1 Modelling , parameters identification and 2 experimental validation of a lead acid battery bank 3 using genetic algorithms 4

Accurate and efficient battery modeling is essential to maximize the performance of 16 isolated energy systems and to extend battery lifetime. This paper proposes a battery model that 17 represents the charging and discharging process of a lead-acid battery bank. This model is 18 validated over real measures taken from a battery bank installed in a research center placed at “El 19 Chocó”, Colombia. In order to fit the model, three optimization algorithms (Particle Swarm 20 Optimization, Cuckoo Search, and Particle Swarm Optimization+Perturbation) are implemented 21 and compared, being the last one a new proposal. This research shows that the model with the 22 proposed algorithm is able to estimate and manage the real battery characteristics as SOC and 23 charging/discharging voltage. The comparison between simulations and real measures shows that 24 the model is able to absorb reading problems, signal delays, and scaling errors. The approach we 25 present can be implemented in other types of batteries especially those used in stand-alone 26 systems. 27


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This paper is organized as follows. In Section 2, a system description of the research center is

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The discharge voltage is simulated through equation (2), while the charge voltage is simulated

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The SOC is understood as the fraction or percentage of the capacity is still available in the 127 battery and is estimated by equation (5). SOCo corresponds to the battery initial SOC and Cbat is 128 battery capacity in Ah.

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and kc_bat is an associated parameter to battery aging. New parameters (kc120, ksoc, kI, kc_bat) have an 133 initial value of 1 to maintain the original model performance. The model is able to choose the discharging equation when the input signal has a positive sign 139 (+A); otherwise, the model chooses the charging equations if the current sign is negative (-A).

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Current and temperature are taken as input signals.
141 Figure 3 shows a comparison between the experimental system performance and the battery 142 model using parameter sets from Table 1. Furthermore, Table 2 A general PS is defined as: where c k i,j represents any parameter of the PS; e.g, , is the value of first parameter (Vbo) in 174 second parameter set of the 8th population PS28. This parameter involve both Vboc and Vbodc.

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In the first step, the initial values coefficients were taken from literature (Table 1). In the second 177 step, the initial population (k=1) of size j is an array of PS that is creates following the bellow rules:

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In third step, each PSj 1 is evaluated in battery model, and for the next step, different error εPSj 1 184 is calculated for charging/discharging operation mode for both system signals, battery Voltage and 185 battery SOC, in first place, the battery voltage error εVPSj 1 is calculated as: where, εVc and εVdc are the battery voltage error (charging/discharging). The mean errors are 187 calculates as: where Vm and Vs are the measured and simulated battery voltage respectively, and j is number 189 of samples. In similar way, the battery SOC error εSOCPSj 1 is calculated as: where, εSOCc and εSOCdc are the battery voltage error (charging/discharging where SOCm and SOCs are the measured and simulated battery SOC respectively, and j is 193 number of samples.

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The goal is to minimize both εPSj k . The optimization process ends when a stop condition is met.

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Each GA uses a particular policy to create the new population from the previous evaluated one.

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The goal is to converge to the optimal solution in a minimum number of steps. In order to perform

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The swarm moves around a particle that finds a possible good place (solution) and explores this 211 location; if another particle finds a better solution, the swarm moves to this new place. The    where ci GBest is coefficient belonging to GBest until iteration l-1, and n is perturbation value.

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Because the use of perturbations this proposal is named PSO+P. The perturbation is introduced into 231 iteration around the stabilization of PSO, and then is necessary run first the PSO.

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The model duality allows the use of different errors for both battery operation modes 241 (charging/discharging). Therefore, we can study separately each behavior. Figure 5 shows both. In

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Results discard CS as a suitable method to fit the battery behavior. Therefore, only PSO and

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PSO+P need a closer look in order to select the best algorithm. A fast analysis over the curves shows 256 that PSO is the faster algorithm but it falls in local minima so it loses search ability, thus confirming 257 its known limitations. PSO+P obtain the best results but it requires more iterations to achieve it. A 258 quantitative comparison is shown in Table 4. Criteria used to evaluate each algorithm are precision,   Table 4 describes PSO as the best algorithm regarding velocity and computational cost.

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However, PSO+P show the most accurate fit. Therefore, the model is fit from PSO+P results. These 276 parameters are shown in Table 5.

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The simulation using the identification of parameters with PSO+P significantly improved the 286 mean errors presented in Table 2. The mean errors achieved with this method were 0.29% and 0.44%

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for the battery voltage signal (discharging/charging, respectively) for an average between the two 288 signals of 0.365%.

Experimental validation 291
Model validation is carried out with data from three months divided into 4-day packages to be   Table 6.

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When the battery model is validated for a 15-day package with the parameters shown in Table 5 308 the SOC mean error was 0.46% and voltage mean error 0.45% (see Figure 8). Again the model 309 presents a good matching with the measured signal. However, on day 4th and 11th overcharging 310 events occurred, and the model only was able to represent the second event.

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The model has been fit to represent the real behavior of a BESS placed at "El Chocó". Colombia.

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Despite, improvements should be made to the model. The complexity of the model implies the use 325 of GA to perform the optimization, so this paper has tested three GA in order to find the best

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The present proposal has the ability to find PS to fit the model no matter the start point from 332 however, to continue the present work is necessary to make a sensibility analysis of the parameters 333 to define the influence of each parameter in the simulation results, and establish its adequate 334 boundaries

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The battery model with the new extracted parameters presents a good matching with the reals 337 measurement obtained from the research center. The main advantage of the developed model is its 338 low computational cost and its ability to absorb reading problems, and scaling errors when the 339 simulation is valid with real measurements. The model and its fitting approach presented in this 340 paper may be applied to other types of batteries especially those used in stand-alone systems.

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The battery model will be used to make a better battery energy management system from 343 research center placed at "El Chocó". Colombia.

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Author Contributions: The present work was developed with following contributions: Conceptualization,