemy, 2020) does not reportresults for the Photowatt-PWP201 PV module.

5.2.2. DDM results

Following the success of interval B&B in the aboveSDM case, we proceed to the DDM case, which is morechallenging due to the additional two parameters (see (5)).In the previous study (Chenouard and El-Sehiemy, 2020),ibexopt failed to generate a tight optimality bound sub-ject to a timeout of 20000 s. More specifically, when op-timizing the seven parameters of the DDM for the RTCFrance dataset, the lower bound of the global minimumobtained by ibexopt in 20000 s was 0. However, the zerolower bound is trivial and useless since we are minimiz-ing a summation of squared terms in (10a). From thisparticular standpoint, we decided to allow ibexopt morerunning time attempting to get tighter optimality bounds.The overall workflow is identical to the one presented inthe above SDM scenario except that the SDM is replacedwith the DDM.

We first examined the DDM for the RTC France datasetwith the search space given by Table 1. The timeout wasset to 86400 s, that is, 24 hours. The obtained resultsare listed in the left side of Table 3. We see that, un-fortunately, the significant increase of the allowed runningtime did not help much. In particular, the lower boundstill remains zero. In a more detailed investigation, wefound that the upper bound was reduced to 9.8657E-4 atthe end of the first hour. In other words, ibexopt madelittle progress in the subsequent 23 hours, only decreas-ing the upper bound from 9.8657E-4 to 9.8358E-4, whilethe lower bound stayed at zero. It thus seems hopelessthat the optimality bound can be tightened substantiallyby allowing even longer execution time5. The consider-able slowdown of the solver at a later stage of the searchprocess is probably caused by the so-called cluster effect,a pain that B&B methods frequently suffer from (Mon-tanher et al., 2018; Floudas, 2013). This effect is roughlycharacterized by the excessive splitting of the search spacein the neighborhood of the global optimum. The resultant

5We have actually run ibexopt for another 5 hours but observedno perceivable change in both lower and upper bounds.

10

Table 3: Optimization results for DDM using interval B&B.The bounds of the global minimum RMSE value are reported, whoseupper bound is attained by the parameter values above it.

Variable RTC France Photowatt-PWP201

Iph(A) 0.760815738919 1.0339286971I01(µA) 0.217867184041 1.86575472010E-23I02(µA) 0.781454995330 0.535399234849n1 1.44827388213 9.58860778809n2 1.98183166760 42.6724488388Rs(Ω) 0.0367359827333 1.63619822583Rp(Ω) 55.8931982861 607.690281231

RMSE[0,9.83581875679E-4]

[0,1.61865668151E-3]

Gap 9.83581875679E-4 1.61865668151E-3Time (s) 86400 43200

boxes accumulate as a cluster around the globally opti-mal solution, and, what is worse, the pattern repeats itselfrecursively at increasingly smaller scales if we want highaccuracy. Upon closer inspection, we notice that the num-ber of boxes that have been explored at the end of 24 hoursis already more than 25 million. In addition, the multipleoccurrences of each parameter incidental to the SSE ob-jective function (10a), known as the dependence problem,can also compromise the efficiency of interval calculationand lead to over-estimation of the value range (Hansen andWalster, 2003; Chenouard and El-Sehiemy, 2020).

Though we did not get an extraordinarily tight opti-mality bound like that in Table 2, which is mainly at-tributed to the zero lower bound, the upper bound is stillinformative: it is very close to the best known result in theliterature. For example, the best known minimum RMSEso far for the RTC France benchmark with the DDM is9.8248E-4 (see, e.g, results listed in (Chen et al., 2016, Ta-ble 6), (Zhang et al., 2020, Table 11), (Zhang et al., 2020,Table 10), (Li et al., 2019, Table 4), (Yu et al., 2019, Ta-ble 3), (Yousri et al., 2019, Table 1), and (Gnetchejo et al.,2019, Table 11)). In view of these facts, the acquired upperbound 9.8596E-4 in Table 3 implies that 9.8248E-4 is likelyto be the global minimum value despite the lack of theoret-ical guarantee. In addition, just like the above SDM case,the seven parameter values reported in previous tables wehave just cited all reside in a small neighborhood of theresult located by ibexopt in Table 3.

As for the Photowatt-PWP201 benchmark, most ex-isting studies only employ the SDM probably due to theincreased difficulty of the DDM. Nevertheless, there isno technical reason that prevents the application of theDDM to a PV module. However, to our best knowledge,the limited existing studies that consider the DDM andthe Photowatt-PWP201 dataset, such as those reportedin (Nunes et al., 2018, Table 12) and (Kler et al., 2019,Table 14), adopt a procedure distinct from the prevailingone: they use the Newton-Raphson method rather than(8) to compute current values iteratively (refer to Section3 for more details). Consequently, unlike previous cases,we did not find any compatible results that can be used

for DDM’s perforamnce comparison with respect to thePhotowatt-PWP201 PV module here. In spite of the lackof references, the result we have obtained in Table 3 looksreasonable if we compare it with its SDM counterpart inTable 2. Besides, unlike the RTC France case, the upperbound calculated by ibexopt did not reduce any longerafter 12 hours in the PV module case.

Remark 4. Apart from the disparate current calculationprocedures as emphasized above, the parameter rangesgiven in (Nunes et al., 2018, Table 2) and (Kler et al.,2019, Table 2), especially ranges of I01 and I02, are to-tally different from the most widely used ones in Table1, which thus blocks reasonable comparison between theseparameter estimation results.

5.2.3. Summary of global optimality analysis

The results of the interval B&B method for the fourrepresentative cases, reported previously in Table 2 andTable 3, are collected here in Table 4 to give a conciseoverview. Notably, the interval B&B algorithm (via ibexopt)is capable of locating the ε-global minimum rigorously forboth benchmark datasets using the five-parameter SDM,though its running time is exceedingly long compared withmetaheuristic methods. For the more complicated seven-parameter DDM, the interval B&B method failed to ob-tain an adequately tight optimality bound in 24 hours,but the acquired upper bound can still play the role ofa good reference value. To demonstrate visually the ac-curacy of the extracted parameters listed in Table 2 andTable 3, we insert them into (8) to reconstruct the I-Vcurves, which are shown in Fig. 3. Note that the negativecurrent and voltage values therein simply imply a reversedirection (Easwarakhanthan et al., 1986). Obviously, thereis excellent consistency between the measured data andthe simulated data, even using the DDM whose parametervalues correspond to the RMSE upper bound in Table 3.Comparing the RMSE values in Table 2 and Table 3 (sum-marized in Table 4), we notice that the RMSE (or its upperbound) of the DDM is consistently smaller than that of theSDM. This situation is expected and makes complete sensebecause the several-parameter DDM is more flexible thanthe SDM and, in particular, the SDM (4) is essentially aspecial instance of the DDM by fixing I02 = 0 in (5).

Remark 5. We emphasize that, to avoid possible mis-understanding, the global minimum value J∗ is defined interms of the objective function value (10a), and it does notimply that there is necessarily a unique global minimumpoint θ∗ in (10b) corresponding to J∗. Moreover, sinceour discussion is conventionally limited to five significantdigits, it is not surprising that many close but not identi-cal parameter values can lead to the same objective value.For example, slightly different values of Rp are identifiedby three methods in (Li et al., 2019, Table 3) though theyall obtain the same global minimum 9.8602E-4 for the firstcase in Table 4. This fact is also noticeable by comparingparameter values in Table 2 and Table 6.

11

Table 4: Overview of the optimization results of interval B&B in four benchmark cases.

SDM DDM

RTC France Photowatt-PWP201 RTC France Photowatt-PWP201

RMSE 9.8602E-4* 2.4250E-3* 9.8358E-4† 1.6186E-3†

Time (s) 13547 38924 86400 43200

* Certified global minimum value with respect to five significant digits† Certified upper bound of the global minimum value in five significant digits

(a) RTC France (b) Photowatt-PWP201

Figure 3: Measured data and calculated data using both the SDM and the DDM with parameters optimized by interval B&B.

5.3. Optimization via simple differential evolution (DE)

As we have mentioned above, despite its capability toguarantee global optimality (or optimality bounds), theinterval B&B algorithm is not suitable for practical es-timation of PV parameters owing to its excessively longexecution time. In addition, a guaranteed global mini-mum may not be necessary for real-world industrial ap-plications. By contrast, a variety of metaheuristic meth-ods with distinct methodology and complexity can obtaina reasonably small objective value in a far shorter time.Nevertheless, in view of the abundance of metaheuristicalgorithms, an interesting but rarely studied question, aswe have asked in Section 1, is whether such PV param-eter estimation really demands the increasingly compli-cated metaheuristics in the recent literature. The answerto this puzzle is of both theoretical implication and prac-tical importance. Unfortunately, it is extremely difficult,if ever possible, to perform theoretical analysis of stochas-tic search based metaheuristics (Yang, 2020). In this sec-tion, we try to get some insights from another perspectiveby examining whether the intentionally simple differentialevolution (DE) method in Algorithm 2 can achieve compa-rable performance for PV parameter estimation using boththe SDM and the DDM. The two widely used benchmarkdatasets introduced in Section 5.1 are considered againfor comparison purposes. In addition to the informativereference values acquired by interval B&B in Section 5.2,the accuracy and efficiency of DE will also be evaluatedagainst other popular metaheuristic methods.

Table 5: Control parameters of simple DE in Algorithm 2.

Np Cr F G

SDM 50 0.6 0.9 800DDM 50 0.6 0.9 1600

Starting with the recommendation in (Das and Sugan-than, 2011, Section III), we quickly determined appropri-ate control parameter values for DE in Algorithm 2 afterseveral preliminary trials, which are listed in Table 5. Notethat these values are kind of canonical in the literature,and little time was spent in parameter tuning thanks tothe simplicity of Algorithm 2. Remarkably, both mod-els share almost the same control parameters on the twodatasets: the only difference lies in the number of gener-ations. Intuitively, the more complicated DDM requiresmore generations of DE than the simpler SDM.

5.3.1. SDM and DDM results

Since DE in Algorithm 2 is a stochastic search algo-rithm, we followed the convention in the literature, e.g.,(Liang et al., 2020b; Jordehi, 2016b; Yousri et al., 2020;Yu et al., 2019), and executed DE 30 times for each casein order to obtain a trustworthy judgment of its accuracyand consistency. The parameter values obtained by DE ina typical run are listed in Table 6. The corresponding I-Vcurves reconstructed with these parameters are illustratedin Fig. 4 to facilitate visual inspection. Since the RMSE

12

Table 6: Parameter values obtained by DE in a typical run. (RT:RTC France; PW: Photowatt-PWP201)

SDM DDMRT PW RT PW

Iph(A) 0.760775 1.03051 0.760781 1.03051I0/I01(µA) 0.323021 3.48226 0.225974 9.77791E-3I02(µA) — — 0.749344 3.47248n/n1 1.481184 48.6428 1.45101 48.64284n2 — — 1.99999 48.64283Rs(Ω) 0.036377 1.20127 0.0367404 1.20127Rp(Ω) 53.71852 981.982 55.4854 981.982RMSE 9.8602E-4 2.4250E-3 9.8248E-4 2.4250E-3

Table 7: Statistics of RMSE values yielded by DE in 30 runs. (RT:RTC France; PW: Photowatt-PWP201)

SDM DDMRT PW RT PW

Min 9.8602E-4 2.4250E-3 9.8248E-4 2.4250E-3Mean 9.8602E-4 2.4250E-3 9.8267E-4 2.4250E-3Max 9.8602E-4 2.4250E-3 9.8602E-4 2.4250E-3Std 4.3929E-17 2.9525E-17 7.1027E-7 2.3955E-17

values are fairly small in all cases (see Table 4 and Ta-ble 6), there is, unsurprisingly, almost no visual differencebetween Fig. 3 and Fig. 4.

The convergence curves of this simple DE in a typicalrun on the RTC France dataset using both models areshown in Fig. 5. As we have observed in many runs, justlike Fig. 5 here, the DE often takes far fewer generationsto converge than the number G specified in Table 5, whichis a conservative choice to ensure better convergence. Theconvergence curves on the other dataset share a similarcharacter and are presented in our online materials.

The statistics of RMSE in the 30 runs are analyzedand reported in Table 7. Overall, we see from Table 7that the performance of our DE algorithm is remarkablystable despite its stochasticity in nature. When apply-ing to the SDM on the two datasets and to the DDM onthe Photowatt-PWP201 dataset, the DE algorithm alwaysyields the same minimal RMSE (in terms of five significantdigits) in all 30 trials. Even in the worst case (DDM andRT), the maximum and minimum RMSE values obtainedin 30 runs are still very close to each other, which is alsoimplied by the small standard deviation in Table 7.

The simple DE algorithm managed to find the globalminimum for the first two cases in Table 7, i.e., for SDMon the two datasets. As for the more challenging DDMcases, DE achieved the best known result in the literaturefor the RTC France PV cell, i.e., 9.8248E-4. Neverthe-less, comparing the last case’s RMSE in Table 7 with itsupper bound in Table 3, we see that DE failed to accom-plish an RMSE value below the upper bound (2.4250E-3vs 1.6186E-3). The obtained minimum RMSE thus can-not be the global minimum value in the last case. Evenso, we should not depreciate the simple DE’s performancehastily in this particular case due to lack of usable com-

parison in the current literature (recall the last paragraphof Section 5.2.2 for elucidation): no comparable results, toour best knowledge, have been reported towards modelingthe Photowatt-PWP201 PV module with the DDM.

More specifically, by focusing on the last column ofTable 6 (i.e., the DDM fitted to the Photowatt-PWP201module), we notice that the obtained RMSE is equal to itsSDM counterpart. This result is not surprising since theSDM is a specialization of the DDM as we have mentionedin Remark 5. The main reason that DE failed to locatean RMSE value within the optimality bounds revealed byinterval B&B in Table 3 is presumably attributed to theextremely small value of I01 therein, whose order of mag-nitude is -23. Such a tiny number is commonly treated aszero in general computation, though it still has perceivableimpact on the RMSE in this specific case. Unfortunately,it is really challenging for DE to pinpoint such a smallvalue within a relatively large search range [0, 50] (see Ta-ble 1). In fact, in the existing yet incompatible studies (seeSection 5.2.2) that consider the last case (“DDM + PW”),a greatly narrowed search range is usually set for I01 andI02 (Nunes et al., 2018; Kler et al., 2019; Jordehi, 2016b),e.g., [1E-15, 1E-3] in (Kler et al., 2019), which facilitatesthe discovery of good parameter values. More interest-ingly, in the two PV module (“PW”) cases in Table 6, theparameter values acquired by DE essentially degrade theDDM to the SDM by n1 = n2 ≈ n and I01 + I02 ≈ I0,which can be figured out easily by comparing Eq. (5) to(4). This argument is also justified by the indistinguish-able overlap between the SDM and DDM curves in Fig. 4b(as opposed to the small but still observable gap betweenthe two curves in Fig. 3b).

5.3.2. Comparison with existing algorithms

The parameter values obtained by DE in Table 6 arequite close to those reported in existing studies and a de-tailed comparison of the values themselves is omitted hereto save space. Interested readers may refer to tables insome latest articles like (Liang et al., 2020b; Long et al.,2020) to examine parameter values obtained in variousstudies. In general, almost identical parameter values areacquired as long as the resultant RMSE values are similar,a fact that verifies the correctness of each other. We devotethe subsequent comparison to different algorithms’ perfor-mance statistics and time efficiency, both being importantfactors in real-world applications.

For fair comparison, we mainly copied the results di-rectly reported in recent papers that have published state-of-the-art algorithms. No existing investigations, however,target the Photowatt-PWP201 module using the DDMthat are compatible with our study here (see Section 5.2.2for an explanation). Besides, the relevant metaheuristicstudies in the literature rarely provide their source code tothe public, which renders it cumbersome and inconvenientto test their algorithms on new cases (Chenouard and El-Sehiemy, 2020). To our best efforts, we only got the sourcecode of two recent methods, namely performance-guided

13

(a) RTC France (b) Photowatt-PWP201

Figure 4: Measured data and calculated data by both the SDM and the DDM with parameter values identified by DE.

(a) SDM

(b) DDM

Figure 5: Convergence curves of DE on the RTC France dataset forboth models in a typical run.

JAYA (PGJAYA) (Yu et al., 2019) and self-adaptive ensemble-based differential evolution (SEDE)6 (Liang et al., 2020a).We ran the two algorithms to deal with the last case, i.e.,to estimate DDM parameters for the PV module, usingtheir original control parameter settings and report theresults at the bottom of Table 8.

6Available at https://github.com/cilabzzu/Publications/tre

e/master/PGJAYA and https://github.com/cilabzzu/Publicatio

ns/tree/master/SEDE, respectively.

Since metaheuristic algorithms are stochastic in na-ture, a common practice is to evaluate their statisticalcharacteristics in 30 independent runs (Liang et al., 2020b;Jordehi, 2016b; Liang et al., 2020a; Li et al., 2019). In lightof the abundance of studies on this topic, we extracted thestatistical results of selected methods, especially the state-of-the-art ones, from those reported in four latest articles,(Li et al., 2019, Table 12), (Yu et al., 2019, Table 3), (Lianget al., 2020b, Table 9), and (Liang et al., 2020a, Table9). The selection was guided by the following key criteria:first, choose state-of-the-art ones according to the RMSEthey have attained; second, select deliberately algorithmsof distinct methodology and particularly DE variants fora comprehensive comparison. Interested readers may re-fer to the cited tables for other well-established methods.The average CPU time of these methods were mainly readfrom (Liang et al., 2020a, Fig. 7) (except TLABC) and(Yu et al., 2019, Fig. 5) (for only TLABC). The resultsare collected and listed in Table 8. We have the followinginteresting observations and discussions regarding Table 8.

• Overall, it is clear that our simple DE in Algorithm2 and several strong competitors like SEDE can at-tain the best RMSE values. Moreover, if we takethe complete statistics into consideration, DE ex-hibits superior performance, e.g., with regard to themean RMSE and the worst RMSE values. This ar-gument is in agreement with the fact that RMSEvalues acquired by DE exhibit tiny standard devia-tions, among the top three smallest ones in all cases.

• We compare the simple DE intentionally with threemore complicated DE variants, namely, SGDE, SEDE,and CoDE. In the two SDM cases, the four DE al-gorithms exhibit almost identical accuracy in termsof RMSE values, while CoDE demonstrates the high-est stability measured by the standard deviation andSGDE is the least stable one. By contrast, in themore challenging seven-parameter DDM cases, our

14

Table 8: Comparison of statistical results of various algorithms in four benchmark case studies. (RT: RTC France; PW: Photowatt-PWP201)

Model Data AlgorithmRMSE

Time (s)Min Mean Max Std

SDM

RT

MLBSA (Yu et al., 2018) 9.8602E-4 9.8602E-4 9.8602E-4 7.0800E-11 28TLABC (Chen et al., 2018) 9.8602E-4 9.9417E-4 1.0308E-3 1.1896E-5 43IJAYA (Yu et al., 2017c) 9.8602E-4 9.8605E-4 9.8684E-4 1.4931E-7 25PGJAYA (Yu et al., 2019) 9.8602E-4 9.8602E-4 9.8603E-4 2.8029E-9 25SATLBO (Yu et al., 2017a) 9.8602E-4 9.8879E-4 1.0067E-3 4.8133E-6 35

SGDE (Liang et al., 2020b) 9.8602E-4 9.8602E-4 9.8603E-4 2.4746E-9 —*

SEDE (Liang et al., 2020a) 9.8602E-4 9.8602E-4 9.8603E-4 4.2000E-17 28CoDE (Wang et al., 2011) 9.8602E-4 9.8602E-4 9.8602E-4 2.3100E-17 25Simple DE 9.8602E-4 9.8602E-4 9.8602E-4 4.3929E-17 0.34

PW

MLBSA 2.4250E-3 2.4253E-3 2.4336E-3 1.5600E-6 28TLABC 2.4250E-3 2.4254E-3 2.4287E-3 8.7464E-7 42IJAYA 2.4250E-3 2.4251E-3 2.4253E-3 5.0766E-8 27PGJAYA 2.4250E-3 2.4251E-3 2.4260E-3 1.7859E-7 24SATLBO 2.4250E-3 2.4254E-3 2.4315E-3 1.1622E-6 38

SGDE 2.4250E-3 2.4250E-3 2.4250E-3 4.1697E-10 —*

SEDE 2.4250E-3 2.4250E-3 2.4250E-3 3.1400E-17 28CoDE 2.4250E-3 2.4250E-3 2.4250E-3 2.1700E-17 27Simple DE 2.4250E-3 2.4250E-3 2.4250E-3 2.9525E-17 0.31

DDM

RT

MLBSA 9.8248E-4 9.8506E-4 9.8613E-4 1.2400E-6 33TLABC 1.0012E-3 1.2116E-3 1.9826E-3 2.1100E-4 44IJAYA 9.8249E-4 9.8686E-4 9.9941E-4 3.2211E-6 26PGJAYA 9.8260E-4 9.8603E-4 9.9599E-4 2.3666E-6 28SATLBO 9.8282E-4 1.0054E-3 1.2306E-3 5.0271E-5 37

SGDE 9.8441E-4 9.8577E-4 9.8602E-4 4.0150E-7 —*

SEDE 9.8248E-4 9.8289E-4 9.8602E-4 9.1700E-7 28CoDE 9.8249E-4 1.0036E-3 1.5496E-3 1.0300E-4 25Simple DE 9.8248E-4 9.8267E-4 9.8602E-4 7.1027E-7 0.80

PWPGJAYA 2.4250E-3 2.4272E-3 2.4485E-3 5.4346E-6 373†

SEDE 2.4250E-3 2.4250E-3 2.4250E-3 6.6661E-17 324†

Simple DE 2.4250E-3 2.4250E-3 2.4250E-3 2.3955E-17 0.84

* Since SGDE extracts three model parameters simultaneously, its runtime is not directly comparable here andthus not reported by (Liang et al., 2020a).

† Time measured on our PC using MATLAB R2020a and the source code provided by the original studies. Allthe other runtime results of competing methods are extracted from the literature.

15

simple DE outperforms both CoDE and SEDE withits enhanced performance stability. The favorableconsistency of DE’s performance across multiple runsis probably attributed to its simplistic mechanismthat introduces randomness in (13) and (15).

• Compared with the other algorithms, the biggest ad-vantage of the simple DE highlighted in Table 8 isdefinitely its substantially reduced running time. DEstands out from these competing techniques by tak-ing virtually only one percent of the CPU time con-sumed by others. This impressive speedup is mainlybrought by its extreme simplicity that includes onlyfour computationally cheap equations in the mainloop of Algorithm 2. Moreover, a common metric toevaluate the efficiency of metaheuristics is the num-ber of objective function evaluations termed NFE .There is no standard value of NFE in the PV pa-rameter estimation literature though a typical valueis 50000 (Liang et al., 2020a; Yu et al., 2019; Li et al.,2019; Chen et al., 2016). We acknowledge that an in-evitable tradeoff of the simplicity of our DE is the(moderately) increased number of function evalua-tions for comparable performance. To be specific,the maximum number of function evaluations in DEis 40000 for the SDM and 80000 for the DDM, re-spectively (see Table 5). Nonetheless, note that thecalculation of the objective (10a) with a few dozensof data points is inexpensive. Consequently, it isthe algorithm’s internal computation burden ratherthan the objective function evaluation that tends todominate the overall time consumption, at least inthe four cases here. This fact is implied particularlyby the significantly longer runtime of the three morecomplex DE variants in Table 8.

• Focusing on the last case (i.e., “DDM+PW”) in Ta-ble 8, we notice that the best RMSE value attainedby the three algorithms all turns to be 2.4250E-3,though this value is certainly not the global mini-mum, since an upper bound of the global minimumhas been identified as 1.6186E-3 in Table 4. A possi-ble reason for this common failure, as we have specu-lated in Section 5.3.1, is that the ideal value of I01 isunreasonably small, which poses a tremendous chal-lenge to these metaheuristic methods since extraor-dinarily high precision is required to get such a valuefor I01 inside a wide range.

Remark 6. It should be pointed out that, among the run-time listed in Table 8, only the bottom two of PGJAYAand SEDE in the “DDM+PW” case were measured on ourdesktop PC. We understand that different computing en-vironments and certain implementation details may affectmore or less the execution time. Note, however, that thecomputing power of the server that produces other refer-ence runtime in Table 8 from (Yu et al., 2019) and (Lianget al., 2020a) is at least as strong as our PC. Overall,

the time reduction of our simple DE is around two ordersof magnitude, whose efficiency advantage is indeed com-pelling regardless of these possible external disturbances.

Remark 7. Usually, far fewer generations are enough forthe simple DE to converge in reality (recall Fig. 5). Tech-niques like early stopping may help reduce further the exe-cution time, though they are not considered in this study.As a side note, another “DE” algorithm (Ishaque andSalam, 2011) is also mentioned in (Liang et al., 2020b,Table 9). However, that DE (Ishaque and Salam, 2011)depends exclusively on the manufacturer’s datasheet in-formation and is distinct from our methodology here.

6. Conclusion

In this paper, we conducted a critical study of solarPV parameter estimation from a perspective distinct fromthe majority of existing work. We tried to address twoessential issues that have seldom been attempted in thecurrent literature. In regard to the two most widely usedbenchmark datasets, the globally minimum RMSE for theSDM and a reasonably tight upper bound for the DDMwere certified rigorously with a deterministic interval anal-ysis based branch and bound (B&B) algorithm. However,the running time of this interval B&B algorithm is overlylong for practical usages despite its theoretical guarantee.Next, we showed through extensive examination that, forthe first time and somewhat surprisingly, a simple DE al-gorithm was able to locate the global minimum or at leastattain the best known result. More importantly, the simpleDE algorithm is indeed striking due to its favorable perfor-mance stability and its unmatched efficiency. Therefore,we suggest that a practitioner should start with the simpleDE outlined in Algorithm 2 as the first off-the-shelf tool,especially for real-time parameter estimation applications,where the environment keeps changing (Chen et al., 2016).

According to the results obtained above in this study,we present the following comments and suggestions regard-ing the two questions in Section 1 and for future work.

(1) The globally minimal RMSE values related to the SDMfor both the RTC France PV cell and the Photowatt-PWP201 PV module have been certified in this study.It has already been reached by various metaheuristics.It is thereupon futile for researchers to develop moresophisticated metaheuristics to pursue even smallerRMSE. On the one hand, this certified global mini-mum may be viewed as a golden reference when exam-ining a metaheuristic’s fundamental capability. On theother hand, since the global minimum can be attainedby many recent metaheuristic methods (see Table 8),the benchmark role of the SDM with the two datasetsin measuring the performance of various methods, stilla de facto standard in the literature, seems to be un-dermined and not fully convincing.

16

(2) The global minimum in the two DDM cases has notbeen identified, since the interval B&B method canonly yield a not so tight optimality bound (see Ta-ble 3). Even so, no metaheuristics can reach the pin-pointed upper bound yet in the “DDM+PW” case (re-call Table 8 and Table 4). Overall, the more challeng-ing DDM cases may serve as better benchmarks thanthe SDM to assess various metaheuristics. Besides,the certification of global optimality demands moreadvanced global optimization techniques (Floudas andGounaris, 2008) and deserves more investigations.

(3) In all the four test cases, the simple DE depicted byAlgorithm 2 performs surprisingly equally well whencompared with a variety of sophisticated metaheuristicmethods apart from its unparalleled efficiency advan-tage. We surely admire the power of advanced meta-heuristics like those in Table 8, but it seems unfortu-nate that the competence of simpler algorithms hasbeen largely underestimated in the current literaturefor PV parameter estimation. We suggest that futureresearchers justify their methods by comparing againstthe simple DE here, which is an appropriate baselinefor forthcoming studies. Besides, further examinationof other classic metaheuristics, like the canonical par-ticle swarm optimization, is also meaningful.

(4) It should be emphasized that we are definitely not un-derrating the value of sophisticated metaheuristics ina general sense. Such complexity is necessary for manydifficult real-world tasks (Yang, 2020). However, notethat the normal PV parameter estimation is essen-tially a straightforward nonlinear least squares prob-lem with only five or seven variables. A reasonablespeculation is that the present problem is not toughenough to highlight the real power of such advancedmetaheuristics. Instead, the simple DE in Algorithm 2appears to be a better match for this specific problemfrom a pragmatic view. We advocate the examina-tion of more challenging scenarios in future investiga-tions, where there is still vast room for improvement,e.g., the DDM with the Photowatt-PWP201 PV mod-ule dataset, datasets of larger sizes, datasets in differ-ent temperatures, and the less commonly studied butmore demanding three diode model (Yang et al., 2020;Yousri et al., 2020; Qais et al., 2019).

(5) Given the abundance of metaheuristics in the litera-ture, we believe that more efforts may be better putinto other aspects apart from fitting accuracy, whichmay include rigorous certification of global optimality,satisfaction of practical industry needs, and construc-tion of new high-quality datasets, among others.

Acknowledgements

We would like to thank Dr. Gilles Chabert for develop-ment of ibex and helpful discussions. This work was sup-ported by the Exploiting Distributed Generation (EDGE)programme grant (No: EDGE-GC-2018-002).

Data Availability

The two benchmark datasets and our code are publiclyaccessible at https://github.com/ShuhuaGao/rePVest.

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