#
Constrained Least-Squares Parameter Estimation for a Double Layer Capacitor^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{−1}[4], or the ability to cycle power by charging and discharging energy at a more rapid rate than batteries. With reference to EVs and HEVs, they contribute significantly by providing short/rapid discharges during starting and acceleration periods. In general, SCs are deployed as a rapid-response buffer between the electric drive system and the primary storage device to improve the performance of the system in terms of cost, system life, and overall efficiency [5,6]. This capability gives SCs a major lead for their application in EVs and HEVs, where they are required to be continually charged/discharged without loss in performance—something which is difficult with batteries (PB, N-Cd, niMH and Li-ion) for their longer time constant [7].

^{®}-Simulink

^{®}environment. Results show agreement between a real SC and the simulated Lagrange model.

## 2. Description and Analysis of the SC Model

#### 2.1. The 2-Branch SC Model

#### 2.2. Analysis of the 2-Branch Model

_{1}= 0 is made, which is reasonable for a DLC [9,15]. Moreover, R

_{3}is assumed known, since this parameter, which is the leakage resistance, is not difficult to find experimentally by using a discharge test without any external load [9]. Moreover, the attempt to retrieve it by using this approach, although theoretically possible, gives rise to an ill-conditioned problem.

## 3. Methodology

#### 3.1. Algorithms Adopted

- By using the Ordinary Least Squares (OLS) regression off-line to solve Equation (3), but without using the constraint as expressed in Equations (4a)–(4e). This has been performed in simulation and experimentally and has been used as starting point for the subsequent minimization,
- By considering the following minimization problem:$$\begin{array}{c}\begin{array}{c}\mathit{min}{\Vert A\alpha -b\Vert}_{2}\\ subjecttotheconstraint:{\alpha}_{2}={\alpha}_{3}{\alpha}_{5}\end{array}\\ \end{array}$$
- By using the Faranda method which is conducted off-line and specific to two-branch models of SC on experimental data for comparison [15].

#### 3.2. Signal Processing System

^{®}has been used to design a 50th order FIR $D\left(z\right)$ tuned to have a passband frequency of 100 Hz and stop band frequency at 180 Hz. The magnitude frequency response and the phase response of the $D\left(z\right)$ filter is shown in Figure 4. The group delay of the Digital Derivative Filter is determined to be 25 samples from the Group Delay plot as shown in Figure 5. The sampling frequency adopted is f

_{s}= 1 kHz.

## 4. Simulation

#### Simulation Results with a Ramp Input Current

^{®}-Simulink

^{®}environment running on a PC with an Intel i5 processor.

## 5. Experimental Verifications

#### 5.1. The Super Capacitor Bank

_{Bank}, becoming 83.33 F. The voltage rating of the SC Bank is calculated to be 16.2 V. A 10 Ω load resistor has been used in the discharge circuit only which results in a time constant, $\tau $, of 833 s.

- ESR @ 1 kHz = 18 mΩ
- ESR in DC = 30 mΩ
- Max. Peak Current = 20 A
- Max. Continuous Current = 167 A
- Rated Voltage = 15 V

#### 5.2. Experimental Rig

_{Bank}.

_{Bank}is 16.2 V, a voltage divider has been used as an intermediary to the dSPACE system. A National Instruments (NI) USB 6211 DAQ system was utilized as a redundant data capture system but did not play any crucial role as dSPACE delivered better results. The sampling frequency of 1 kHz has been selected for the charge and discharge tests. The Experimental Test Rig is shown in Figure 11.

#### 5.3. Experimental Determination of R_{3}

_{3}, a basic approach has been applied [44]. The RC time constant is given by: $\tau ={R}_{3}{C}_{Bank}$.

#### 5.4. The Bessel Filter

#### 5.5. The Ramp Current Generator

#### 5.6. Experimental Charge Curves

_{Bank}with an input step current and an input ramp current, respectively. The step current has a constant value of 6 A. For the input ramp current, the ramp starts at 0 A and rises with constant slope to a value of 5.6 A. Notice that for both Figure 12 and Figure 13, the current rolls down exponentially once the C

_{Bank}if fully charged. These are the values adopted for the experimental verification of the method.

**Figure 12.**Experimental Current and Voltage charge curve of a 16.2 V 83.33 F EDLC with a step current of 6 A.

**Figure 13.**Experimental Current and Voltage charge curve of a 16.2 V—83.33 F EDLC with a ramp current 0–5.59 A.

## 6. Results and Discussion for Experimental Data

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ehsani, M.; Gao, Y.; Emadi, A. Fundamentals, Theory, and Design, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2017; ISBN 978-1-315-21940-0. [Google Scholar]
- Dey, S.; Mohon, S.; Pisu, P.; Ayalew, B.; Onori, S. Online State and Parameter Estimation of Battery-Double Layer Capacitor Hybrid Energy Storage System. In Proceedings of the 2015 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, 15–18 December 2015; pp. 676–681. [Google Scholar]
- Cao, J.; Emadi, A. A New Battery/UltraCapacitor Hybrid Energy Storage System for Electric, Hybrid, and Plug-In Hybrid Electric Vehicles. IEEE Trans. Power Electron.
**2012**, 27, 122–132. [Google Scholar] [CrossRef] - Devillers, N.; Jemei, S.; Péra, M.-C.; Bienaimé, D.; Gustin, F. Review of Characterization Methods for Supercapacitor Modelling. J. Power Sources
**2014**, 246, 596–608. [Google Scholar] [CrossRef] - Farhadi, M.; Mohammed, O. Energy Storage Technologies for High-Power Applications. IEEE Trans. Ind. Appl.
**2016**, 52, 1953–1961. [Google Scholar] [CrossRef] - Shen, X.; Chen, S.; Li, G.; Zhang, Y.; Jiang, X.; Lie, T.T. Configure Methodology of Onboard Supercapacitor Array for Recycling Regenerative Braking Energy of URT Vehicles. IEEE Trans. Ind. Appl.
**2013**, 49, 1678–1686. [Google Scholar] [CrossRef] - Rafik, F.; Gualous, H.; Gallay, R.; Crausaz, A.; Berthon, A. Frequency, Thermal and Voltage Supercapacitor Characterization and Modeling. J. Power Sources
**2007**, 165, 928–934. [Google Scholar] [CrossRef] - Zubieta, L.; Bonert, R. Characterization of Double-Layer Capacitors for Power Electronics Applications. IEEE Trans. Ind. Appl.
**2000**, 36, 199–205. [Google Scholar] [CrossRef] - Pucci, M.; Vitale, G.; Cirrincione, G.; Cirrincione, M. Parameter Identification of a Double-Layer-Capacitor 2-Branch Model by a Least-Squares Method. In Proceedings of the IECON 2013—39th Annual Conference of the IEEE Industrial Electronics Society, Vienna, Austria, 10–13 November 2013; pp. 6770–6776. [Google Scholar]
- Kitahara, A.; Watanabe, A. (Eds.) Electrical Phenomena at Interfaces (Fundamentals, Measurements and Applications); Marcel Dekker: New York, NY, USA, 1984. [Google Scholar]
- Morrison, S.R. The Chemical Physics of Surfaces; Springer Science & Business Media: Berlin, Germany, 2013; ISBN 978-1-4899-2498-8. [Google Scholar]
- Maundy, B.J.; Elwakil, A.; Freeborn, T.; Allagui, A. Improved Method to Determine Supercapacitor Metrics from Highpass Filter Response. In Proceedings of the 2016 28th International Conference on Microelectronics (ICM), Giza, Egypt, 17–20 December 2016; pp. 25–28. [Google Scholar]
- Buller, S.; Karden, E.; Kok, D.; De Doncker, R.W. Modeling the Dynamic Behavior of Supercapacitors Using Impedance Spectroscopy. IEEE Trans. Ind. Appl.
**2002**, 38, 1622–1626. [Google Scholar] [CrossRef] - Halper, M.S. Supercapacitors: A Brief Overview; MITRE: McLean, VA, USA, 2006; pp. 1–41. [Google Scholar]
- Faranda, R. A New Parameters Identification Procedure for Simplified Double Layer Capacitor Two-Branch Model. Electr. Power Syst. Res.
**2010**, 80, 363–371. [Google Scholar] [CrossRef] - Sakka, M.A.; Gualous, H.; Omar, N.; Mierlo, J.V.; Sakka, M.A.; Gualous, H.; Omar, N.; Mierlo, J.V. Batteries and Supercapacitors for Electric Vehicles; IntechOpen: London, UK, 2012; ISBN 978-953-51-0893-1. [Google Scholar]
- IEC 62576:2018|IEC Webstore. Available online: https://webstore.iec.ch/publication/28801 (accessed on 23 January 2023).
- Logerais, P.O.; Camara, M.A.; Riou, O.; Djellad, A.; Omeiri, A.; Delaleux, F.; Durastanti, J.F. Modeling of a Supercapacitor with a Multibranch Circuit. Int. J. Hydrogen Energy
**2015**, 40, 13725–13736. [Google Scholar] [CrossRef] - Drummond, R.; Howey, D.A.; Duncan, S.R. Parameter Estimation of an Electrochemical Supercapacitor Model. In Proceedings of the 2016 European Control Conference (ECC), Aalborg, Denmark, 29 June–1 July 2016; IEEE: Piscataway, MJ, USA, 2016; pp. 1–6. [Google Scholar]
- Reichbach, N.; Kuperman, A. Recursive-Least-Squares-Based Real-Time Estimation of Supercapacitor Parameters. IEEE Trans. Energy Convers.
**2016**, 31, 810–812. [Google Scholar] [CrossRef] - Eddahech, A.; Ayadi, M.; Briat, O.; Vinassa, J.-M. Online Parameter Identification for Real-Time Supercapacitor Performance Estimation in Automotive Applications. Int. J. Electr. Power Energy Syst.
**2013**, 51, 162–167. [Google Scholar] [CrossRef] - Freeborn, T.J.; Maundy, B.; Elwakil, A.S. Measurement of Supercapacitor Fractional-Order Model Parameters From Voltage-Excited Step Response. IEEE J. Emerg. Sel. Top. Circuits Syst.
**2013**, 3, 367–376. [Google Scholar] [CrossRef] - Oukaour, A.; Pouliquen, M.; Tala-Ighil, B.; Gualous, H.; Pigeon, E.; Gehan, O.; Boudart, B. Supercapacitors Aging Diagnosis Using Least Square Algorithm. Microelectron. Reliab.
**2013**, 53, 1638–1642. [Google Scholar] [CrossRef] - Vitale, G. Supercapacitor Modelling by Lagrange’s Equations; EEEJ: Las Palmas de Gran Canaria, Spain, 2016; Volume 1, pp. 127–132. [Google Scholar]
- Alonge, F.; Rodonò, G.; Cirrincione, M.; Vitale, G. Supercapacitor Diagnosis Using an Extended Kalman Filtering Approach. In Proceedings of the 2016 IEEE 16th International Conference on Environment and Electrical Engineering (EEEIC), Florence, Italy, 7–10 June 2016; pp. 1–6. [Google Scholar]
- Nadeau, A.; Sharma, G.; Soyata, T. State-of-Charge Estimation for Supercapacitors: A Kalman Filtering Formulation. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4–9 May 2014; pp. 2194–2198. [Google Scholar]
- Zhang, L.; Wang, Z.; Sun, F.; Dorrell, D.G. Online Parameter Identification of Ultracapacitor Models Using the Extended Kalman Filter. Energies
**2014**, 7, 3204–3217. [Google Scholar] [CrossRef] - Nonlinear Extension of Battery Constrained Predictive Charging Control with Transmission of Jacobian Matrix|Elsevier Enhanced Reader. Available online: https://reader.elsevier.com/reader/sd/pii/S014206152200758X?token=D6E214EE9322B88D938DA48FE1FC341540098929F043EA1C45991DAE9C0EEA8BCBD06100680774AED9CE8425C8AB38C6&originRegion=us-east-1&originCreation=20230426011722 (accessed on 26 April 2023).
- Saha, P.; Dey, S.; Khanra, M. Modeling and State-of-Charge Estimation of Supercapacitor Considering Leakage Effect. IEEE Trans. Ind. Electron.
**2020**, 67, 350–357. [Google Scholar] [CrossRef] - Naseri, F.; Farjah, E.; Ghanbari, T.; Kazemi, Z.; Schaltz, E.; Schanen, J.-L. Online Parameter Estimation for Supercapacitor State-of-Energy and State-of-Health Determination in Vehicular Applications. IEEE Trans. Ind. Electron.
**2020**, 67, 7963–7972. [Google Scholar] [CrossRef] - El Mejdoubi, A.; Chaoui, H.; Gualous, H.; Sabor, J. Online Parameter Identification for Supercapacitor State-of-Health Diagnosis for Vehicular Applications. IEEE Trans. Power Electron.
**2017**, 32, 9355–9363. [Google Scholar] [CrossRef] - Shi, Z.; Auger, F.; Schaeffer, E.; Guillemet, P.; Loron, L. Interconnected Observers for Online Supercapacitor Ageing Monitoring. In Proceedings of the IECON 2013—39th Annual Conference of the IEEE Industrial Electronics Society, Vienna, Austria, 10–13 2013; IEEE: Piscataway, MJ, USA, 2013; pp. 6746–6751. [Google Scholar]
- Chaoui, H.; El Mejdoubi, A.; Oukaour, A.; Gualous, H. Online System Identification for Lifetime Diagnostic of Supercapacitors With Guaranteed Stability. IEEE Trans. Control Syst. Technol.
**2016**, 24, 2094–2102. [Google Scholar] [CrossRef] - Dănilă, E.; Livint, G.; Lucache, D.D. Dynamic Modelling of Supercapacitor Using Artificial Neural Network Technique. In Proceedings of the 2014 International Conference and Exposition on Electrical and Power Engineering (EPE), Iasi, Romania, 16–18 October 2014; pp. 642–645. [Google Scholar]
- Marie-Francoise, J.-N.; Gualous, H.; Berthon, A. Supercapacitor Thermal- and Electrical-Behaviour Modelling Using ANN. IEE Proc.-Electr. Power Appl.
**2006**, 153, 255–262. [Google Scholar] [CrossRef] - Eddahech, A.; Briat, O.; Ayadi, M.; Vinassa, J.-M. Modeling and Adaptive Control for Supercapacitor in Automotive Applications Based on Artificial Neural Networks. Electr. Power Syst. Res.
**2014**, 106, 134–141. [Google Scholar] [CrossRef] - Farsi, H.; Gobal, F. Artificial Neural Network Simulator for Supercapacitor Performance Prediction. Comput. Mater. Sci.
**2007**, 39, 678–683. [Google Scholar] [CrossRef] - Miniguano, H.; Barrado, A.; Fernández, C.; Zumel, P.; Lázaro, A. A General Parameter Identification Procedure Used for the Comparative Study of Supercapacitors Models. Energies
**2019**, 12, 1776. [Google Scholar] [CrossRef] - Fathy, A.; Rezk, H. Robust Electrical Parameter Extraction Methodology Based on Interior Search Optimization Algorithm Applied to Supercapacitor. ISA Trans.
**2020**, 105, 86–97. [Google Scholar] [CrossRef] [PubMed] - Jannif, N.I.; Ram, K.; Bangalini, K.; Loli, A.; Mohammadi, A.; Cirrincione, M. Supercapacitor Parameter Estimation and Hybridyzation with PEMFC for Purge Compensation. In Proceedings of the 2022 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), Sorrento, Italy, 22–24 June 2022; pp. 70–75. [Google Scholar]
- Prasad, R.; Mehta, U.; Kothari, K.; Cirrincione, M.; Mohammadi, A. Supercapacitor Parameter Identification Using Grey Wolf Optimization and Its Comparison to Conventional Trust Region Reflection Optimization. In Proceedings of the 2019 International Aegean Conference on Electrical Machines and Power Electronics (ACEMP) & 2019 International Conference on Optimization of Electrical and Electronic Equipment (OPTIM), Istanbul, Turkey, 27–29 August 2019; pp. 563–569. [Google Scholar]
- Jannif, N.I.; Cirrincione, G.; Cirrincione, M.; Mohammadi, A.; Vitale, G. Experimental Application of Least-Squares Technique for Estimation of Double Layer Super Capacitor Parameters. In Proceedings of the 2017 20th International Conference on Electrical Machines and Systems (ICEMS), Sydney, Australia, 11–14 August 2017; pp. 1–5. [Google Scholar]
- Samwha Datasheet & Applicatoin Notes—Datasheet Archive. Available online: https://www.datasheetarchive.com/Samwha-datasheet.html (accessed on 24 January 2023).
- Alexander, C.K.; Sadiku, M.N.O. Fundamentals of Electric Circuits; McGraw-hill Education: New York, NY, USA, 2017; ISBN 978-1-259-25132-0. [Google Scholar]
- Max275 Datasheet—4th- and 8th-Order, Continuous-Time Active Filters. Available online: https://www.digchip.com/datasheets/parts/datasheet/280/MAX275.php (accessed on 24 January 2023).
- Solano, J.; Hissel, D.; Pera, M.-C. Modeling and Parameter Identification of Ultracapacitors for Hybrid Electrical Vehicles. In Proceedings of the 2013 IEEE Vehicle Power and Propulsion Conference (VPPC), Beijing, China, 15–18 October 2013; IEEE: Piscataway, MJ, USA, 2013; pp. 1–4. [Google Scholar]

**Figure 4.**The magnitude and phase frequency response of the Digital Derivative Filter $D\left(z\right)$.

Circuit Parameters | Unit | Faranda Parameters of the DLC (Used for Simulation) |
---|---|---|

${C}_{0}$ | $F$ | 43.95 |

${k}_{v}$ | $F/V$ | 1.69 |

${R}_{2}$ | $\mathsf{\Omega}$ | 40.9 |

${C}_{2}$ | $F$ | 6.51 |

${\tau}_{2}={R}_{2}{C}_{2}$ | $s$ | 299.72 |

Alpha Parameters | Estimated (OLS) | True | Error (%) |
---|---|---|---|

${\alpha}_{1}$ | 40.78 | 50.46 | 19.18 |

${\alpha}_{2}$ | 728.30 | 506.52 | 43.79 |

${\alpha}_{3}$ | 1.65 | 1.69 | 2.37 |

${\alpha}_{4}$ | 18,566 | 13,172 | 40.95 |

${\alpha}_{5}$ | 417.50 | 299.72 | 39.30 |

Circuit Parameters | Unit | Estimated (OLS) | True | Error (%) |
---|---|---|---|---|

${C}_{O}$ | $F$ | 44.46 | 43.95 | 1.16 |

${k}_{v}$ | $F/V$ | 1.65 | 1.69 | 2.37 |

${\tau}_{2}$ | $s$ | 417.56 | 299.72 | 39.32 |

Alpha Parameters | Estimated (CMM) | True | Error (%) |
---|---|---|---|

α1 | 48.49 | 50.46 | 3.90 |

α2 | 453.69 | 506.52 | 10.43 |

α3 | 1.66 | 1.69 | 1.78 |

α4 | 12,000 | 13,172 | 8.90 |

α5 | 272.9 | 299.72 | 8.95 |

Circuit Parameters | Unit | Estimated (CMM) | True | Error (%) |
---|---|---|---|---|

${C}_{O}$ | $F$ | 43.95 | 43.95 | 0 |

${k}_{v}$ | $F/V$ | 1.66 | 1.69 | 1.78 |

${\tau}_{2}$ | $s$ | 272.94 | 299.72 | 8.94 |

**Table 6.**Alpha Parameter Estimation Results by OLS with Experimental Data using an Input Ramp Current and compared against Faranda.

Alpha Parameters | Estimated (OLS) | Estimated (Faranda) | Relative Error with Respect to Faranda (%) |
---|---|---|---|

${\alpha}_{1}$ | 58.28 | 50.46 | 15.50 |

${\alpha}_{2}$ | 18 | 506.52 | 96.45 |

${\alpha}_{3}$ | 1.97 | 1.69 | 16.57 |

${\alpha}_{4}$ | 938.87 | 13,172 | 92.87 |

${\alpha}_{5}$ | 21 | 299.72 | 92.99 |

**Table 7.**DLC Parameter Estimation Results by OLS with Real Experimental Data and a Ramp Current as Input together with comparison to Faranda.

Circuit Parameters | Unit | Estimated (OLS) | Estimated (Faranda) | Relative Error with Respect to Faranda (%) |
---|---|---|---|---|

${C}_{O}$ | $F$ | 44.55 | 43.95 | 1.37 |

${k}_{v}$ | $F/V$ | 1.97 | 1.69 | 16.57 |

${\tau}_{2}$ | $s$ | 21.07 | 299.72 | 92.97 |

**Table 8.**Alpha Parameter Estimation Results by CMM with Real Experimental Data compared against Faranda.

Alpha Parameters | Estimated (CMM) | Lower Bound | Upper Bound | Estimated (Faranda) | Relative Error with Respect to Faranda (%) |
---|---|---|---|---|---|

α_{1} | 40 | 10 | 60 | 50.46 | 20.73 |

α_{2} | 400 | 300 | 700 | 506.52 | 21.03 |

α_{3} | 1.78 | 0.5 | 2 | 1.69 | 5.33 |

α_{4} | 10,000 | 1000 | 20,000 | 13,172 | 24.08 |

α_{5} | 224 | 100 | 400 | 299.72 | 25.26 |

**Table 9.**DLC Parameter Estimation Results by CMM with Real Experimental Data and a Ramp Current as Input together with comparison to Faranda.

Circuit Parameters | Unit | Estimated (CMM) | Estimated (Faranda) | Relative Error with Respect to Faranda (%) |
---|---|---|---|---|

${C}_{O}$ | $F$ | 44.64 | 43.95 | 1.57 |

${k}_{v}$ | $F/V$ | 1.78 | 1.69 | 5.33 |

${\tau}_{2}$ | $s$ | 224 | 299.72 | 25.26 |

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## Share and Cite

**MDPI and ACS Style**

Jannif, N.I.; Kumar, R.R.; Mohammadi, A.; Cirrincione, G.; Cirrincione, M.
Constrained Least-Squares Parameter Estimation for a Double Layer Capacitor. *Energies* **2023**, *16*, 4160.
https://doi.org/10.3390/en16104160

**AMA Style**

Jannif NI, Kumar RR, Mohammadi A, Cirrincione G, Cirrincione M.
Constrained Least-Squares Parameter Estimation for a Double Layer Capacitor. *Energies*. 2023; 16(10):4160.
https://doi.org/10.3390/en16104160

**Chicago/Turabian Style**

Jannif, Nayzel I., Rahul R. Kumar, Ali Mohammadi, Giansalvo Cirrincione, and Maurizio Cirrincione.
2023. "Constrained Least-Squares Parameter Estimation for a Double Layer Capacitor" *Energies* 16, no. 10: 4160.
https://doi.org/10.3390/en16104160