UPS and Battery State Prediction [/ˈəps ənd ˈbætəri ˈsteɪt priˈdɪkʃən/] n - The worth of a Uninterruptible power supply rests wholly upon the battery bank behind it. A UPS that reports full capacity while its batteries have silently degraded is a liability wearing the guise of an asset. State-of-charge estimation and capacity fade prediction are the two problems that decide whether the UPS will do its office when it is needed.

Of the State of Charge

State of charge (SOC) is the fraction of the battery’s energy that remains available:

$$\text{SOC}(t)=\text{SOC}(t_0)-\frac{1}{C_n}{\int }_{t_0}^{t}I(\tau )d\tau$$

where $C_n$ is the nominal capacity in amp-hours and $I$ the discharge current. This is Coulomb counting: integrate current to track charge. It is simple and runs in real time, yet error gathers with the years, for $C_n$ itself drifts as the battery ages.

The Coulomb counting estimate is corrected against the voltage-SOC relationship, which holds relatively stable at rest. A Kalman filter weds the two signals: the state transition follows the Coulomb counting equation; the observation is the open-circuit voltage taken at known rest intervals.

The state-space form:

$${\text{SOC}}_t={\text{SOC}}_{t-1}-\frac{I_{t-1}\Delta t}{C_n}+w_t$$ $$V_t=f({\text{SOC}}_t)+v_t$$

where $f$ is the nonlinear OCV-SOC curve and $v_t$ the measurement noise. Because $f$ is nonlinear, an extended Kalman filter or unscented Kalman filter is required: the same estimation framework as State Space Models and the Kalman Filter, extended to admit the nonlinear observation equation.

Of Capacity Fade

Battery capacity diminishes with cycling and calendar aging; the chief mechanisms are electrolyte decomposition, lithium plating, and active material loss. For VRLA (valve-regulated lead-acid) batteries, common in data center UPS systems, the fade rate depends upon temperature and depth of discharge.

A simple capacity fade model:

$$C(n)=C_0\exp (-\alpha n)$$

where $n$ is the cycle count and $\alpha$ a fade rate that rises with temperature. More refined models apply an Arrhenius correction for the temperature dependence:

$$\alpha (T)={\alpha }_0\exp \left(-\frac{E_a}{RT}\right)$$

where $E_a$ is an activation energy and $R$ the gas constant. From historical capacity measurements taken during periodic discharge tests, the fade rate is estimated by fitting the exponential model to the data.

Of the Runtime Foretold

Available runtime under a given load:

$$t_{\text{run}}=\frac{C_n\cdot \text{SOC}}{\bar{I}}$$

where $\bar{I}$ is the average discharge current under the expected IT load. This holds only if $C_n$ is current rather than nominal; a facility that trusts a nameplate capacity in place of a measured one will receive an optimistic runtime.

The full uncertainty in runtime ought to be propagated. The variance of the runtime estimate is a function of the variance of the SOC estimate and of the load forecast:

$$\text{Var}(t_{\text{run}})\approx {\left(\frac{\partial t_{\text{run}}}{\partial \text{SOC}}\right)}^2\text{Var}(\text{SOC})+{\left(\frac{\partial t_{\text{run}}}{\partial \bar{I}}\right)}^2\text{Var}(\bar{I})$$

An interval serves operators better than a point estimate when they must decide whether to transfer to generator or await the utility’s return.

Of the Decision to Replace

Replacement is an economic reckoning: replace the battery bank when the cost of a failure during a true outage exceeds the replacement cost, weighted by the probability of a long outage and the probability of battery failure.

$$\text{Replace when: }{P}_{\text{fail}}\cdot C_{\text{incident}}>C_{\text{replace}}$$

The probability of failure is drawn from the capacity fade model and from the distribution of outage durations in historical utility data. The incident cost includes hardware loss, SLA penalties, and reputational harm. This framing is in essence a threshold policy under the Value at Risk and Expected Shortfall framework, applied to battery lifetime rather than financial loss.

See Predictive Generator Maintenance for the complementary problem: keeping the backup generator ready to take over when the batteries run out.