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Modeling and Experiment Validation of the DC/DC Converter for Online AC Impedance Identification of the Lithium-Ion Battery.

Introduction

Nowadays the energy crisis and environment pollution problem have become more and more serious. Attention from all over the world has been paid to the lithium ion batteries due to properties like high energy density [1][2] and low emissions and the lithium ion battery are used in various applications such as battery packs in electric vehicles and hybrid electric vehicles [3][4]. The electrochemical impedance spectroscopy, abbreviated as EIS, has been widely used in the lithium ion battery research related to aging states and its influence on the application as well as the temperature estimation and expansions [5][6][7].

The principles of the EIS method are that a small voltage or current signal at some frequencies is superimposed on the battery and then after acquiring the voltage and current signal simultaneously the resistance can be obtained by performing FFT processing of the two signals. The EIS method has three basic assumptions, namely linearity, casualty and stability [8]. The linearity of the battery is ensured by controlling the amplitude of the added current or voltage fluctuation signal. The casualty is obvious because if no excitation signal is added, no corresponding signal can be measured. The stability means the system can return to the normal state when the added signal is removed and this is satisfied.

In general, the impedance acquisition of the battery can be classified into two groups, namely the impedance measurement approach and the online estimation approach. The impedance measurement approach relies on the experiment equipment and after being measured, the important work is to estimate physical parameters of the battery. The impedance characteristic of the battery depends on the battery conditions such as the state of charge, the temperature, the current rate and eventually the previous history [5][9][10]. The state of charge of the battery has a high variation of the impedance in this frequency range, so the comparison and accuracy analysis of the impedance-based method is proposed in [11]. The relation between physical parameters and the impedance spectroscopy can be established by the equivalent circuit model such as the Randles' model and the transmission line model [12][13]. In [14], it is stated that the real part of the battery impedance is related to the battery temperature and the sensitivity analysis reveals that a high variation of impedance with temperature can be found for low frequencies. The impact of the charging methodology on the battery lifetime is investigated and the comparative study between charging techniques is presented through characterizing the batteries using the EIS method [15].

The online estimation approach of the impedance parameters considers the significant change of the battery characteristics during its lifetime due to aging [16]. In this group, the Kalman filter technique is a common used method and it is a recursive estimator [15]. The single Kalman filter can be used to estimate the battery state of charge based on the equivalent circuit model [17]. For consideration of nonlinearities in the battery model, the advanced versions of Kalman filter are employed, such as the extended Kalman filter or the sigma point Kalman filter. Some recent publications have considered the nonlinearity between the state of charge and the open circuit voltage [18][19][20].

Though much progress has been made in the impedance acquisition, the application of these methods to the actual battery packs is limited in many aspects. The impedance measurement needs the laboratory equipment which is usually of high price and its cost cannot be tolerated in practical systems. The online estimation requires a lot of calculations involving matrix operations and the implementation on an ordinary low-cost microcontroller is in fact difficult to achieve. Furthermore, the complex matrix operations can lead to numeric instabilities [16]. Hence, the combination of online measurement of the impedance and the online estimation method would be very meaningful. The first thing that needs to be done towards this target is to design the device for online measurement of the impedance. However, there is not much known literature in this aspect.

The research in [21] puts forward a solution which uses the power converter to excite the battery for impedance spectroscopy at any working point and the impedance spectroscopy is performed by the control system without additional power hardware. The limitation of this method is that the output current of the battery is much smaller than the practical current of the battery packs in vehicles. The similar work of the online impedance measurement can also be found in the area of the hydrogen polymer electrolyte membrane fuel cell system. The system for AC impedance measurement of fuel cell in vehicles is proposed in [22] and via the on-board system the function of this system has been validated. However, no more information is provided about the controllable high voltage DC/DC converter except the key frequency point of the excitation frequency of 300Hz.

The power converter does not exist in all the applications of the lithium-ion battery so in this paper, a new impedance measurement method is proposed that yields a wide frequency range and this method is applicable to all kinds of batteries system without much increase in the system cost. To do so, the topology of the impedance measurement system is given consisting of a power converter for excitation signal generation and a cell voltage monitor device for acquiring signals and calculating impedance. The average state space equations of the power converter are deduced to check the controllability and observability of the target input current of the converter. Based on these state space equations, the small signal model is derived to calculate the gain amplitude of the fluctuated current signal against the only control variable namely the duty cycle of the PWM of the power semiconductor. This helps to design the control algorithm and will allow for the verification of the proposed system topology. By setting up the model of power systems in Matlab/Simulink, the system function is verified. After design this power converter, the experiment is conducted on a high power LiMnO2 battery and the feasibility of the converter in online impedance measurement is validated in a wide frequency range.

The organization of the paper is as follows. The system configuration and the mathematical model of the power converter are presented in detail in Section 2. Then the function of the system is verified with respect to the gain amplitude of the excitation current signal and the comparison of the calculated impedance is made with the theoretical equivalent circuit model in Section 3. Subsequently, the power converter and the cell voltage monitor are designed and implemented to a high power battery pack and the validation of the converter model and the online impedance measurement is obtained in Section 4. Conclusions are drawn in Section 5.

System Modeling

To start with, the system configuration is described, including the DC/DC converter, the lithium-ion battery pack and the load. Then the modeling of the lithium-ion battery and the converter is presented successively.

System Configuration

The system for online AC impedance identification of lithium-ion battery, as shown in Figure 1, is made up of a Lithium-ion battery pack with BMS (Battery Management System), a cell voltage monitor, a shunt resistor, an electronic load and a DC/DC converter as well as a resistor for its power consumption. The lithium-ion battery pack with BMS is regarded as the voltage source of the DC/DC converter and the electronic load. The BMS helps to monitor the normal operation state of the battery pack, which provides information of temperature, state of charge and state of health and so on. The electronic load is controlled to draw constant current from the battery pack and is like a kind of constant current sink.

Usually, the DC/DC converter is a power conversion device but in this system it is implemented to generate current excitation signal which will be superimposed on the battery pack. This current signal is the summation of a sinusoidal current part and a direct current part because the input current of the converter cannot be below zero. The controller of the converter is responsible for the function realization. For the power conversion of the DC/DC converter is not effectively utilized but just to produce heat, the operation time should be as shorter as possible and the direct current part should be as lower as possible if to be used in a practical system.

The cell voltage monitor is developed for measuring the voltage of each cell. The shunt resistor measures the output current of the battery pack and transforms the current signal to a voltage signal which can be sampled by the cell voltage monitor. Combining the cell voltage monitor and the shunt resistor, the output voltage and current of each cell can be acquired simultaneously and after recording these signals, the impedance spectroscopy can be calculated using FFT technique. Different from the conventional analog signal processing circuit, the cell voltage monitor must be capable of extracting weak fluctuated signals from the large direct current and voltage at the stable operating point. On basis of the cell voltage monitor and the DC/DC converter, the work in this paper can go on smoothly.

Lithium-ion Battery Model

When studying the electrochemical impedance spectroscopy, the lithium-ion battery is thought to be a linear system around its operating point. To fit the impedance spectroscopy, many equivalent circuits have been tried and there are two commonly used models, namely the Randles' model and the model with a Warburg element as shown in Figure 2(a) and Figure 2(b) respectively. The term [E.sub.n] is the Nernst voltage based on thermodynamics and the term [R.sub.m] is sum of the contact resistance, the internal ion transfer resistance and the wire resistance. The term [R.sub.C] means faradic resistance at the electrochemical interface and the term [C.sub.d1] means the double layer capacitor. When considering the finite diffusion process of the lithium-ion inside the battery, the double layer capacitor is replaced with a Warburg element [Z.sub.w].

DC/DC Converter Model

The DC/DC converter, as shown in Figure 3(a), is a common boost converter comprised of an inductor [L.sub.1], a switching power-semiconductor module [G.sub.1], a diode [D.sub.1] and a capacitor [C.sub.1]. The resistor [R.sub.L1] and [R.sub.1C] are considered to approximate the parasitic power consumption of the inductor [L.sub.1] and capacitor [C.sub.1] respectively. The turn-on and turn-off power losses of the module [G.sub.1] and the diode [D.sub.1] are both included in the resistor [R.sub.L1] because these processes are fairly complicated and in fact, no exact equation can be used to describe them quantitatively. A resistor R represents the load demand of the converter.

The converter is controlled to inject a fluctuated current signal to the battery pack. The fluctuated current signal may be of any wave form but to ensure high signal noise ratio (SNR), the sinusoidal current disturbance signal is chosen to be the target wave form. The frequency of the disturbance signal is extended to be as widely as possible while

there is inherent limitation of DC/DC converter, one of which is the switching frequency of module [G.sub.1], to realize the full range of frequency spectroscopy.

The principle of the DC/DC converter is founded on the pulse width modulation (PWM) and the duty cycle D and the switching frequency are two key parameters of PWM. When the module [G.sub.1] is on as shown in Figure 3(b), the converter operates in one way and the state equations are described by Equation 1. When the module [G.sub.1] is off as shown in Figure 3(c), the converter operates in the other way and the state equations are described by Equation 2.

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

Where,

[u.sub.in] = input voltage of the converter

[i.sub.in] = input current of the converter

[u.sub.o] = output voltage of the converter

[u.sub.c1] = voltage over the capacitor [C.sub.1]

t = time

After taking the duty cycle D into consideration, the average state equations are deduced as Equation 3. The re-arranged equations are described by Equation 4 in order to acquire input variables X, state variables U and the output variable iin as well as corresponding coefficient matrix A, B, C and T.

[mathematical expression not reproducible] (4)

According to Equation 4, the state space equations can be described by Equation 5 and all variables and coefficient matrices can be described by Equation 6.

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

The controllability matrix Q is described by Equation 7 and the observability matrix R by Equation 8. It is easy to find that this system is completely controllable but not observable because the theoretical voltage [u.sub.C1] of the capacitor [C.sub.1] is not measurable.

For the DC/DC converter, the directly adjustable parameter is the duty cycle and to realize the generation of the periodic current wave, the small signal model of the DC/DC converter needs to be setup at first. Then the amplitude gain of periodic current signal against the periodically changed duty cycle around their stable values is to be analyzed, which contributes to having a better knowledge of system properties in this special application.

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

To derive the small signal model, small dynamic signals are added to all the related parameters and variables as Equation 9. The term [l.sub.in] stands for the dynamic signal of small amplitude compared to the stable value [L.sub.in]. The term [u.sub.0], [u.sub.C1], d and [u.sub.in] are dynamic small signals of the output voltage, the capacitor voltage, the duty cycle and the input voltage respectively. Meanwhile, the term [U.sub.o], [U.sub.C1], [D.sub.0] and [U.sub.in] represent the stable value of the output voltage, the capacitor voltage, the duty cycle and the input voltage accordingly.

When the Equation 9 is substituted into Equation 4, the summation of stable variables in every equation is zero and the small signal model can be proposed by neglecting the twice order infinitely small items. However, through analyzing Equation 4 simply, the duty cycle D is part of dominators of all coefficient matrices and to do the Taylor expansion of these matrices needs a lot of efforts but it has no much influence on what to be emphasized, which will be analyzed later. In consequence, to simplify the state space equations, an assumption is made that the equivalent internal resistor of capacitor [C.sub.1] can be neglected, which means the term [R.sub.C1] is equal to zero. Then the term [u.sub.C1] is equal to the term [u.sub.o] and the state space is re-stated as Equation 10. After simplification, the small signal model is obtained as Equation 11.

By doing Laplace transform of Equation 11, the relation between the small input current [l.sub.in] and the small input voltage [u.sub.in] as well as the small output voltage [u.sub.0] is presented as Equation 12.

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

If the term [u.sub.in](s) is zero, the Equation 12 becomes Equation 13. This case is reasonable when the input voltage source of the DC/DC converter is an ideal device. The gain [G.sub.id] (s) is defined as the ratio of [l.sub.in] (s) to d(s).

If the term [u.sub.in] (s) is equal to--[R.sub.in][l.sub.in](s), the Equation 12 becomes Equation 14. This case is reasonable when the input voltage source is not an ideal device and the internal resistor exists. In fact, if the input source is a lithium-ion battery, the term [R.sub.in] is a function of angular frequency which results in the impedance spectroscopy of the battery and this is just the research object of our work. Even though the term [R.sub.in] represents the characteristic of the voltage source, the only contribution of it can be thought mathematically to be part of the internal resistor of the inductance.

In the actual application, the value of load resistor R is chosen to be much larger than the internal resistance [R.sub.in] and the resistor [R.sub.L1], that is to say, R[(1-D).sup.2] >>[R.sub.L1]+[R.sub.in] As a result, even though the term [R.sub.in] changes with angular frequency, it has little effect on amplitude and phase of the gain [G.sub.id](s). When the excitation frequency of the small signal duty cycle approximates to zero or is infinitely large, the gain [G.sub.id](s) is described by Equation 15. At other angular frequencies, substituting s = j[omega] = j2[pi]f into Equation 14 and another Equation 16 will be obtained.

To obtain the variation trend of the amplitude of the gain [G.sub.id](j2[pi]f), the first-order derivation as Equation 17 of it can be solved to figure out the maximum and the minimum.

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

It can be concluded that the solution to Equation 17 reaches the extreme value only when the frequency f is close to the critical value [f.sub.C] described by Equation 18 based on magnitudes of these parameters. Equation 18 shows that the DC/DC converter changes the resonant frequency of conventional circuit composed of an inductance and a capacitor in serial by being multiplied by the ineffective duty cycle.

At the critical frequency [f.sub.C], the amplitude of the gain [G.sub.id](j2[pi]f) is described by Equation 19. Comparing the Equation 19 with the Equation 15, the derivation result as Equation 20 is acquired.

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible] (25)

When designing the DC/DC converter, some basic principles described by Equation 21 are adhered to ensure that the DC/DC converter works in continuous current mode and the ripple current and voltage are relatively small. The term [f.sub.s] stands for the switching frequency. The function Ratio ([f.sub.C]) increases monotonously with the increase of the variable [f.sub.C].

If the function Ratio ([f.sub.C]) is greater than one, the following inequality as Equation 22 is established and the solution as Equation 23 to it can be achieved. After transforming Equation 21 to get close to the form of Equation 23, the Equation 24 is obtained. According to principles of the DC/DC converter, the frequency [f.sub.of] of fluctuation current signal must be lower than the switching frequency f of module [G.sub.1]. The disturbance current signal will have a good SNR(signal noise ratio) when the Equation 25 is satisfied. The other key point, on which the small signal model is based, is that in each period of the current excitation signal the more switching periods are included, the better the quality of the current signal is.

In this paper, it is assumed that the Equation 23 is satisfied. In consequence, when the frequency is away from the critical point, the amplitude of the gain [G.sub.id](j2[pi]f) decreases greatly, as shown in Figure 4.

The gain function of the periodic disturbance current signal against the periodic fluctuated duty cycle forms the basis of applying the DC/DC converter to generate the target current signal. As the critical frequency is lower than the switching frequency, it cannot be avoided that the critical frequency is within the concerned range of the impedance spectroscopy. Hence, the amplitude of the current disturbance signal will be very high around the critical frequency and this may influence the stability of the converter control, which needs to be paid attention to. However, measures can be made to change the critical frequency, such as adjusting the capacitor value online according to the target current fluctuation signal at the cost of additional hardware.

Verification of the DC/DC Converter Model

To verify the derived small signal model of the DC/DC converter, the open-loop control is utilized to adjust the duty cycle command. To demonstrate the expecting function of the converter, the closed-loop control is utilized to adjust the amplitude the current excitation signal within the achievable frequency range. The work is shown as follows.

Control Algorithm Design

The state space equations of the DC/DC converter are not very complicated and they are setup from the viewpoint of an average model which does not pay too much attention to the dynamic response within every switching period because the ripple current cannot be avoided. The proposed control algorithm is shown in Figure 5. The term [i.sub.in_ref], [D.sub.Model], [D.sub.PID], D and [i.sub.in] stand for the target input current of the DC/DC converter, the stable duty cycle calculated by feed-forward control, the closed-loop duty cycle calculated by PID controller, the actual duty cycle for module [G.sub.1] and the actual input current of the converter respectively.

To guarantee fast dynamics, the model-based nonlinear feed-forward control is used to calculate the stable target duty cycle. Usually, the feed-forward control method is not related to time and it relies on accuracy of the mathematical model by assuming the dynamic terms in the state space equations to be zero. When the accuracy is high enough, the error of the actual input current of the DC/DC converter away from the target will be very little.

However, limited by the imprecise physical model of module [G.sub.1] and diode [D.sub.1], the PID controller is applied to ensure the stable convergence to the target value. According to the control theory, the step response of the PID controller always has some overshoot. On the one hand, to reduce the overshoot it will be at the sacrifice of slow dynamic response. On the other hand, to achieve fast dynamic response it will be at the sacrifice of large overshoot. Therefore, the combination of the model-based nonlinear feed-forward and the PID controller is more acceptable taking the dynamic response and static error into consideration. In this way, the time constant of the PID controller can be a little longer.

The control system, like a micro controller unit, of the converter is usually discretized and the highest control frequency is the switching frequency of module [G.sub.1]. Naturally, the PID controller is in the discrete form.

System Simulation

To verify the function of the DC/DC converter, the system is setup in Matlab/Simulink and modeled as shown in Figure 6. For simplification, the Randles' model, which is a common circuit in electrochemical area, is used to emulate the electrical property of the Lithium-ion battery. The electronic load in the system is treated as a constant current source while this component needs to be controlled in the software, so a signal source is added as the control unit. Using power electronics and passive elements, the model of the converter is established. After taking the actual operation process into consideration, some other elements are placed such as a resistor [R.sub.1], the contactor [S.sub.1] and [S.sub.2].

Parameters of all components are listed as in Table 1. The internal resistors of capacitor C and L are usually at the magnitude of milliohm as described by manufacturers. The constant current drawn by the controllable current source is 100 A and the equivalent voltage of the battery is 100 Volts.

According to the mathematical model of the DC/DC converter and component parameters aforementioned, the variation of theoretical amplitude of the gain function [G.sub.id](2[pi]f) with frequency is calculated and plotted as in Figure 7. The critical frequency is 145.42Hz and the approximation value based on Equation 18 is 145.28Hz, so these two values are much closed to each other and the error is within 0.1%. In Equation 18, the term [R.sub.L1] + [R.sub.in] contributes only a small part to the summation R[(1-D).sup.2] + [R.sub.L1] + [R.sub.in] when considering the magnitude of the term [R.sub.L1] + [R.sub.in] and R[(1-D).sup.2]. As a result, this approximation is meaningful to quickly find the critical frequency with high precision.

In the small signal model derivation of the converter, to reduce computation efforts, an assumption is made that the internal resistor of the capacitor is zero. Now based on system parameters, this assumption is verified to be reasonable and the description is as follows. In the coefficient matrices, the term [R.sub.C1][C.sub.1]+[DRC.sub.1] exists in the dominator. The term DR is about 10[OMEGA] and the term [R.sup.C1] is just 0.001[OMEGA], so the term [R.sub.C1][C.sub.1] can be ignored with little influence.

The simulation procedure consists of five parts as shown in Figure 9 just for example. Firstly, the current source starts to draw a linearly-increasing current from battery and then the battery stabilizes at the target current for some time during which the contactor [S.sub.1] and [S.sub.2] are all switched off. No input current flows through the DC/DC converter. Secondly, the contactor [S.sub.1] is turned on and the battery begins to charge the inductor [L.sub.1] and the capacitor [C.sub.1] as well as the resistor R for some time. The resistor R1 functions as a damping device to weaken the serious oscillation resulting from the serial inductor [L.sub.1] and capacitor [C.sub.1] but there is no way to avoid this oscillation.

Thirdly, the contactor S is closed and the duty cycle D of module [G.sub.1] increases gradually until the input current of the converter reaches the target stable value upon which the fluctuated current signal is superimposed. Fourthly, after the third part operates for some time, the target of fluctuated current signal is added and the duty cycle D changes periodically. Fifthly, the input current returns to the stable value with the cancellation of the fluctuated current signal.

The stable value of the input current is 20 A and the amplitude of the disturbance current signal is 2.5 A with the frequency ranging from 0.1Hz to 1k Hz. In the following, the current fluctuation signal at 0.1Hz, 1Hz, 10Hz, 100Hz, 200Hz, 500Hz and 1k Hz is simulated and results are presented.

Another important thing to check is whether the ratio of the amplitude of the current fluctuation signal to the amplitude of the periodically-changed duty cycle is consistent with the theoretical value calculated using the mathematical model of the converter. Hence, in the model built in Matlab/Simulink, the closed-loop feedback controller is ineffective and only the model-based feed-forward control is implemented to see the detailed current response of the converter. In this way, the amplitude of the fluctuated duty cycle is almost the same for all the frequency range.

The simulation result at 100Hz is shown in Figure 10. By doing FFT of the current signal from 0.5s to 0.7s, the calculated amplitude of the disturbance current signal is 10.4 A. The amplitude of the duty cycle fluctuation is about 0.0315. Then the ratio between the upper two values is about 330.16 and the error between the simulation result and the theoretical one is about 7.27% against the theoretical value at 307.8.

The results at different frequencies are listed in Table 2. It is illustrated that in a wide range of frequency, the theoretical gain ratio is close to the simulated value with error less than 8%. The exception at 1k Hz is because this frequency is just 10% of the switching frequency of module [G.sub.1] and it uses only ten points to emulate a sinusoidal wave. In addition, the ripple current is in the same magnitude of the fluctuated input current when considering the theoretical gain ratio.

At the same time, when the frequency like 100Hz and 200Hz is around the critical point, the simulated amplitude of the input current is almost 50% of the stable value and this is away from the basis of the small signal model. However, the disturbance current signal can still be produced very well and it proves the feasibility of the proposed function of the converter.

Except for the three frequencies aforementioned, the simulated input current in most frequencies is around 2.5 A which is the control target of the converter. To reach the target current, the PI controller becomes effective again and the simulation result will be given as follows.

Based on these research results, the target current of the model-based feed-forward controller is adjusted with respect to the amplitude gain ratio at various frequencies. Things to be noticed in actual application are that the amplitude of the fluctuated duty cycle should not be too large to result in the failure of the DC/DC converter, especially when the change rate of current is very fierce.

Figure 11 shows the simulation result of the output current and voltage of the battery at the frequency of 100Hz. For simplicity, only the wave of part 4 as described in Figure 9 is illustrated and the filtered output current is also included. As can be seen, the stable output current of the battery is combined with a fluctuated current and there exists a corresponding voltage response in the stable output voltage. To calculate the impedance, FFT technique is applied and the result is listed in Table 3.

Figure 12 displays the simulation result at the frequency of 10Hz. As for the specific output signals at these frequencies won't be displayed here for convenience. The figured impedance at 10Hz and other frequencies is also listed in Table 3. The data indicates that the DC/DC converter can be controlled well to inject a sinusoidal current signal into the battery.

Experiment Validation of System Function

In the experiment, the LiMn[O.sub.2] battery with 5 cells of 35Ah paralleled and 24 such components in serial is selected as the research object. The battery capacity is 175Ah and the nominal voltage is 100Volts. Parameters of the DC/DC converter are listed in Table 1. The electronic load BTS600 from Digatron Power Electronics Corporation serves as the constant current load and the current drawn from the battery is 75 A. The other electronic load form ITECs Corporation is operated in constant resistance mode of which the value is 20[OMEGA] The initial state of charge of the battery is around 75% when the experiment starts and then it is around 55% when the AC impedance identification experiment is activated. The ambient temperature is around 23 degrees centigrade.

Gain Amplitude Comparison

To compare the theoretical gain amplitude with the experiment result, the target duty cycle which is figured out offline based on the mathematical model is set in the micro controller unit and the output of PWM to the DC/DC converter is controlled by means of the look-up table of the duty cycle. The duty cycle is organized in the following form as Equation 26 and the fluctuated amplitude of duty cycle is 7.38% of its stable value, which satisfies the assumption of the small signal model.

[mathematical expression not reproducible] (26)

[mathematical expression not reproducible] (27)

The stable input current of the converter at D=0.592 is about 30 A on condition that the input voltage is 100Volts and the efficiency of the converter is 100%. When the efficiency is just 90%, the input current will decrease to 27 A. Furthermore, when the input voltage also decreases to 90Volts, the input current will decrease to about 24 A. This can be explained by Equation 27 and the term [eta] stands for the efficiency. The measured stable current is 25.7 A.

The experiment is conducted at the frequency of 1000Hz, 500Hz, 200Hz, 100Hz, 40Hz, 10Hz, 6.4Hz and 1Hz. The actual signal of the current is sampled with the help of the embedded signal analyzer and the stable current is filtered to extract the weak and varied part. The phase shift and amplitude attenuation caused by the analyzer need to be calibrated. Taking the practical parameters of the analyzer, only the amplitude decay of the high-pass filter should be compensated and other parts of the analyzer have little effect on the amplitude. In consequence, the gain amplitude at 200Hz whose value is 245.712 is treated as the standard to which only the ratio at other frequencies is calculated in this paper for simplicity. The acquired current signals at these frequencies are shown in Figure 13 and Figure 14. At most frequency points, the sample number by the micro controller unit is set as 500 except for the frequency of 1Hz whose sample number is just 400. The frequency revolution is 0.2Hz for the target frequency range from 1Hz to less than 10Hz, 2Hz for the range from 10Hz to less than 100Hz and 20Hz for the range from 100Hz to 1k Hz.

The comparison between experiment results and theoretical ones is listed as Table 4. The experiment results are calculated using FFT technique. The THD (Total Harmonic Distortion) value is defined as the ratio of root mean square value of all frequencies except the value at fundamental frequency to the root square value of all frequencies. It can be found that only the error at 1000Hz and 500Hz is within 7% and the error becomes larger than 20%. The reason is that according to the predefined amplitude 0.0437 of the fluctuated duty cycle, only when the theoretical gain amplitude is smaller than 60 is the obtained amplitude of the disturbed current signal smaller than 2.5 A which is just 10% of the stable current. It is obvious that the gain amplitude at other frequencies is much larger than 60. When the frequency is below 40Hz, the error remains around 24% and in this frequency range the gain amplitude decreases very little. However, when the frequency like 100Hz and 200Hz is next to the critical frequency 118.62Hz, the error becomes very significant and the nonlinear characteristic of the converter dominates.

Though the error exists between the experiment and theoretical results, the THD (Total Harmonic Distortion) value at most frequency points is less than 5%. As for the THD value at 1k Hz and 500Hz, it results from the basic principles of the converter and the switching frequency is just 10k Hz. To generate a sinusoidal wave of high SNR, the target frequency should be away from the switching frequency and it is easy to be understood. The relatively low THD value demonstrates the capability of the signal analyzer and the feasibility of using the converter to inject sinusoidal current signal into the battery. Even though the amplitude of the measured fluctuated current at most frequency is larger than 2.5 A, all of them are lower than 10 A and this meets the basic assumption of performing AC impedance identification method.

AC Impedance Identification Result

The measured current and voltage after being filtered by the signal analyzer at 10Hz and 100Hz are shown in Figure 15 and Figure 16. As can be seen, the fluctuated current signal is well injected into the battery and the signal analyzer is capable of measuring the voltage response of the battery. The period of these signals is very definite and the noise has no much effect on the quality of target signals. For the right choice of the shunt resistor, the converted voltage signal of the current and the voltage response of each single cell are of the same magnitude. In this way, the transfer function of the signal analyzer can be omitted while calculating the impedance because the transfer function exists in the measurement of the single cell voltage and current at the same time.

The current fluctuation signal is conducted at the frequency of 500Hz, 400Hz, 320Hz, 200Hz, 100Hz, 80Hz, 64Hz, 50Hz, 40Hz, 32Hz, 20Hz, 10Hz, 8Hz, 6.4Hz, 5Hz and 4Hz. When the frequency is higher than 500Hz, the current change is too fast to be tolerated by power semiconductors. When the frequency is lower than 4Hz, it takes a long time to sample these signals and the internal state of the battery changes so the measured impedance is very scattered.

The Figure 17 shows the measured AC impedance of cell 9, cell 14, cell 16 and cell 22 respectively. The experiment data is fitted using the Zview Test software and the equivalent circuit is the circuit with a Warburg element. The fitted line and the experiment data are compared and to some extent the experiment data is in consistent with the fitted line. The non-uniformity of these single cells can be seen from this figure.

Summary/Conclusions

This work is focused on identifying the AC impedance of lithium-ion battery online by means of a DC/DC converter which is paralleled to the output of the battery. The main function of the DC/DC converter is to inject a fluctuated current signal into the battery while the battery is at work. The operating principles of the boost DC/DC converter are analyzed and the average state space equations are also built. It is found that the converter is fully controllable but partially observable.

Based on the average state space equations, the small signal model is derived to figure out the gain function of the fluctuated current signal against the fluctuated duty cycle of the converter. The resonant frequency exists in this gain function when the frequency increases from zero to the switching frequency of power semiconductor and attention should be paid to avoid operation around the resonant frequency. Then the equivalent circuit model of the battery and the model of the converter is established in Matlab/Simulink to verify the state space equations. Simulation results demonstrate the possibility of applying the converter to identify the AC impedance in a wide frequency range which is tolerable to the converter.

After the function verification of the system, the converter is designed and the experiment is conducted on a lithium-ion battery of high voltage and current. The fluctuated current signal and the voltage response of single cells are measured using a signal analyzer. The largest obstacle is that the weak AC signals should be extracted from the significant direct current signal and this question is solved by implementing a special signal processing circuit. The AC impedance of a single cell is measured and it is fitted to the equivalent circuit model of the battery. The experiment results also indicate the non-uniformity of each single cell. In consequence, the converter can be used online to identify the AC impedance of the lithium-ion battery and the proposed method is feasible. Furthermore, on this basis the research method of estimate other parameters of the battery can be enriched especially in the actual application such as the transport system. The future work is to refine the signal quality and optimize the control algorithm of the converter.

References

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Contact information

Po Hong, Ph.D. Candidate

Department of Automotive Engineering, Tsinghua University State Key Laboratory of Automotive Safety and Energy, Room 213, Tsinghua University, Haidian District, Beijing City, China hongp09@163.com

Acknowledgments

This work was supported by NSFC (National Natural Science Foundation of China) under Grant No. 51576113 and U1564209, by Most (Ministry of Science and Technology of China) under Grant No. 2013BAG16B01, No. 2014DFG71590 and No. 2015BAG06B01, by the independent research plan Z02-1 of Tsinghua University under Grant No. 20151080411 and the State Key Laboratory of Automotive Safety and Energy under the contract of No. ZZ2014-034.

Po Hong, Hongliang Jiang, Jian qiu Li, Liangfei Xu, and Minggao Ouyang

Tsinghua University

doi:10.4271/2017-01-1198
Table 1. System parameters

Load Resistor R                   20 [OMEGA]
Inductor [L.sub.1]                 0.001 H
Internal Resistor [R.sub.L1]       0.001 [OMEGA]
Capacitor [C.sub.1]              300 [mu]F
Internal Resistor [R.sub.C1]       0.001 [OMEGA]
Resistor RS1                       0.01 [OMEGA]
Resistor RS2                       0.01 [OMEGA]
Capacitor CS1                      0.4 F
Resistor of Contactor S1 and S2    0.001 [OMEGA]
Resistor R1                       20 [OMEGA]
On-state Resistor of D1 and G1     0.000 [OMEGA]
Switching Frequency of G1         10 kHz

Table 2. Comparison between simulation and small signal model

Frequency  Input Current  Simulated Gain Ratio  Theoretical Gain Ratio

   0.1 Hz      2.59 A             83.37                  78.87
   1 Hz        2.60 A             82.84                  78.89
  10 Hz        2.66 A             84.85                  80.63
 100 Hz       10.4 A             330.16                 307.8
 200 Hz       10.01 A            321.35                 330.8
 500 Hz        2.11 A             67.74                  69.11
1000 Hz        0.91 A             29.17                  32.23

Frequency  Error

   0.1 Hz   5.7%
   1 Hz     5.0%
  10 Hz     5.23%
 100 Hz     7.27%
 200 Hz    -2.86%
 500 Hz    -1.98%
1000 Hz    -9.49%

Table 3. Calculated results of voltage and current by FFT

Frequency  Current Amplitude  Current Phase   Voltage Amplitude

   0.5 Hz       2.51 A         71.2[degrees]       50.1 mV
   1 Hz         2.41 A        164.2[degrees]       48.23 mV
  10 Hz         2.47 A        182.2[degrees]       48.28 mV
 100 Hz         2.49 A          3.3[degrees]       29.59 mV
 200 Hz         2.72 A        -72.0[degrees]       28.75 mV
 500 Hz         2.63 A        269.9[degrees]       26.59 mV
1000 Hz         2.43 A        252.4[degrees]       24.31 mV

Frequency   Voltage Phase

   0.5 Hz  250.8[degrees]
   1 Hz    -16.5[degrees]
  10 Hz     -4.8[degrees]
 100 Hz    166.5[degrees]
 200 Hz     97.5[degrees]
 500 Hz     85.5[degrees]
1000 Hz     70.1[degrees]

Table 4. Comparison between experiment and theoretical results

Freq. /Hz  Experiment Result  Compensated Experiment Result

1000            0.1205                   0.121
 500            0.2815                   0.281
 200            1                        1
 100            1.1819                   1.182
  40            0.8431                   0.8437
  10            0.5867                   0.5941
  64            0.5892                   0.6072
   1            0.3159                   0.5937

Freq. /Hz  Theoretical Result  Error   THD Value

1000             0.130          6.91%   11.45%
 500             0.273          2.93%    8.73%
 200             1             /         1.87%
 100             2.842         58.4%     0.86%
  40             0.676         24.8%     1.78%
  10             0.493         20.5%     3.92%
  64             0.486         24.9%     0.82%
   1             0.481         23.4%     2.71%
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Author:Hong, Po; Jiang, Hongliang; Li, Jian qiu; Xu, Liangfei; Ouyang, Minggao
Publication:SAE International Journal of Alternative Powertrains
Article Type:Report
Date:Jul 1, 2017
Words:7847
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