Accurate estimation of state-of-charge of supercapacitor under uncertain leakage and open circuit voltage map
Graphical abstract
Introduction
In Wireless Sensor Network (WSN) applications, deployment of Supercapacitors (SCs), which are also known as electric double layer capacitors, as powering devices has a great benefit in terms of life-span improvement of the network. The sensor nodes of the WSNs are installed to acquire data for monitoring the physical and/or environmental conditions in many applications [[1], [2], [3]]. Essentially, high power density and long life-span of SC (typically, 50,000–1,00,000 cycles) make it an attractive alternative over battery as an energy buffer for the sensor nodes [4]. The life-span of a SC equipped sensor node can be extended up to 20 years [5]. While powering the sensor nodes, SCs are affected by the physical factors and environmental conditions of the location where the nodes are installed. Leakage is one of such factors and need to be tracked accurately for effective control and monitoring of SCs. The leakage effect in SCs originates from: (i) Ohmic leakage between the electrodes, and (ii) redox reaction with the electrolyte impurities. In recent years, several experimental studies have explored the effect of SC voltage and temperature on the leakage dynamics of the device [[6], [7], [8], [9], [10]]. On a Maxwell BCAP350 SC, it has been observed that an increase in terminal voltage by 0.2 V and temperature by 20 °C amplify the leakage current flow up to 2.5 times and 7.5 times, respectively [9,10]. Furthermore, the leakage current changes as the SC ages [11,12]. Therefore, accurate tracking of leakage current is critical for effective management of SC energy, particularly for low duty-cycle applications; and health monitoring [13]. Motivated by this issue, we focus on estimating supercapacitor SOC while tracking the leakage effect in real-time.
Real-time monitoring of SC requires an accurate and computationally efficient model of the device. In this study, we adopt an Equivalent Circuit Model (ECM) of SC due to its computational simplicity over electrochemical models [14]. A number of research works have been performed on modeling and parameter identification of SCs [[14], [15], [16], [17], [18]]. These ECMs are mostly suitable for fast charge-discharge operation as the leakage effect has not been considered. However, it is essential to consider leakage for low duty-cycle applications [13]. A number of ECMs have been proposed in the literature that consider the leakage effect of SC [7,[19], [20], [21], [22], [23], [24], [25]]. However, none of the aforementioned models is completely observable when SOC is considered as a state of the SC. Hence, these models are not suitable for explicit SOC estimation as observability is a key requirement for designing estimation schemes [26]. In our previous work [27], we proposed an Open Circuit Voltage (OCV) based ECM with the following features: (i) the model offers complete observability, and (ii) explicitly captures the leakage effect. In this paper, we will adopt that ECM to design our estimation scheme.
Regarding SOC estimation, several algorithms such as linear Kalman filter [28], extended Kalman filter (EKF) [16,29,30], Unscented Kalman filter (UKF) [31], H observer [18], sliding mode observer [[32], [33], [34]], Luenberger observer [15], artificial neural network (ANN) [35] and practical model based approach [36] have been proposed in the past few years. However, none of these aforementioned approaches considers leakage while estimating the SOC. As shown in our previous work [27], such ignorance of leakage can lead to large errors in SOC estimation. In Ref. [27], the proposed estimation algorithm considers the effect of the leakage current for SOC estimation, where leakage current is identified and characterized offline at the beginning of SC life. In summary, none of the aforementioned approaches explicitly co-estimates SOC and leakage effect which may significantly vary in real-time. Therefore, the offline identified leakage may not be sufficient and it is important to estimate the leakage current in real-time for accurate energy management of SCs.
In case of the OCV based ECM of SCs, the OCV-SOC map of the device plays an important role in predicting the output voltage of the device. The OCV-SOC map is typically derived based on a set of OCV-SOC data, obtained by performing charge-discharge operation of the device. The OCV is commonly computed from measured terminal voltage of the SC under no-load condition and the SOC is calculated from the integration of the terminal current minus leakage current of the device [27]. Consequently, incorrect leakage current information leads to incorrect computation of the SOC vector which in turn leads to incorrect OCV-SOC function computation. In addition, the offline identified OCV-SOC map might get affected by one or more of the following factors: i) The assumption on the time required to complete SC charge redistribution may not be accurate. ii) Magnitude of the terminal current used for OCV-SOC mapping [14]. iii) The average OCV-SOC profile taken in order to neglect the hysteresis between the charge and discharge OCV profile. iv) The current and voltage sensor noises. Although the above mentioned reasons are not very significant individually, but the combined effect may lead to significant inaccuracy in the SOC estimation which is highly sensitive to change in OCV-SOC map [27]. In summary, inaccurate leakage and OCV-SOC map information may affect the accuracy of SOC estimation significantly. Furthermore, accurate information about leakage and OCV-SOC map are useful for health monitoring of SC. Hence, real-time estimation of leakage and OCV-SOC map can be beneficial for real-time energy management of SC.
Based on the literature review, it is found that none of the existing estimation framework has considered simultaneous explicit estimation of SOC, leakage and OCV-SOC map. In this paper, we address this gap by proposing an online algorithm for co-estimation of SOC, leakage current and OCV-SOC map. This work is an extension of our previous work [27] where we proposed an observable ECM with leakage effect and corresponding SOC estimation algorithm. The main difference between this work and [27] is the following: the algorithm in Ref. [27] estimates the SOC in real-time based on offline identified leakage current whereas the current work co-estimates SOC, leakage effect, and OCV-SOC map simultaneously in real-time. Specifically, we explore the following in this work: (i) We experimentally identify the adopted ECM parameters based on a Maxwell 25 F commercial supercapacitor (BCAP0025 T01). (ii) We design an Unscented Kalman Filter (UKF) based co-estimation algorithm which takes system nonlinearities accurately into account. (iii) We test and validate the estimation algorithm via simulation and experimental studies. (iv) We study the robustness of the proposed algorithm in the presence of parametric and measurement uncertainties.
This paper is organized as follows. Section 2 describes the supercapacitor model. Section 3 details the UKF-based co-estimation scheme. In section 4, the experimental identification of the ECM parameters is presented. Section 5 discusses the simulation and experimental results. Section 6 concludes the work.
In this study, we adopt the ECM from our previous work [27], shown in Fig. 1. The series resistance RE captures the equivalent series resistance of the device, the RD−CD branch captures the self-discharge dynamics of the SC due to charge redistribution, vOCV represents the Open Circuit Voltage (OCV) as a function of SOC, the variable resistance RL captures leakage effect and characterizes the charge loss in the SCs.
Based on Kirchoff's laws, the mathematical model corresponding to the ECM in Fig. 1 is presented as:where i(t) = iD(t) + iL(t) is the input current, iD(t) is the current flow due to charge redistribution, iL(t) is the leakage current, vD(t) is the voltage drop across the capacitor CD and v0(t) is the terminal voltage. The OCV (vOCV) and the variable resistors (RD, RL) are given bywhere m, n and p are scalar parameters; q(.) is the nonlinear OCV-SOC function which is chosen to be a N-th order polynominal. Now, according to the Coulomb-counting method, the SOC can be modelled aswhere CAh is the capacity of the supercapacitor in Ampere-hour and is defined as: , where Cmax is the maximum capacitance and Vmax is the maximum voltage of the SC. Note that i(t) < 0 denotes the charging and i(t) > 0 denotes the discharging in our formulation. The identification approach of the aforementioned ECM parameters through experiments is discussed in section 4.
This section discusses the design of unscented Kalman filter based online estimation scheme. Our main objective is to update both the parameters, leakage and OCV-SOC function in real-time to compensate for the error in SOC caused due to incorrect information about them. Specifically, we update the parameter q1 in (6), a coefficient of the OCV-SOC function q(.). Note that we have chosen the coefficient associated with the linear term in (6) instead of all the coefficients. This is due to the following reasons: (i) the linear term is the dominating term on the OCV-SOC function, and (ii) choosing to update all the coefficients will lead to the observability problem in the estimation. Finally, we assume that the leakage current iL and the OCV-SOC model coefficient q1 are slowly varying, i.e.
First, we formulate the following state-space model based on (1)–(9).where is the state vector, y = v0 is the terminal voltage, and u = i is the terminal current. The functions f(.) and g(.) are given by:
Before elaborating the estimator design, we must verify the observability of the state-space model (10). The observability of a nonlinear system can be verified by the following local observability test matrix [37]:where n denotes the order of the nonlinear system. The observability of the nonlinear state-space model (10) is verified for the full operating region and . At all the aforementioned operating points, , i.e., the system is locally observable.
Next, we modify the state-space model (10) in order to introduce the uncertainties in the state and output equations:where γ and υ are the uncertain terms arising from the modeling and parametric uncertainties.
Now, by discretizing the model (12) using Euler's method, we can getwhere F(.), G(.), Γn and are the functions of the discretization time-step and f(.), g(.), γ and υ, respectively. We treat the variables Γn and as process and measurement noises respectively. Furthermore, we assume that the Γn and are zero mean white Gaussian noises with covariances Q and R, respectively.
In this subsection, we discuss the generalized steps involved in the UKF estimator based on the model (13) [38]:
- 1.
Initialization: To initialize the estimated state vector , process covariance matrix Φ+(k), process noise covariance matrix Q(k) and the measurement noise covariance R(k), for the time instant k = 0.
- 2.
Prediction or Time update:
- (a)
To generate a set of sigma points , with each points being associated with a weight wj, for j = 1, 2, …, 2n where n is the number of states and .
- (b)
To propagate the sigma points, through the nonlinear state equation.
- (c)
To calculate the mean of the predicted state, :
- (d)
To predict the process covariance matrix:
- 3.
Correction or Measurement update:
- (a)
To propagate the measurement sigma points:
- (b)
To calculate the mean of the measurements:
- (c)
To calculate the measurement covariance matrix:
- (d)
To calculate the cross covariance of the state and measurement:
- (e)
To update the Kalman gain:
- (f)
To correct the prior mean of the predicted state, :
- (g)
To correct the predicted process covariance matrix, Φ−(k):
Note that, we choose the UKF as the integrated state and parameter estimator because it captures the system nonlinearities more accurately over the linear and extended variants of the Kalman filter [27]. We will demonstrate the performance of the proposed co-estimation scheme in section 5.
Section snippets
Experimental identification of ECM
In this section we discuss the experimental identification of the ECM parameters. We have conducted experiments on a Maxwell BCAP0025 T01 25 F supercapacitor cell. The specifications of the SC cell are listed in Table 1. All the charge-discharge experiments are conducted using Bitrode FTV supercapacitor testing module with current and voltage measurement accuracy of 1 mA and 1 mV, respectively. The experimental data acquisition is performed at ∼25oC with a sampling frequency of 1 Hz.
Following
Results: simulation and experimental studies
In this section, we evaluate the proposed estimator by performing simulation and experimental studies. We choose the following parameters for UKF: Φ(0) = diag(10−07, 10−07, 10−10, 10−03); Q = diag(4.1 × 10−03, 5 × 10−03, 3.29 × 10−04, 1.38 × 10−02) and R = 10−02. These parameters are chosen heuristically to achieve better estimation accuracy and least convergence time. We analyze the performance of the proposed estimator via a set of case studies. The current profile used for these studies is
Conclusion
In this paper, a co-estimation scheme is presented for accurate SOC estimation of SC by tracking the leakage current and OCV-SOC map in real-time. The adopted OCV based ECM captures the leakage effect of supercapacitor and is suitable for explicit SOC estimation of the device. We identify the ECM parameters using the experimental data obtained from a charge-discharge experiments on a Maxwell 25 F supercapacitor. An UKF based scheme is designed under simultaneous state-parameter estimation
Acknowledgements
This work is supported in part by Science for Equity, Empowerment & Development (SEED) Division, Department of Science & Technology, Ministry of Science and Technology, India under SYST; Ref. SP/YO/054/2016.
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2022, Renewable and Sustainable Energy ReviewsCitation Excerpt :The advantage of this method is its high accuracy, which is achieved by incorporating the correlation between the supercapacitor states and parameters in the estimation process. The UKF has also been used in Refs. [89,149], in which more complicated battery models are used to further increase the estimation accuracy. Also, in Ref. [150], a comparison between the EKF-based and UKF-based SoC estimation mechanisms is fulfilled, where it is concluded that the UKF excels the EKF in terms of accuracy and convergence time.
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