A Dense Explorative ElectroStatic Discharge optimization Algorithm for Photovoltaic Parameters Estimation

A Dense Explorative ElectroStatic Discharge optimization Algorithm for Photovoltaic Parameters Estimation

Item Type Conference Paper

Authors Pervez, Imran;Antoniadis, Charalampos;Massoud, Yehia Mahmoud

Citation Pervez, I., Antoniadis, C., & Massoud, Y. (2022). A Dense

Explorative ElectroStatic Discharge optimization Algorithm for Photovoltaic Parameters Estimation. 2022 IEEE 65th International Midwest Symposium on Circuits and Systems (MWSCAS). https://

doi.org/10.1109/mwscas54063.2022.9859366 Eprint version Post-print

DOI 10.1109/MWSCAS54063.2022.9859366

Publisher IEEE

Rights Archived with thanks to IEEE Download date 2023-11-29 20:02:47

Link to Item http://hdl.handle.net/10754/680508

A Dense Explorative ElectroStatic Discharge

Optimization Algorithm for Photovoltaic Parameters Estimation

Imran Pervez, Charalampos Antoniadis, and Yehia Massoud Innovative Technologies Laboratories (ITL),

King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia {imran.pervez, charalampos.antoniadis, yehia.massoud}@kaust.edu.sa

Abstract—Photovoltaic (PV) systems are becoming one of the most emerging systems for power generation owing to their low cost and clean operation. Due to the high demand for PVs in sev- eral small and large-scale applications, their accurate modeling is essential in power systems simulation to obtain reliable results.

In this work, an improved ElectroStatic Discharge Algorithm with Dense Explorative Search (ESDADES) is proposed that estimates the parameters of the most popular PV cell models with more accuracy than previous works. More specifically, the dense explorative search proposed in this work enhances the ESDA capability to search around the regions closer to the best solution found by ESDA more extensively, thus improving convergence accuracy. The proposed ESDADES was compared to two recently proposed optimization algorithms, namely the Self-adaptive Ensemble-based Differential Evolution (SEDE), the Directional Permutation Differential Evolution (DPDE), and the simple ESDA. The experimental results demonstrated that the proposed algorithm arrives faster at more accurate estimates of the examined PV cell models parameters

Index Terms—Metaheuristic algorithms, ESDADES, parame- ter estimation, optimization, PV models

I. INTRODUCTION

The rapid depletion of the limited conventional energy sources has made it necessary to shift to sustainable renew- able energy sources, with PVs attracting the most attention.

However, several issues remain with PV electricity generation, such as less conversion efficiency and power losses under partial shading over the PV surface. Therefore, modeling and simulation of PV are performed to design PV systems and improve their performance [1].

For correct simulation and performance analysis, it is essen- tial to know the accurate non-linear current versus voltage (I- V) characteristics of a PV. Due to faults, aging, and changing operation conditions, experimental current and voltage data for modeling these I-V characteristics may not be accurate [2].

Therefore, compact parameterized I-V models are necessary to be adjusted according to the certain conditions, which gives rise to the accurate PV model parameter estimation (PE) problem. The objective of the PE is to minimize the root mean square error (RMSE) of the experimental current values and those resulting from the PV current modeling. The smaller the error, the more accurate the calculated parameters are leading to more accurately estimated I-V characteristics.

Several optimization algorithms searching for the optimal parameters have been proposed in the literature [2]- [18]

that can be categorized into two classes [2]: metaheuristic algorithms [2]- [15] and analytical methods [16]- [18]. The analytical methods try to estimate parameters through mathe- matical programming [2]. However, they are computationally expensive, and they may converge to local minima because the problem is non-convex [2]. Metaheuristic techniques, on the other hand, spread solutions within the parameters bounds emulating phenomena from physics, biology, nature etc. Although, they cannot guarantee the global convergence, they explore more extensively the search space and perform a simple evaluation of the objective function, making more likely to converge to the global maximum, and reducing the computational complexity. Several metaheuristic techniques have been applied so far for this problem [2]- [15].

All these works were devoted to making the PV parameter estimation more accurate. In very recent works like [4], a self-adaptive ensemble-based differential evolution (SEDE) was proposed. The algorithm used three different differential evolution (DE) based mutation strategies divided into two groups with respect to the self-adaptive scheme in [4]. In [2], a directional permutation differential evolution (DPDE) algorithm improved the accuracy by exploiting the information of selective individuals from an external archive through random permutation and the differential vector deciding in which direction to move based on the fitness of agents.

Though these algorithms contribute to improved accuracy, they still need improvement to estimate the PV parameters more accurately. To this end, in this work, an Electrostatic Discharge Algorithm based on Dense Explorative Search (ES- DADES) is proposed as an extension to the simple Electro- Static Discharge Algorithm (ESDA) [19]. The algorithm takes advantage initially of the high search space exploration and the fast exploitation of the simple ESDA, and then conducts a denser search confining its exploration in a compact area of the search space to estimate parameters more accurately.

The rest of the paper is organized as follows. Section II provides the problem formulation. Section III first describes the simple ESDA and then our improved version of it for PV parameters estimation. Section IV demonstrates our experi- mental results, while section V concludes the work.

(a) (b) I^L I^D I^st 𝐈^𝑺𝑫𝑴

𝐕_𝐨 R^st

R^se

I^D1 I^D2 I^st I^L

R^st R^se 𝐈^𝑫𝑫𝑴

𝐕_𝐨

Fig. 1: Circuit models of (a) SDM, and (b) DDM

II. PROBLEMFORMULATION

The most popular PV cell models in the literature are the Single Diode Model (SDM) and the Double Diode Model (DDM). The parameters characterizing these models are op- timized by minimizing the root MSE between some real experimental-current values and the current values calculated using these models.

A. Single Diode Model

The SDM circuit model is shown in Fig. 1(a). It consists of a resistance (Rse) connected in series with the parallel combination of a diode, a PV light current source (IL), and a shunt resistance (Rst). The output current (I_o^SDM) is given as:

I^SDM(V_o, I_o) =I_L−I_D−I_st

=IL−Is×

exp

qc(Vo+RseIo) akbT

−1

−Vo+RseIo

Rst

(1) where Vo is the output voltage of the PV, Is is the diode saturation current, ID is the diode current, qc is the electron charge, k_b is the Boltzmann constant, and T is the cell temperature.

B. Double Diode Model

The DDM (see Fig. 1(b)) adds one more diode parallel to the SDM diode to consider the current recombination losses in the depletion region. The output current (I^DDM) in this model is given as follows:

I^DDM(Vo, Io) =IL−ID1−ID2−Ist

=IL−Is×

exp

qc(Vo+RseIo) ak_bT

−1

−I_s2×

exp

q_c(V_o+R_seI_o) bkbT

−1

−V_o+R_seI_o Rst

(2) whereI_s,I_s2,aandbare the diode diffusion current, the diode recombination current, the ideality factor of the diode diffusion current and the ideality factor of the diode recombination current, respectively.

C. Objective Function

The optimal values for the unknown parameters of each model are obtained from the solution of the root MSE min- imization between the true experimental-current values and the theoretical-current values computed using the two models (error function).

The error function (f_P) using SDM is (f_P(V_o, I_o) =I^SDM(V_o, I_o)−I_o

P={IL,Is, Rse, Rst, a} (3) and using DDM is

(f_P(V_o, I_o) =I^DDM(V_o, I_o)−I_o

P={IL,Is, Is2, Rse, Rst, a, b} (4) respectively, with P ¹ being the set of unknown parameters in each model. Finally, the objective function (root MSE) is given as follows:

RM SE(P) = v u u t

1 Nexp

Nexp

X

i=1

f_P²(V_oⁱ, I_oⁱ) (5) whereV_oⁱ andI_oⁱ are thei-th real voltage and current experi- mental samples, respectively, andN_expis the total number of these samples.

III. PROPOSEDALGORITHM

The objective function (see eq. (5)) is non-convex [2], and thus mathematical programming techniques are not suitable for solving the minimization problem because they may con- verge to local minima. On the other hand, although they cannot guarantee global convergence, metaheuristic techniques explore the search space more extensively, making it more likely to converge (close) to the global maximum. This section first describes the metaheuristic algorithm ESDA and then the proposed DES technique, which improves ESDA convergence accuracy.

A. ElectroStatic Discharge Algorithm (ESDA)

ESDA is a nature-inspired (NI) algorithm drawing intu- ition from ElectroStatic Discharge (ESD) phenomena between objects. ESD phenomena can be either direct or indirect.

The former concerns the interaction between the intruder and the receptor objects, while the latter concerns the impact on a third object due to the interaction between the intruder- receptor objects. Inspired by the above phenomena, ESDA randomly initializes objects (solutions) in different locations in the search space. The object’s location determines its immunity in ESD phenomenon (based on the evaluation of the fitness function). In each iteration, three objects are considered randomly, and the one with the worst fitness is updated to make it come closer to the objects with the better fitness values. The number of objects participating in ESD interaction depends

1We denote vectors with bold letters.

TABLE I: New lower and upper bounds used in ESDADES.

Parameters SDM DDM

C_LB C_{U B} C_LB C_{U B}

I_L(A) 1.59412556185012e-4 9.05874438149601e-05 1.991810536254e-3 1.39181053625403e-3 Is(µA) 1.92470166058799e-09 3.1924701660588e-08 1.90889515325939e-07 1.50889515325939e-07

a 1.52323706616402e-2 2.27676293383599e-2 7.878881761137e-2 5.878881761137e-2 Rse(Ω) 2.1275120940202e-3 2.8724879059798e-3 3.4130448376353e-3 4.413044837635e-3 Rst(Ω) 0.383507819644699 1.6864921803553 8.78196093472599 10.781960934726

I_s2(µA) NA NA 2.4217102165069e-07 3.4217102165069e-07

b NA NA 0.19672718721977 0.39672718721977

on the value of a random number. If that value is larger than 0.5, two objects take part in the ESD interaction; otherwise, all three are involved. If two objects are involved, i.e., direct interaction, the position of the object with the worse fitness value (let us assume X2) is updated as shown below:

Xⁱ⁺¹₂ =Xⁱ₂+ 2∗r∗(Xⁱ₁−Xⁱ₂) (6) On the other hand, if all three are involved, i.e., indirect interaction, the location of the object with the lowest fitness value (let us assume X₃) is updated as follows:

Xⁱ⁺¹₃ =Xⁱ₃+ 2∗r₁∗(Xⁱ₁−Xⁱ₃) + 2∗r₂∗(Xⁱ₂−Xⁱ₃) (7) whereXⁱ₁,Xⁱ₂andXⁱ₃are the objects’ locations at the current iteration,Xⁱ⁺¹₂ andXⁱ⁺¹₃ are the objects’ locations at the next iteration, and r,r1 andr2 are random values between 0 and 1. After objects’ locations update, any object falling out of the search space bounds or being updated more than three times is reinitialized randomly inside these bounds. For objects with fewer than three updates, a random number is generated for each location coordinate (vector dimension), and if that number is smaller than 0.2, the corresponding coordinate is reinitialized within its bounds.

B. Electro Static Discharge Algorithm with Dense Explorative Search

Object re-initialization in ESDA enables search space ex- ploration because new random object locations are examined.

However, the convergence accuracy of ESDA for PV param- eters estimation is not comparable to the results of recent works, as shown in the experimental results. We improve the convergence accuracy of the simple ESDA by introducing the Dense Explorative Search (DES) technique.

The idea of DES is to regenerate objects (solutions) more densely near the best solution found so far. In this study, the number of regenerated solutions was 2.5 times the size of the initial population (Ω) for both models. The dense search near the optimum global increases the likelihood of finding solutions with better fitness values (RMSE in our case) significantly, thus improving accuracy. The DES verifies if the current iteration count has exceeded a threshold in each iteration. Once that threshold is exceeded, 2.5×|Ω| new solutions are densely generated nearby the current best object position (X^∗) within the new bounds given as:

Magnified

(a)

ESDA SEDE DPDE ESDADES

(b)

Magnified ESDA SEDE DPDE ESDADES

Fig. 2: Convergence results of ESDA, SEDE, DPDE, and ESDADES for (a) SDM and (b) DDM. The discrepancy in iterations until convergence is owed to different population sizes, which means more fitness evaluations occur, and the maximum number of fitness evaluations we allowed in our experiments.

UBnew=X^∗+CU B

LBnew=X^∗−CLB

(8) , and with each new solution initialized as follows:

Xk =LBnew+r∗(UBnew−LBnew) (9) where X_k is the new reinitialized object position, r is a random scalar between 0 and 1, and C_{U B} and C_LB are the new bounds confining subsequent search aroundX^∗(see Table I). Finally, the threshold iteration count for SDM, and DDM are 156, and 112 respectively.

IV. EXPERIMENTALEVALUATION

This section compares the proposed ESDADES to DPDE and SEDE as well as the simple ESDA. We compare these algorithms in the accuracy of PV parameters estimation for SDM and DDM and their convergence speed. The R.T.C France commercial solar cell [20] was considered to obtain 26 experimental voltage and current values at 33^◦C with a cell diameter of57mm.

The values of lower bound (LB) and upper bound (UB) for both models were selected similarly to [2], [4]. In addition, the population size of ESDADES, DPDE, SEDE, and ESDA was set to 60, 18, 30, and 60, respectively, and the probability and historical memory size of DPDE was set to 0.11 and 5, respectively. We performed our experiments on a system equipped with an Intel core i7-1165G7 processor @ 2.80GHz 1.69 GHz and 16 GB RAM.

TABLE II: SDM parameters estimates using ESDADES, DPDE, SEDE and ESDA algorithm.

Parameters Models Algorithm

ESDA SEDE DPDE ESDADES

I_L(A) SDM 0.7607755 0.7607755 0.7607755 0.7607755

DDM 0.760768167194272 0.760780877737228 0.760781094359775 0.760781235839245

Is(µA) SDM 0.32302086741 0.323020868100 0.32302081524 0.32302081204

DDM 3.17112492144775e-07 2.25880840491376e-07 7.52627717781459e-07 2.65803623229658e-07

Rse(Ω) SDM 0.036377091 0.036377091 0.036377092 0.036377092

DDM 0.0363752280954649 0.0367366100327704 0.0367407312134622 0.0366318621663932

Rst(Ω) SDM 53.7185292 53.7185278 53.7185235 53.7185249

DDM 53.9481441763902 55.4550398931927 55.4891752861524 55.2008826513980

a SDM 1.480173344 1.480173344 1.480173328 1.480173327

DDM 1.47870036943218 1.45006071756636 1.99999999987772 1.46283634694150

I_s2(µA) SDM NA NA NA NA

DDM 5.12121698315915e-08 7.26429145558331e-07 2.26058644183031e-07 9.99999661478501e-07

b SDM NA NA NA NA

DDM 1.98150692086449 1.99043582445024 1.45004711548943 2.23955898640544 RM SE SDM 9.860218778916055e-04 9.860218778916910e-04 9.860218778915200e-04 9.860218778915072e-04

DDM 9.856746441885779e-04 9.825795794586624e-04 9.824690432402288e-04 9.807673270804725e-04

Table II demonstrates the convergence values of the SDM and DDM unknown parameters and the corresponding RMSE using the examined algorithms. As shown from that Table, the proposed algorithm converges to lower RMSE²than the other algorithms. In particular, for the DDM, the degree of improve- ment is much higher as the RMSEs start to differ from much lower decimal places. Moreover, in Fig. 2, we demonstrate the convergence behavior of all algorithms. Again, it can be seen that the proposed algorithm converges faster than other algorithms while reaching lower RMSE values.

V. CONCLUSION

This work proposed an ElectroStatic Discharge Algorithm with Dense Explorative Search (ESDADES). The idea relies on exploiting the initial diverse exploration of the simple ESDA and then exploring the nearby region to the opti- mum solution found by ESDA more extensively to search for better solutions. Our experimental results demonstrated that the proposed algorithm arrives faster at more accurate PV SDM and DDM parameters estimates than two recently proposed algorithms (SEDE and DPDE) and the simple ESDA.

Finally, we plan to extend the proposed algorithm to estimate parameters for PV module models in the future.

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