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Ultracapacitors (UCs) are the focus of increasing attention in electric vehicle and renewable energy system applications due to their excellent performance in terms of power density, efficiency, and lifespan. Modeling and parameterization of UCs play an important role in model-based regulation and management for a reliable and safe operation. In this paper, an equivalent circuit model template composed of a bulk capacitor, a second-order capacitance-resistance network, and a series resistance, is employed to represent the dynamics of UCs. The extended Kalman Filter is then used to recursively estimate the model parameters in the Dynamic Stress Test (DST) on a specially established test rig. The DST loading profile is able to emulate the practical power sinking and sourcing of UCs in electric vehicles. In order to examine the accuracy of the identified model, a Hybrid Pulse Power Characterization test is carried out. The validation result demonstrates that the recursively calibrated model can precisely delineate the dynamic voltage behavior of UCs under the discrepant loading condition, and the online identification approach is thus capable of extracting the model parameters in a credible and robust manner.

In order to address serious concerns over energy sustainability and environmental impact, governments, the automotive industry, and academia are endeavoring to expedite a paradigm shift to a green transportation system [

Ultracapacitors, also known as supercapacitors or double-layer capacitors, have high power density, low internal resistance, high efficiency, and exceedingly long cycle life, while possessing the merits of wide operating temperature range and fast charging [

A model that can simulate the dynamics of an UC with high precision is vital for energy management design in electric vehicles equipped with UCs or HESSs. The modeling of UCs has a rich history. This research can be generally grouped into three categories: electrochemical models, artificial neural networks (ANNs), and equivalent circuit models. Electrochemical models are developed from first principles and depict the real physical-chemical reactions within an ultracapacitor utilizing partial differential equations [

Equivalent circuit models have been carefully developed, especially for the energy management design and power control; these have been extensively reported in the literature [

This paper is arranged as follows: Section 2 reviews the equivalent circuit model used to describe the voltage response of UCs. Then, the formulation for the extended Kalman filter for model parameter estimation is detailed. Section 3 describes a test rig that was specially developed in order to carry out the experimental UC tests. Section 4 discusses the modeling results, followed by key conclusions in Section 5.

There are a variety of equivalent circuit model structures for UCs as reported in the literature. This paper selects the one that consists of a bulk capacitor, a second-order capacitance-resistance network and a series resistance. The second-order capacitance-resistance network is composed of two parallel _{0} denotes the voltage across the bulk capacitor _{1} and _{2} denote the voltages of the two _{s} denotes the series resistance.

According to basic electrical circuit principles, the continuous state equation can be derived as:

The state equation can be further transformed into the discrete state equation so that:

Where Δ

Recently, the Kalman filtering has gained more popularity in the field of state estimation, parameter estimation and dual estimation due to its inherent merits [_{k}_{k}_{k}_{k}_{k}

Time update
_{kk}_{−1} is the priori estimate of the parameter vector _{k}_{k}_{k}_{|}_{k}_{−1} represents the _{k}_{−1} represents the posteriori estimation error of parameter vector

Measurement update:
_{k}_{k}_{−1} and _{k}_{−1} according to _{k}_{k}_{k}_{k}

It is obvious that the derivative calculation is recursive, and can be initiated by:

In order to collect experimental data for the parameter estimation, a test rig was developed. A block diagram is shown in

In order to validate the proposed estimation algorithm, a transient power test based on the standard Dynamic Stress Test (DST) was conducted on the established test rig. The DST-based test can represent the dynamic load conditions of a UC during daily driving of an electric vehicle with UCs as the single or complementary energy storage. The voltage and current of the UC in the DST test are shown in the

It is well-known that Kalman filters require

Given the specified parameters, the proposed extended Kalman filter was implemented in order to estimate online the model parameters of the UC. The evolution of the estimated model parameters is shown in _{1} is bigger than that of _{2} while _{1} is just slightly smaller than _{1}. It means that the first _{s}_{s}

The evolution of the measured and estimated voltages is shown in

In order to validate the derived model, a Hybrid Pulse Power Characterization (HPPC) was conducted. The voltage and current profiles in the HPPC test are shown in

This paper presents an online model identification method for a UC model based on the well-known Kalman filter. An equivalent circuit model was used to represent the dynamics of a UC. It was composed of a bulk capacitor, a second-order

Lei Zhang is grateful to the funding from the China Scholarship Council as well as the University of Technology Sydney and Beijing Institute of Technology for his studies.

The authors declare no conflict of interest.

The ultracapacitor model structure.

The block diagram of the test rig.

Pictures of the test rig. (

The measured voltage (

The evolution of the estimated model parameters in the DST test: _{1}; (_{1}; (_{2}; (_{2}; (_{s}

The measured and estimated voltages in the DST test.

The relative voltage error in the DST test.

The measured voltage (

The simulated and measured voltages in the HPPC test.

The error between the simulated and measured voltages in the HPPC test.

The estimation results of the extended Kalman filter.

_{1} |
_{1} |
_{2} |
_{2} |
_{s} | |
---|---|---|---|---|---|

0.9677 | 1.55 × 10^{−5} |
0.8767 | 8.77 × 10^{−6} |
3.85 × 10^{−4} |
6.93 × 10^{−4} |

The estimated model parameters.

_{1}(F) |
_{2}(F) |
_{1}(Ω) |
_{2}(Ω) |
_{s} | |
---|---|---|---|---|---|

2601 | 628 | 1065 | 4.85 × 10^{−4} |
7.14 × 10^{−4} |
6.93 × 10^{−4} |