Reference

NICELABE Seminar Lecture Note

Algorithms for Battery Management Systems course, Prof. Gregory Plett

What is the Importance of a good SOC estimator?

• BMS must be able to estimate two fundamentally different types of nonmeasurable battery-pack quantity
  • States (values that change quickly) : e.g., SOC, Diffusion voltage, hysteresis voltage
  • Parameters (values that change slowly) : e.g., cell capacities, resistances(내부저항), aging effects
• SOC estimate for all cells is important input to balancing, energy, power calculations

🗯️ Balancing

• 각각의 cell이 동일 level SOC를 가지도록 하는 기술

Additional benefits

• Longevity
• Performance
• Reliability
• Density
• Economy

Fully charged

• DEFINITION : A cell is fully charged when its OCV equals $v_h(T)$, a manufacturer-specified voltage (may be function of T (Temperatrue))
• e.g., $v_h(300K) = 4.2V$ for LMO; $v_h(300K) = 3.6V$ for LFP
  • LMO & LFP are kind of Cathode Active Materials (양극성 활물질, 리튬 베터리 종류)
• Define the SOC of a fully charged cell to be 100%

Fully discharged

• DEFINITION : A cell is fully discharged when OCV equals $v_l(T)$, a manufacturer-specified voltage (may be function of T (Temperature))
• e.g., $v_l(300K) = 3.0V$ for LMO; $v_l(300K) = 2.0V$ for LFP
• Define the SOC of a fully discharged cell to be 0%

🗯️ OCV? Terminal Voltage? Assume with Diffusion voltage!!

• OCV (Open Circuit Voltage) → General Idea
  • Open Circuit에서 측정한 voltage
  • Battery 관점에서는 내부저항을 고려해야 함!
• Terminal voltage → General Idea
  • 외부 회로가 연결된 상태에서의 전압
  • 내부저항 뿐만 아니라 회로에 따른 여러 영향들까지 고려
• Diffusion voltage
  • Diffusion
    • 시간에 따라 평행상태에 이르기 위해 리튬 농도가 확산되어 균일해지는 현상
    • 리튬이 가지는 고유 특성
• OCV & Terminal voltage → Lithium Ion Battery Idea
  • So, OCV is additionally depend on powder 표면 리튬 농도 (Diffusion)
  • Also Terminal voltage must be considered with diffusion voltage

Total capacity

• DEFINITION : Cell total capacity $Q$ is quantity of charge removed as cell is brought from fully charged state to fully discharged state
• SI unit for charge is coulombs (C), but more common to use units of ampere hours(Ah) to measure the total capacity of a battery cell
• Total capacity of a cell is not a fixed quantity
  • It generally decays slowly over time as the cell degrades c.g. aging effects
  • (느린) 시간에 대한 함수로 정의할 수 있음

Discharge capacity

• DEFINITION : Discharge capacity $Q_{rate}$ is quantity of charge removed as cell discharged at constant rate from fully charged state until terminal voltage $v(t)$ reaches $v_l(t)$
• Strongly dependent on cell’s internal resistance, and so also rate and temperature
  • e.g., rate 높게 방전 → 내부저항으로 인한 $v(t)$ ↓ → Discharge capacity ↓
• The discharge capacity of a cell at a particular rate and temperature is not a fixed quantity: It also generally decays slowly over time as the cell degrades
  • Discharge capacity 만큼 방전 후, cell 휴식상태 → diffusion 현상에 의한 cell 전압 ↑ ⇒ 낮은 rate으로 방전 가능!
  • Discharge capacity 만큼 방전된 상태 $\ne$ fully discharged 상태!
  • 즉 Discharge capacity는 SOC 정의하는 데 적합하지 않음!

Nominal capacity

• DEFINITION : Cell nominal capacity $Q_{nom}$ is a manufacturer-specified quantity intended to be representative of 1C-rate discharge capacity $Q_{1C}$ of a particular manufactured lot of cells at room temperature
• 표준 정격 용량
• The nominal capacity is a constant value
• But, each of a cell’s total capacity is different!
  • $Q_{nom} \ne Q_{1C}$, $Q_{nom} \ne Q$
  • 따라서 Nominal capacity는 SOC 정의하는 데 적합하지 않음!

Residual capacity & state of charge

• DEFINITION : Cell residual capacity $Q_{res}$ is quantity of charge that would be removed from cell if it were brought from its present state to a fully discharged state
• DEFINITION : Cell state-of-charge is ratio of residual capacity to total capacity, $Q \over Q_{res}$

Voltage-based method to estimate SOC

• OCV is a deterministic function of SOC, $OCV(z(t))$
• Measure cell terminal voltage, v(t), and look up on OCV versus SOC curve

  

• Ignores effects of $i(t) \times R_0$ losses (내부저항), diffusion voltages, and hysteresis on $v(t)$

  

• Allow effect of $i(t) \times R_0$ losses (내부저항)
  • $v(t) = OCV(z(t)) - i(t)R_0$
  • Better, but still ignores effects of diffusion voltages, hysteresis and so is still noisy
  • Filtering helps but adds delay, which must be accounted for

    

Current-based method to estimate SOC

• Okay for short periods of operation when initial conditions are known or can be frequently reset (for long periods)
• Subject to drift due to current sensor’s fluctuations, current-sensor bias, incorrect capacity estimate, other losses
• Uncertainty/error bounds grow (without limit) over time until estimate is reset

Model-based state estimation

• Model-based estimators implement algorithms that use sensed measurements to infer internal hidden state of dynamic system
• Same input propagated through true system, model, measured and predicted outputs compared; error used to update model’s state estimate
• Combination of Voltage-based method and Current-based method
• Apply this estimation to Battery, parameters would be like this:
  • Input : Cell current
  • Output : Cell terminal voltage
  • Model : e.g., ESC
  • Output error 요소 : State 추정 오차, 측정 오차 (Process, Sensor), Model 오차

Linear Kalman filter

• Kalman filter gives optimal state estimate

  

  $u_k$ : measured input signal - Cell current $w_k$ : process-noise random input $v_k$ : sensor-noise random input $x_k$ : SOC $y_k$ : Cell terminal voltage Functions $f()$ and $h()$ may be time-varying, also can be non-linear (→ In this case, use nonlinear Kalman filter)

👀 Vector notation

• Superscript “-” indicates a predicted quantity based only on past measurements
• Superscript “+” indicates an estimated quantity based on both past and present measurements
• Symbol “^” indicates a predicted or estimated
• quantity : $\widehat x^+$ or $\widehat x^-$

Cost function to minimize (optimize)

• $\widehat x_k^{MMSE}(\mathbb Y_k) = \underset{\widehat x_k}{\operatorname{argmin}}(\mathbb E[\;\mid\mid x_k - \widehat x_k^+\mid\mid_2^2\mid \mathbb Y_k\;])$

  $\qquad\qquad\quad\;\, = \underset{\widehat x_k}{\operatorname{argmin}}(\mathbb E[\;(x_k - \widehat x_k^+)^T(x_k - \widehat x_k^+)\mid \mathbb Y_k\;])$

  $\qquad\qquad\quad\;\, = \underset{\widehat x_k}{\operatorname{argmin}}(\mathbb E[\;x_k^Tx_k - 2x_k^T \widehat x_k^+ + (\widehat x_k^+)^T \widehat x_k^+ \mid \mathbb Y_k\;])$

• $0 = {d \over d\widehat x_k^+} \mathbb E[\; x_k^+ x_k - 2x_k^T \widehat x_k^+ + (\widehat x_k^+) \widehat x_k^+ \mid \mathbb Y_k \;]$
• $0 = \mathbb E[-2(x_k - x_k^+) \mid \mathbb Y_k] = 2 \widehat x_k^+ - 2\mathbb E[x_k \mid \mathbb Y_k]$
• $\therefore \widehat x_k^+ = \mathbb E[x_k \mid \mathbb Y_k]$

🗯️ Additional vector calculus

• ${d\over dX}Y^T X = Y$, ${d \over dX}X^T Y = Y$, and ${d \over dX}X^T A X = (A + A^T)X$

Prediction error and Innovation

• Prediction error

  $\tilde x_k^- = x_k - \widehat x_k^- \; where \; \widehat x_k^- = E[x_k \mid Y_{k-1}]$

• Innovation

  $\tilde y_k = y_k - \widehat y_k \; where \; \tilde y_k = E[y_k \mid Y_{k-1}]$

• Properties can be like this:

  $\mathbb E[\tilde x_k^-] = \mathbb E[x_k] - \mathbb E[\mathbb E[x_k \mid \mathbb Y_{k-1}]] = \mathbb E[x_k] - \mathbb E[x_k] = 0 \ \mathbb E[\tilde y_k] = \mathbb E[y_k] - \mathbb E[\mathbb E[y_k \mid \mathbb Y_{k-1}]] = \mathbb E[y_k] - \mathbb E[y_k] = 0$

Predict / correct solution

• Properties can be like this: $\mathbb E[\tilde x_k^- \mid \mathbb Y_{k-1}] = \mathbb E[x_k - \mathbb E[x_k \mid \mathbb Y_{k-1}] \mid \mathbb Y_{k-1}] = 0 = \mathbb E[\tilde x_k^-] \ \mathbb E[\tilde x_k^- \mid \mathbb Y_k] = \mathbb E[\tilde x_k^- \mid \mathbb Y_{k-1},y_k] = \mathbb E[\tilde x_k^- \mid y_k]$ $\mathbb E[\tilde x_k^- \mid \mathbb Y_k] = \mathbb E[x_k \mid \mathbb Y_k] - \mathbb E[\widehat x_k^- \mid \mathbb Y_k] = \widehat x_k^+ - \widehat x_k^-$

• Predict/correct sequence of steps

  $\therefore \widehat x_k^+ = \widehat x_k^- + \mathbb E[\tilde x_k^- \mid y_k]$

State update equation

• General equation

  $\mathbb E[x\mid y] = \mathbb E[x] + \Sigma_{\tilde x \tilde y}\Sigma_{\tilde y}^-(y - \mathbb E[y]), \quad \Sigma \; is \; covariance \; matrix$

• Applying this equation to our problem, we get

  

  $\therefore \widehat x_k^+ = \widehat x_k^- + L_k\tilde y_k, \quad L \; is \; Kalman \; gain$

Uncertainty of state estimate

• Calculation of $\Sigma_{\tilde x, k}^+$,

  

• $\widehat x_k^+ \pm 3 \sqrt{diag(\Sigma_{\tilde x,k}^+)}$, 3 is something specific scaling factor of probability(%)
• Boundary를 형성하여 추정의 정확성을 판단

Visualizing the linear Kalman filter

• $A, \; B$ matrix → State space model notation
• $C, \; D$ matrix → Model notation of representing output from state and input

  

Kinds of nonlinear Kalman filters

• Three basic KF generalizations for nonlinear systems
  • Extended Kalman filter (EKF)
    • Analytic linearization of model at each point in time
  • Sigma-point (Unscented) Kalman filter (SPKF/UKF)
    • Statistical/empirical linearization of model at each point in time: Can be much better than EKF at same computational complexity
  • Particle filters
    • Most precise, but often thousands of times more computations required than either EKF/SPKF

🗯️ Idea of nonlinear Kalman filters

• Nonlinear 부분을 linear로 근사화해서 정리!

Prediction steps of nonlinear Kalman filters

• EKF and SPKF both follows same set of general steps as KF

  

  

• EKF and SPKF simply have different expressions from KF for evaluating the expectation operations (in Step 1a, 2a) → Because of different system model

Extended Kalman filter (EKF)

• EKF makes two simplifying assumptions when adapting general sequential inference equations to a nonlinear system:
  • When computing estimates of the output of a nonlinear function, EKF assumes $E[fn(x)] \approx fn(E[x])$, which is not true in general
  • When computing covariance estimates, EKF uses Taylor-series expansion to linearize the system equations around the present operating point

EKF on ESC results

• EKF was executed for a test having dynamic profiles from 100% SOC down to around 10% SOC
  • RMS SOC estimation error = 0.46%
  • Percent of time error outside bounds = 0%

    

Pack-average SOC

• While “pack SOC” does not make sense, concept of pack-average SOC is a useful one
• Since all cells in series experience same current, their SOC values will
  • Move in the same direction for any given applied current, by a similar amount (but different because of unequal cell capacities)
• We take advantage of this similarity by creating:
  • Algorithm to determine the composite average behavior of all cells in battery pack
  • Algorithm to determine the individual differences between specific cells and that composite average behavior

Defining the pack-average state / cell-difference states

• We define pack-average state x-bar as $\bar x = {1\over N_s}\sum_{i=1}^{N_s}x_k^{(i)}$
• Can then write an individual cell’s state vector as $x_k^{(i)} = \bar x_k + \Delta x_k^{(i)}$

  

• Complexity
  • Bar filter is of same computational complexity as individual state estimators used as a basis
  • But, delta filters can be made very simple
  • Also, delta states change much more slowly than average state, so delta filters can be run less frequently, down to $1/N_s$ times rate of bar filter

Summary

• SOC를 추정하면 가질 수 있는 Good advantages → Battery SOC 추정이 매우 중요
• SOC를 추정하기 위한 여러가지 방법론들 (Voltage-based, Current-based, Model-based)
• Classical한 SOC approach - using Kalman Filter (linear/non-linear)