Adaptive, multi-parameter battery state estimator with optimized time-weighting factors

Abstract

We derive and implement a battery control algorithm that can accommodate an arbitrary number of model parameters, with each model parameter having its own time-weighting factor, and we propose a method to determine optimal values for the time-weighting factors. Time-weighting factors are employed to give greater impact to recent data for the determination of a system’s state. We employ the (controls) methodology of weighted recursive least squares, and the time weighting corresponds to the exponential-forgetting formalism. The output from the adaptive algorithm is the battery state of charge (remaining energy), state of health (relative to the battery’s nominal performance), and predicted power capability. Results are presented for a high-power lithium ion battery.

Appendix

We provide a brief recapitulation of the model described in Reference [5] that is appropriate for the purposes of this work. The state of charge is taken as a weighted average (weight factor w _SOC ) of values extracted by coulomb integration and voltage-based modeling:

$$SOC=w_{SOC}(SOC_C)+(1-w_{SOC})(SOC_V).$$

(17)

For the coulomb-based state of charge, SOC _C ,

$$SOC_C(t)=SOC(t-\Delta t)+\mathop{\int}\limits_{t-\Delta t}^{t}\left[100 \frac{\eta_I I}{Ah_{\rm nominal}}-S_D\right]\frac{dt}{3600}.$$

(18)

The voltage-based state of charge, SOC _V , can be determined by inverting a voltage expression for the cell derived from the equivalent circuit depicted in Fig. 1 so as to extract the open-circuit potential,

$$V=V_{\rm oc}+IR-A\mathop{\int}\limits_{\zeta=t}^{\zeta=0}I(\zeta)\exp[-B(t-\zeta)d\zeta.$$

(19)

These two equations can be recast in recursive forms as

$$SOC_C(t)=SOC(t-\Delta t)+\left[ \frac{100}{Ah_{\rm nominal}}\left\{\frac{{(\eta_I I)}_{t-\Delta t}+ {(\eta_I I)}_{t}}{2}\right\}-S_D\right]\frac{\Delta t}{3600}$$

(20)

and

$$V_t={(V_{oc}+IR)}_t+\left(\frac{I_{t-\Delta t}+I_{t}}{2}\right) A_dr\Delta t+\exp(-B\Delta t){(V-V_{oc}-IR)}_{t-\Delta t}.$$

(21)

Time is represented by t and I denotes current; discharge currents are taken as negative. The nominal capacity Ah _nominal corresponds to the ampere-hours of capacity the battery delivers when discharged from 100% SOC to 0% SOC at low rates of discharge. The self-discharge rate S _D and the current efficiency η _I are expected to vary with both temperature and SOC. The factor 3,600 has units of s/h, and the factor 100 is employed to keep a consistent percent basis. The parameters A and B correspond to A = 1/C _D and B = 1/(R _ct C _D ) = 1/τ, where τ can be viewed as a time constant. A _d is the inverse of the capacitance on discharge, and r is the ratio of A for charge to that of discharge; i.e.,

$$ r(T,SOC)=A_c/A_d=C_{D,{\rm discharge}}/C_{D,{\rm charge}}. $$

(22)

The open-circuit potential V _oc is a function of temperature, SOC _V , and a hysteresis function:

$$ V_{oc}=V_o(T,SOC_V)+V_H. $$

(23)

A look-up table can be used to determine the SOC _V once the value of V _o is obtained. For the hysteresis contribution, we construct the following first-order differential equation to calculate a hysteresis voltage V _H :

$$ \frac{\partial V_H}{\partial t}=\beta(\eta_II-S_D)\left[V_{H,{\rm max}} -\hbox{sign}(I)V_H\right], $$

(24)

or

$$ {(V_H)}_t\approx{(V_H)}_{t-\Delta t}+\beta\Delta t{\{(\eta_II-S_D)[V_{H,{\rm max}} -\hbox{sign}(I)V_H]\}}_{t-\Delta t}. $$

(25)

For prolonged charge currents, or short but very large charge currents, the hysteresis voltage tends to about V _H,max. The exact opposite holds for discharge currents, in which case the hysteresis voltage tends to −V _H,max. Note also that if the current remains at zero for a long time, the hysteresis voltage tends to the charge-decreasing condition through self-discharge. The parameters in this equation (including V _H,max) can be temperature and SOC dependent. While hysteresis plays a critical role in NiMH batteries, it is far less important in lead acid and lithium ion systems.

By combining the hysteresis and cell voltage expressions, we obtain

$$ \begin{aligned} V_t&=V_o+{(V_H)}_{t-\Delta t}+\beta\Delta t{\left((\eta_II-S_D)[V_{H,{\rm max}} -\hbox{sign}(I)V_H]\right)}_{t-\Delta t}\\ &+I_tR+\left(\frac{I_{t-\Delta t}+I_{t}}{2}\right) A_dr\Delta t+E{(V-V_{oc}-IR)}_{t-\Delta t},\\ \end{aligned} $$

(26)

where E = exp(−Δt/τ). Equation 26 is the basis for the assignments provided in Eq. 12.

We now construct the power-projection capability. First, note that the max discharge power can be expressed as:

$$ P_{\rm max,discharge}=IV=IV_{\rm min}. $$

(27)

That is, when the battery voltage obtains its lowest acceptable value, the max discharge power results. We shall refer to the ohmic battery power capability as

$$ P_{\rm max,discharge}=IV_{\rm min}=\frac{(V_{\rm min}-V_{oc})}{R}V_{\rm min}, $$

(28)

consistent with V = V _oc  + IR for an ohmic battery. Similarly, the max charge power of the ohmic battery is given by

$$ P_{\rm max,charge}=IV_{\rm max}=\frac{(V_{\rm max}-V_{oc})}{R}V_{\rm max}, $$

(29)

For the maximum ohmic resistance, obtained at long times (low frequency), R is replaced by R + R _ct , where R _ct is different for charge and discharge (cf. Fig. 1).

The ohmic battery does not address transient effects such as those correlated by the superposition integral. To improve the estimate, we employ Eq. 26 and calculate the maximum charge and discharge powers available for the time interval Δt:

$$ \begin{aligned} &I|_t=-\frac{{(V_{oc}-V)}_t+(AI_{t-\Delta t}\Delta t/2)+\exp(-B\Delta t){[V-(V_{oc}+IR)]}_{t-\Delta t}} {R+(A_dr\Delta t/2)}\\ &P_{\rm max,discharge}(\Delta t)=IV_{\rm min}= \left[-\frac{{(V_{oc}-V_{\rm min})}_t+(A_dI_{t-\Delta t}\Delta t/2)+\exp(-B\Delta t){[V-(V_{oc}+IR)]}_{t-\Delta t}} {R+(A_drI_{t-\Delta t}\Delta t/2)}\right]V_{\rm min}\\ &P_{\rm max,charge}(\Delta t)=IV_{\rm max}= \left[-\frac{{(V_{oc}-V_{\rm max})}_t+(A_cI_{t-\Delta t}\Delta t/2)+\exp(-B\Delta t){[V-(V_{oc}+IR)]}_{t-\Delta t}} {R+(A_drI_{t-\Delta t}\Delta t/2)}\right]V_{\rm max}\\ \end{aligned} $$

(30)

where it is recognized that r =  1 on discharge. To implement these equations, the respective powers are calculated immediately after the algorithm has been employed to finish the SOC determination at time t. In this case, quantities calculated or measured at time t are then stored in the variables listed in the respective power expressions at time t−Δt. Then one must state the duration corresponding to the desired estimate for power. For example, if we want to know the power estimates 3 s from “now”, then the measured and extracted values are placed in the t−Δt quantities, Δt is set to 3 s, and the right sides of the above equations yield the desired power estimates.

The final topic of this Appendix is the skewness test of the current source. Following the procedure described in References [4] and [5], we restrict the skewness test to the actual current values I(t) and do not incorporate the charge-discharge weighting. The subscript s is added to indicate quantities associated with the skewness calculation:

$$ \begin{array}{l} s_{w,s}\left|_{N}\right.=\sum\limits_{j=1}^{N}\lambda^{N-j}=1+\lambda \sum\limits_{j=1}^{N-1}\lambda^{N-1-j}=1+\lambda\left(s_{w,s}\left|_{N-1}\right.\right)\\ s_{I,s}\left|_{N}\right.=\frac{1}{\sum\limits_{j=1}^{N}\lambda^{N-j}}\sum\limits_{j=1}^{N}\lambda^{N-j} I_j=\frac{I_N+\lambda\left(s_{I,s}\left|_{N-1}\right.\right)\left(s_{w,s}\left|_{N-1}\right.\right)} {\left(s_{w,s}\left|_{N}\right.\right)}\\ s_{II,s}\left|_{N}\right.=\frac{1}{\sum\limits_{j=1}^{N}\lambda^{N-j}}\sum\limits_{j=1}^{N}\lambda^{N-j} I_j^2=\frac{I_N^2+\lambda\left(s_{II,s}\left|_{N-1}\right.\right)\left(s_{w,s}\left|_{N-1}\right.\right)} {\left(s_{w,s}\left|_{N}\right.\right)}\\ \end{array} $$

(31)

and

$$ skewness|_{N}=\left|\frac{{(I_N-s_{I,s}|_{N})}^3}{{\left[s_{II,s}|_{N}-{(s_{I,s}|_{N})}^2\right]}^{3/2}} \right|\frac{1}{N}+(skewness|_{N-1})\left(1-\frac{1}{N}\right). $$

(32)

For all of the skewness calculations employed in this work, λ =  0.99 for Eqs. 31. To start the recursive calculations for skewness, the following conditions are used.

$$ \begin{array}{l} s_{w,s}|_1=1\\ s_{I,s}|_1=I_1\\ s_{II,s}|_1=I_1^2\\ \end{array} $$

(33)

For the first time step, skewness| _t=0 is set to skew_cal(we employ a value of skew_cal =  10). Last, we define

$$ skew\_test=\left\{ \begin{array}{l} 0\;\;\hbox{if}\;\;skewness\ge skew\_cal,\\ 1\;\;\hbox{if}\;\;skewness < skew\_cal.\\ \end{array}\right. $$

(34)

The adaptive parameter regression analysis is not employed if the skew test is not passed, in which case:

$$ \hbox{if}\;skew\_test=0,\;\hbox{then}{(m_t)}_t={(m_t)}_{t-\Delta t}. $$

(35)

In summary, the parameter vector m is not updated skew_test = 0; however, the approach allows one to calculate the open-circuit voltage, and SOC _V can always be calculated through the use of previously extracted parameters.