Rest the battery until terminal voltage equals OCV (typically 30 min – 2 hr depending on cell chemistry). Look up SOC from the OCV–SOC characterisation table. This is the only moment Coulomb counting uses voltage — and only to initialise. Alternative: if the cell was fully charged last cycle and SOC(end) was reached, use that as the starting point.

Use the cell datasheet capacity at room temperature. For an aged cell, Q_n must be updated — a 20% capacity fade means Q_n shrinks by 20%, and if the old Q_n is kept, SOC will be overestimated. This is the SOH interaction: Coulomb counting accuracy degrades as the battery ages unless Q_n is recalibrated periodically.

Read the current sensor (Hall effect or shunt resistor) at the sampling instant k. Apply any known offset calibration. Current sensor accuracy is the single biggest factor in Coulomb counting accuracy. A shunt-based measurement with a high-quality ADC (16-bit, ±0.05% accuracy) gives substantially better results than a Hall sensor (±1–3% typical).

Multiply the instantaneous current by the time step. For a 10Hz BMS (Δt = 0.1s), a 100A discharge transfers 100 × 0.1 = 10 As = 0.00278 Ah per step. For high-accuracy systems, use the trapezoidal rule: ΔQ = (I(k) + I(k−1)) / 2 × Δt, which better approximates the integral between samples.

During discharge, every Coulomb that flows out reduces SOC by a full Coulomb (η = 1.0 assumed). During charge, not all charge going in is stored — lithium plating, electrolyte decomposition, and heat absorb some fraction. Typical Li-ion η at room temperature is 0.99–0.995. At low temperatures (−20°C), η can drop to 0.90 or lower, requiring temperature-dependent η(T) tables.

Subtract the normalised charge increment from the previous SOC. The negative sign for discharge follows from the convention that positive current means current leaving the battery. After this step, clamp SOC to [0, 1] to avoid physically impossible values caused by sensor noise or model errors.

Report SOC(k) to the BMS supervisor. Go to S-1 at the next sampling instant. The loop continues for the entire battery operating lifetime.

When the charger terminates the CC-CV charge cycle (current drops below I_cutoff, typically C/20), the cell is fully charged and SOC can be hard-reset to 100%. This resets all accumulated drift. Similarly, if the BMS detects a low-voltage cutoff event, SOC can be reset to 0%. These “anchor points” limit the maximum error between resets to the drift accumulated in a single cycle.

If the battery has been at rest for long enough that V_terminal ≈ OCV, use the OCV table to recalibrate SOC. This is especially valuable for LFP cells where the OCV plateau is flat — wait until the cell has fully relaxed, then the voltage reading gives a reliable SOC reference. The model-based filters (EKF, UKF, PF) do this correction continuously at every step, which is their primary advantage over standalone Coulomb counting.

The filter uses the linear state-space model to roll the previous best estimate x̂(k−1) forward by one time step. Matrix A describes how the state evolves on its own (e.g., SOC drains slowly even at zero current), while B·u(k) injects the known input effect — the measured current I(k) multiplied by the Coulomb-counting factor (−η·Δt/Q_n). The superscript ⁻ denotes this is still a prior estimate: the measurement at time k has not yet been used. Think of this as the model saying “given where we were, here is where we expect to be now.”

Uncertainty must also be propagated forward. P(k−1) is the covariance matrix describing how uncertain we were at the last step. Multiplying by A and Aᵀ transforms that uncertainty through the system dynamics. Adding Q (process noise covariance) accounts for the fact that the model is imperfect — battery aging, temperature effects, and unmeasured disturbances all inject additional uncertainty at every step. After this step, P⁻(k) is always larger than P(k−1): prediction always increases uncertainty.

The Kalman Gain K is the heart of the filter — it decides the optimal blend between the model prediction and the new voltage measurement. The numerator P⁻·Cᵀ represents prediction uncertainty projected onto the measurement space. The denominator adds measurement noise R to that projected uncertainty. When prediction uncertainty P⁻ is large relative to R, K is large → heavily trust the measurement. When the sensor is noisy (large R), K is small → rely more on the model prediction. This ratio is computed optimally, minimising the mean-squared estimation error.

The term [y(k) − C·x̂⁻(k)] is called the innovation or residual — it is the difference between what the voltage sensor actually measured and what the model predicted the voltage would be. If the innovation is zero, the measurement perfectly confirms the prediction and no correction is needed. If it is nonzero, K scales that residual and adds a correction to the predicted state. For SOC estimation, a higher measured voltage than predicted pushes the SOC estimate upward; a lower voltage pulls it down.

After incorporating the measurement, uncertainty always decreases. The factor [I − K·C] shrinks P⁻ by the amount of information gained from the sensor. If K is large (measurement was very trusted), P(k) becomes much smaller than P⁻(k). If K ≈ 0 (sensor was unreliable), P(k) ≈ P⁻(k) and little was learned. This updated P(k) is then carried forward as P(k−1) into the next predict step, completing the recursive loop.

Unlike the KF which uses a fixed linear matrix A, the EKF propagates the state through the actual nonlinear model equations f(·). For the Thevenin battery model this means computing SOC via the full Coulomb-counting equation and V_RC via the full RC exponential decay — no approximation here. The nonlinearity only becomes a problem in the next step when we must characterise how uncertainty propagates, which requires a linearisation.

The Jacobian F(k) is the matrix of all first-order partial derivatives of f with respect to the state x, evaluated at the current estimate x̂(k−1). It is the best linear approximation of f at that point — a local tangent plane. For the Thevenin model, F is a 2×2 matrix: the SOC row has ∂(SOC)/∂SOC = 1 and ∂(SOC)/∂V_RC = 0; the V_RC row has ∂(V_RC)/∂V_RC = e^(−Δt/τ) and ∂(V_RC)/∂SOC = 0. This linearisation is valid only near the current operating point.

Uncertainty is propagated using the locally linearised dynamics F(k) rather than the true nonlinear f. This is the key EKF approximation: while the mean is propagated exactly through f, the covariance is only propagated approximately through the Jacobian. If the true nonlinearity is mild near the operating point, this is accurate. If the system is highly nonlinear (steep OCV curve slope changes), the true covariance grows faster than F·P·Fᵀ predicts, causing P to be underestimated — a precursor to filter divergence.

The measurement function h(x) = OCV(SOC) − R₀·I − V_RC is nonlinear due to OCV(SOC). Its Jacobian H(k) is a 1×2 row vector: the first element is dOCV/dSOC (the slope of the OCV curve at the predicted SOC), and the second is −1 (∂h/∂V_RC = −1 directly). The slope dOCV/dSOC is typically obtained by differentiating a polynomial or spline fit to the OCV lookup table. This value is the single most important parameter in the EKF: it determines how strongly voltage measurements can correct the SOC estimate.

The structure is identical to the standard KF, but now H(k) varies at every time step as the OCV slope changes with SOC. When the OCV curve is steep (e.g., near 0% or 100% SOC for NMC cells), H is large, the innovation covariance is large, and K is large — meaning measurements strongly correct the state. When the OCV curve is flat (e.g., 30–70% SOC for LFP cells), H ≈ 0, K ≈ 0, and the filter effectively ignores voltage measurements and relies purely on Coulomb counting — accumulating drift.

A critical EKF design choice: the innovation [y(k) − h(x̂⁻(k))] uses the full nonlinear h function, not the linearised H·x̂⁻. This means the predicted terminal voltage is computed properly as OCV(SOC⁻) − R₀·I − V_RC⁻, and the difference from the actual measured voltage is the true residual. Only the gain K and covariance propagation use the linearised approximation — this hybrid approach preserves accuracy in the correction step even when linearisation is imperfect.

The updated covariance P(k) reflects the reduction in uncertainty after the measurement was incorporated. Note that this formula (the “simple form”) can occasionally produce non-positive-definite P matrices when K is computed with finite-precision arithmetic. The numerically more stable Joseph form P(k) = (I−KH)·P⁻·(I−KH)ᵀ + K·R·Kᵀ is preferred in embedded BMS implementations to prevent covariance matrix corruption over long operating cycles.

For a 2-state battery model (SOC, V_RC), n = 2, so we generate 2n+1 = 5 sigma points. The centre point χ₀ is simply the current mean estimate x̂. The remaining 4 points are spread symmetrically around the mean: the i-th column of √((n+λ)P) is added to or subtracted from x̂ to form χᵢ and χᵢ₊ₙ. The matrix square root √P is computed via Cholesky decomposition — a stable, efficient numerical method. The spread is controlled by λ: a larger λ places sigma points further from the mean, capturing more of the distribution’s tails at the cost of potentially sampling regions where the nonlinearity is stronger.

Each of the 5 sigma points is independently propagated through the full nonlinear battery model f(·). For each point this means computing: SOC_new = SOC_old − (η·Δt/Q_n)·I and V_RC_new = e^(−Δt/τ)·V_RC_old + R₁(1−e^(−Δt/τ))·I. No linearisation occurs — we evaluate the true nonlinear function at each point. This is the key innovation of the UKF: by sampling around the mean, we capture how the nonlinearity distorts the distribution, something the EKF’s single tangent-line approximation cannot do.

The predicted mean x̂⁻ is a weighted average of all 5 propagated sigma points, using mean weights Wᵐᵢ. The predicted covariance P⁻ is a weighted sum of outer products of the deviations from the mean, using covariance weights Wᶜᵢ. The centre point (i=0) gets a different weight than the outer points (i=1…2n) — this asymmetry is how the UKF captures skewness and higher-order moments of the transformed distribution. Adding Q at the end accounts for process noise, just as in the KF and EKF.

A fresh set of 5 sigma points is generated from the predicted mean x̂⁻(k) and predicted covariance P⁻(k). These are different from the P-1 sigma points because the predicted covariance P⁻ has changed (it grew due to process noise Q). Re-generating ensures the sigma points used for the measurement update accurately represent the current predicted uncertainty — an important consistency requirement of the algorithm.

Each sigma point is mapped to a predicted terminal voltage using the full nonlinear h: V_t = OCV(SOC) − R₀·I − V_RC. Since OCV(SOC) is nonlinear, each of the 5 sigma points produces a slightly different predicted voltage. This captures how uncertainty in SOC translates to uncertainty in terminal voltage — accounting for the curvature of the OCV curve at the current operating point, which the EKF completely misses by using only the slope.

ŷ(k) is the predicted terminal voltage — a weighted mean of the 5 mapped measurements. S(k) is the innovation covariance: a weighted sum of squared deviations of each mapped point from ŷ, plus sensor noise R. A large S(k) means the filter has high uncertainty in what voltage it should see; a small S(k) means it has a narrow expectation. The innovation S(k) plays the role that (C·P⁻·Cᵀ + R) played in the standard KF, but is computed entirely from nonlinear propagation — no Jacobian involved.

Pₓᵧ is the cross-covariance between the predicted state and predicted measurement — it quantifies how correlated our state uncertainty is with our measurement uncertainty. In the KF this was P⁻·Cᵀ; here it is computed directly from the sigma points without needing C or any Jacobian. Dividing by S(k) gives the Kalman Gain K, which has the same interpretation as in the KF: it is the optimal weight for the measurement correction.

The state correction follows the same intuition as the KF: the actual measured voltage y(k) minus the predicted voltage ŷ(k) forms the innovation, which K scales and adds to the predicted state. The covariance update subtracts the information gained from the measurement. Because both ŷ and S were computed through the nonlinear h (not linearised), the UKF’s correction is inherently more accurate than the EKF’s when the OCV curve has significant curvature — for example, near 20% or 90% SOC on an NMC cell.

Each particle xᵢ is a complete hypothesis about the current battery state — a specific (SOC, V_RC) pair drawn randomly from the prior belief p(x₀). If the battery has just been fully charged, the prior might be a narrow Gaussian centred at SOC = 1.0. If the initial SOC is unknown, particles are drawn from a uniform distribution over [0, 1]. All N particles start with equal weight 1/N because we have no reason to prefer one hypothesis over another before seeing any data. With N = 100 particles, there are 100 different “candidate” SOC values simultaneously being tracked. The filter works by progressively weighting down incorrect hypotheses and concentrating probability mass on the correct one.

Every particle is evolved forward using the battery model f — the same Thevenin equations as EKF/UKF. However, each particle receives its own independently drawn process noise sample wᵢ from p(w). This is the Monte Carlo principle in action: by drawing different noise realisations for each particle, the set collectively represents the full spread of where the true state could be after the time update. Crucially, p(w) can be any distribution — uniform, Laplacian, bimodal — not just Gaussian. This is the first place where the PF surpasses all KF variants: it handles non-Gaussian process noise naturally by sampling from it.

For every particle, we ask: “if the true SOC were xᵢ(k), how likely is it that the battery would produce the measured terminal voltage y(k)?” This is evaluated using the likelihood function p(y|x). For Gaussian sensor noise with variance σ²: p(y|xᵢ) = exp(−(y − h(xᵢ))² / 2σ²) / √(2πσ²), where h(xᵢ) = OCV(SOCᵢ) − R₀·I − V_RC_i. Particles whose predicted voltage closely matches the actual measurement get high likelihood and therefore high weight. Particles that predict a wildly different voltage get a weight near zero — they are being “voted down” by the data. This step is entirely nonlinear and requires no Jacobian.

The unnormalised weights w̃ᵢ are divided by their total sum to ensure the weights form a valid probability distribution (Σwᵢ = 1). After normalisation, each wᵢ represents the relative probability that particle i contains the true state. A well-performing filter at this point will have a few particles with high weights (near the true SOC) and many with weights approaching zero (wrong SOC hypotheses). The weighted mean Σwᵢ·xᵢ is the SOC estimate — not a single particle, but the centre of mass of all particles weighted by their evidence.

The SOC estimate is the weighted average of all N particle states. High-weight particles (those consistent with the latest voltage measurement) pull the estimate toward their SOC value; low-weight particles barely contribute. This is a direct approximation of the posterior mean E[x|y(1:k)]. Unlike KF variants, the PF can represent any posterior shape — if particles cluster in two groups (bimodal distribution, possible after sensor loss), the weighted mean correctly sits between them, and the variance P(k) reflects the genuine multi-hypothesis uncertainty.

The variance P(k) measures how spread out the particle cloud is. A narrow cluster (small P) means the filter is confident in the SOC estimate. A wide cloud (large P) means high uncertainty — perhaps because no measurement was available for several seconds, or the voltage measurement was ambiguous. Unlike the KF’s covariance matrix which is propagated analytically (and can become inconsistent), the PF’s uncertainty is always derived directly from the particle distribution, making it inherently self-consistent.

Over time, a catastrophic phenomenon called particle degeneracy occurs: nearly all weight concentrates on a single particle, while the rest carry negligible weight. The effective sample size N_eff measures how many particles are “usefully contributing” — it ranges from 1 (fully degenerate: only one particle matters) to N (all particles equally weighted, ideal). When N_eff drops below a threshold (typically N/2), the filter is wasting computation on irrelevant particles and its estimate becomes unreliable. Resampling is triggered to correct this.

Resampling draws N new particles from the current weighted set, where each particle’s probability of being selected equals its weight wᵢ. High-weight particles are copied multiple times; low-weight particles are likely discarded entirely. The result is N particles that are all equally weighted (1/N) but are concentrated near where the high-weight particles were. Systematic resampling — using a single random number and a regular grid — is preferred over simple multinomial resampling because it has lower variance and avoids duplicating the same particle too many times. After resampling, the particle diversity is restored and the algorithm continues. However, repeated resampling can cause sample impoverishment — loss of diversity when too many copies of the same particle are made — which is countered by ensuring adequate process noise in step P-1.