The widespread integration of photovoltaic (PV) systems, often coupled with battery energy storage systems (BESS), is a cornerstone of the modern energy transition. These grid-tied power electronic interfaces, however, are susceptible to power quality disturbances. A primary concern is the injection of low-order harmonic currents, particularly the second harmonic (2ω, typically 100 Hz or 120 Hz), into the DC link of the inverter. This harmonic originates from grid imbalances and can propagate into the li ion battery pack, which serves as the system’s core energy reservoir. The impact of this superimposed alternating current (AC) ripple on the long-term health and safety of the li ion battery is a critical but not yet fully settled issue in engineering. Some studies suggest accelerated degradation, while others indicate negligible or even potentially beneficial effects. This ambiguity complicates the design of DC-side filters, as over-design increases cost, and under-design risks premature li ion battery failure. Therefore, establishing clear, quantifiable safe operating boundaries for li ion batteries under second harmonic stress is paramount for optimizing system reliability and economics. This article, from an engineering perspective, investigates the influence of second harmonic current components on the internal and external characteristics of li ion batteries. Through electrochemical modeling, simulation, and experimental validation, we delineate three distinct operational regimes and propose a framework for defining the safe operating range of a li ion battery in harmonically distorted environments.
The analysis begins with a suitable mathematical representation of the li ion battery. While various models exist, an enhanced electro-thermal model based on the Shepherd formulation offers an effective balance between accuracy and analytical tractability for studying low-frequency dynamics. The terminal voltage of the li ion battery during discharge and charge can be expressed as follows.
Discharge (i* > 0):
$$V_{batt}(T) = E_0(T) – K(T) \frac{Q(T_a)}{Q(T_a) – it} (it + i^*) + A e^{(-B \cdot it)} – R(T) \cdot i^*$$
Charge (i* < 0):
$$V_{batt}(T) = E_0(T) – K(T) \frac{Q(T_a)}{Q(T_a) – it} \cdot it – K(T) \frac{Q(T_a)}{it – 0.1Q(T_a)} i^* + A e^{(-B \cdot it)} – R(T) \cdot i^*$$
The temperature-dependent parameters are given by:
$$E_0(T) = E_0|_{T_{ref}} + \frac{dE}{dT}(T – T_{ref})$$
$$Q(T_a) = Q + \frac{dQ}{dT}(T_a – T_{ref})$$
$$K(T) = K \cdot e^{\alpha(1/T – 1/T_{ref})}$$
$$R(T) = R \cdot e^{\beta(1/T – 1/T_{ref})}$$
Where:
- $V_{batt}$ is the terminal voltage.
- $E_0$ is the constant voltage (open-circuit voltage).
- $K$ is the polarization constant.
- $Q$ is the battery capacity.
- $it$ is the extracted capacity ($\int i , dt$).
- $i^*$ is the filtered current.
- $R$ is the internal resistance.
- $A, B, \alpha, \beta$ are model constants.
- $T$ and $T_a$ are the internal and ambient temperatures, respectively.
The current excitation containing a DC component and a second harmonic component is defined as:
$$i_{2\omega}(t) = I_{dc} + I_{2\omega,m} \sin(2\omega t)$$
for discharge, and its negative for charge cycles. The core of the investigation lies in varying $I_{2\omega,m}$ while keeping $I_{dc}$ constant to isolate the harmonic’s effect.
External Characteristics: Voltage and Temperature
The immediate external response of the li ion battery to harmonic current is observed in its terminal voltage and temperature. Substituting the current expression into the voltage equations and simplifying by grouping harmonic-independent terms into functions $V_1(T)$ and $V_2(T)$ yields:
$$V_{batt}(T) \approx \begin{cases}
V_1(T) – R(T) \cdot I_{2\omega,m} \sin(2\omega t), & \text{for discharge} \\
V_2(T) + R(T) \cdot I_{2\omega,m} \sin(2\omega t), & \text{for charge}
\end{cases}$$
This derivation clearly shows that the terminal voltage of the li ion battery will exhibit a sinusoidal ripple proportional to the harmonic amplitude $I_{2\omega,m}$ and the internal resistance $R(T)$. Consequently, the peak voltage stress on the li ion battery increases linearly with the harmonic content.
The temperature rise within the li ion battery is governed by power loss, which is primarily ohmic. The loss and subsequent temperature update are coupled:
$$P_{loss} = |E_0 – V_{batt}| \cdot |i| + \frac{dE}{dT} \cdot |i| \cdot T$$
$$T(t) = \mathcal{L}^{-1} \left( \frac{P_{loss} R_{th} + T_a}{1 + s T_c} \right)$$
Since $P_{loss}$ is a function of the squared current to a first approximation, and the RMS value of $i_{2\omega}(t)$ increases with $I_{2\omega,m}$, the internal temperature of the li ion battery is also expected to rise with increasing harmonic amplitude.
Internal Characteristics: Capacity Fade and Resistance Growth
The long-term health of a li ion battery is characterized by its maximum capacity fade and internal resistance growth. A semi-empirical aging model captures these effects based on operational stressors. The model updates the maximum available capacity $Q$ and ohmic resistance $R$ after the completion of a full cycle (a discharge and charge phase). The key equations are:
Capacity Update: $$Q(n) = \begin{cases} Q_{Bol} – \epsilon(n) \cdot (Q_{Bol} – Q_{Eol}), & n/2 \neq 0 \\ Q(n-1), & n/2 = 0 \end{cases}$$
Resistance Update: $$R(n) = \begin{cases} R_{Bol} + \epsilon(n) \cdot (R_{Eol} – R_{Bol}), & n/2 \neq 0 \\ R(n-1), & n/2 = 0 \end{cases}$$
The aging stressor $\epsilon(n)$ accumulates based on the severity of each half-cycle:
$$\epsilon(n) = \epsilon(n-1) + \frac{0.5}{N(n-1)} \left[ 2 – \frac{\gamma_{DOD}(n-2) + \gamma_{DOD}(n)}{\gamma_{DOD}(n-1)} \right]$$
The cycle life $N(n)$ for a given depth-of-discharge (DOD) is the critical link to operating current:
$$N(n) = H \left[ \frac{\gamma_{DOD}(n)}{100} \right]^{-\xi} e^{-\varphi \left( \frac{1}{T_{ref}} – \frac{1}{T_a(n)} \right)} \left( I_{dis,ave}(n) \right)^{-\gamma_1} \left( I_{ch,ave}(n) \right)^{-\gamma_2}$$
This model reveals the pivotal insight: the aging progression of the li ion battery is dominantly governed by the average currents during the discharge ($I_{dis,ave}$) and charge ($I_{ch,ave}$) phases, not by the RMS current. The exponents $\gamma_1$ and $\gamma_2$ are battery chemistry-specific. The average current in a period $T$ for a waveform $i_{2\omega}(t)$ is:
$$I_{ave} = \frac{1}{T} \int_0^T |i_{2\omega}(t)| , dt$$
The behavior of this average current divides the harmonic impact into distinct regimes.
Three Regimes of Harmonic Impact
Based on the relationship between $I_{2\omega,m}$ and $I_{dc}$, and its subsequent effect on $I_{dis,ave}$ and $I_{ch,ave}$, we can define three operational regimes for the li ion battery.
Regime I: Insignificant Impact ($I_{2\omega,m} \le I_{dc}$)
In this regime, the current never crosses zero. The absolute value operation simply integrates the full sinusoidal waveform over a period. The result is that the average current equals the DC component: $I_{dis,ave} = I_{ch,ave} = I_{dc}$. Therefore, from the perspective of the aging model, the li ion battery experiences the same stress as if it were under pure DC current. The harmonic component may cause minor increases in voltage ripple and temperature, but it does not accelerate the fundamental aging mechanisms tied to capacity fade. This implies that within this bound, filtering requirements for the li ion battery’s protection could potentially be relaxed.
Regime II: Life-Extending Potential ($I_{dc} < I_{2\omega,m} < I_{2\omega,m0}$)
When the harmonic amplitude exceeds the DC bias, the current waveform periodically crosses zero, creating short intervals of alternating charge and discharge within what is nominally a longer discharge or charge phase. This significantly alters the average current. For example, during a nominal discharge period, the current dips into a negative (charging) region, reducing the net average discharge current $I_{dis,ave}$. Conversely, during a nominal charge period, the current peaks into a positive (discharging) region, reducing the net average charge current $I_{ch,ave}$. Since both $I_{dis,ave}$ and $I_{ch,ave}$ are less than $I_{dc}$, the product $(I_{dis,ave})^{-\gamma_1} (I_{ch,ave})^{-\gamma_2}$ becomes smaller than its pure-DC counterpart, leading to a higher predicted cycle life $N(n)$. Thus, in this specific window, the second harmonic can actually slow down the aging rate of the li ion battery compared to a pure DC current of the same RMS value.
Regime III: Life-Accelerating Degradation ($I_{2\omega,m} \ge I_{2\omega,m0}$)
As $I_{2\omega,m}$ increases further, the zero-crossing events become more pronounced, and the time spent in the opposite current direction increases. Eventually, the average currents $I_{dis,ave}$ and $I_{ch,ave}$ begin to increase again, surpassing $I_{dc}$ in effect. The critical threshold $I_{2\omega,m0}$ is defined as the harmonic amplitude where the aging stress product equals that of the pure DC case:
$$(I_{dis,ave})^{-\gamma_1} (I_{ch,ave})^{-\gamma_2} \bigg|_{I_{2\omega,m0}} = (I_{dc})^{-\gamma_1 – \gamma_2}$$
For $I_{2\omega,m} > I_{2\omega,m0}$, the average currents are larger than $I_{dc}$, leading to a larger stress product and a lower predicted cycle life $N(n)$. In this regime, the harmonic component unequivocally accelerates the degradation of the li ion battery. Furthermore, the substantially higher voltage ripple and temperature rise in this regime can induce additional secondary degradation mechanisms, such as accelerated solid-electrolyte interphase (SEI) growth or lithium plating, further compromising the li ion battery’s safety and longevity.
The following table summarizes the key characteristics of the three regimes for a li ion battery.
| Regime | Condition ($I_{2\omega,m}$) | Current Waveform Characteristic | Effect on Avg. Currents $I_{dis/ch,ave}$ | Impact on Li-ion Battery Aging | Typical Engineering Implication |
|---|---|---|---|---|---|
| I. Insignificant | $\le I_{dc}$ | No zero-crossing; unidirectional. | $I_{dis,ave} = I_{ch,ave} = I_{dc}$ | Negligible acceleration. Aging comparable to pure DC. | Filtering requirements can be relaxed, reducing system cost. |
| II. Life-Extending | $I_{dc} < I_{2\omega,m} < I_{2\omega,m0}$ | Periodic zero-crossing; bidirectional pulses within a nominal phase. | $I_{dis,ave} < I_{dc}$, $I_{ch,ave} < I_{dc}$ | Potentially slows down capacity fade and resistance growth. | Optimal harmonic management zone. May allow for minimal or strategic filtering. |
| III. Life-Accelerating | $\ge I_{2\omega,m0}$ | Pronounced zero-crossing; significant bidirectional current flow. | $I_{dis,ave} > I_{dc}$, $I_{ch,ave} > I_{dc}$ | Accelerates degradation significantly. Increases voltage/temperature stress. | Harmonic content must be strictly filtered to protect the li ion battery investment and ensure safety. |
Simulation Analysis and Results
To validate the theoretical analysis, a simulation model of a 12.8 V, 40 Ah Lithium Iron Phosphate (LiFePO4) li ion battery was constructed. The battery was subjected to a constant DC current of 40 A ($I_{dc}$) superimposed with varying amplitudes of second harmonic current ($I_{2\omega,m}$). The external and internal parameters were monitored over multiple charge-discharge cycles. The following table presents a subset of the key simulation results at the end of a standardized test cycle, highlighting the trends across the three regimes.
| $I_{2\omega,m}$ (A) | Regime | Peak Voltage (V) | Max Temp. Rise (°C) | Capacity Fade (Ah) | Resistance Growth (mΩ) | $I_{dis,ave}$ (A) (approx.) |
|---|---|---|---|---|---|---|
| 0 | I (Baseline) | 14.90 | 2.1 | 0.0020 | 0.00002 | 40.0 |
| 20 | I | 15.05 | 2.3 | 0.0020 | 0.00002 | 40.0 |
| 40 | I / II Boundary | 15.45 | 2.7 | 0.0020 | 0.00002 | 40.0 |
| 60 | II | 16.05 | 3.4 | 0.0018 | 0.00001 | 38.5 |
| 90 | III | 17.25 | 5.1 | 0.0023 | 0.00003 | 42.5 |
| 120 | III | 18.45 | 7.2 | 0.0030 | 0.00005 | 46.0 |
The data confirms the theoretical predictions. In Regime I (0-40 A), capacity fade remains identical to the baseline despite increasing voltage ripple. The critical observation is at $I_{2\omega,m}$ = 60 A (Regime II), where the capacity fade is actually lower than the baseline, demonstrating the life-extending potential. The corresponding average discharge current is calculated to be lower than $I_{dc}$. In Regime III (90 A and 120 A), capacity fade and resistance growth accelerate significantly, accompanied by a sharp increase in peak voltage and temperature, confirming the hazardous nature of this operating zone for the li ion battery.
Experimental Validation with Coin Cells
To provide empirical evidence, a controlled aging experiment was conducted using 2.2 mAh LiNiMnCoO2 (NMC) coin cells. Twelve cells with nearly identical initial impedance were divided into three groups (A, B, C) corresponding to the proposed regimes. A custom pulse sequence was used to emulate a DC bias with a superimposed second harmonic. The cells underwent 1000 cycles at 30°C, with periodic reference performance tests (RPT) to measure capacity and impedance. The key experimental design and results after 600 cycles are summarized below.
| Cell Group | Cell ID | $I_{2\omega,m}$ (mA) | Regime | Approx. $I_{dis,ave}$ (mA) | Capacity Fade after 600 cycles (%) |
|---|---|---|---|---|---|
| A (Regime I) | A1 | 0.0 | I | 0.400 | -8.5 |
| A2 | 0.1 | I | 0.400 | -8.7 | |
| A3 | 0.3 | I | 0.400 | -8.6 | |
| A4 | 0.4 | I/II Boundary | 0.400 | -8.8 | |
| B (Regime II) | B1 | 0.5 | II | 0.065 / 0.518* | -7.9 |
| B2 | 0.6 | II | 0.131 / 0.592* | -8.1 | |
| B3 | 0.7 | II | 0.196 / 0.662* | -8.4 | |
| B4 | 0.8 | II | 0.260 / 0.730* | -8.6 | |
| C (Regime III) | C1 | 0.9 | III | 0.325 / 0.795* | -9.2 |
| C2 | 1.0 | III | 0.389 / 0.860* | -9.8 | |
| C3 | 1.1 | III | 0.454 / 0.926* | -10.5 | |
| C4 | 1.2 | III | 0.517 / 0.991* | -11.4 | |
| *Average currents are bimodal due to periodic polarity reversal; the smaller value corresponds to the reduced effective average during a nominal phase. | |||||
The experimental results strongly support the three-regime theory. Group A (Regime I) shows consistent capacity fade regardless of the small harmonic addition. Group B (Regime II, $I_{2\omega,m}$ = 0.5 to 0.8 mA) exhibits slower capacity fade compared to the baseline A1 cell. This is the clearest evidence of the life-extending effect, where the calculated average currents are lower than $I_{dc}$. Group C (Regime III, $I_{2\omega,m}$ > ~0.9 mA) shows accelerated capacity fade, with the degradation rate increasing monotonically with harmonic amplitude. The crossover point $I_{2\omega,m0}$ for this specific li ion battery chemistry and format appears to lie between 0.8 mA and 0.9 mA. This experiment confirms that the fundamental aging mechanism is tied to the phase-average current, not RMS, and validates the practical existence of the three operational zones.
Engineering Implications for Safe Operation
The delineation of three distinct regimes provides a concrete framework for defining the safe operating range of a li ion battery in applications like PV-storage systems. The goal shifts from simply minimizing all harmonic content to strategically managing it within safe bounds. This has direct implications for system design and operation.
1. Relaxed Filtering in Regime I: If system analysis confirms that the induced second harmonic on the DC bus will not exceed the magnitude of the DC current drawn/ supplied by the li ion battery ($I_{2\omega,m} \le I_{dc}$), then the filtering requirements can be significantly relaxed. The li ion battery’s internal impedance and double-layer capacitance provide sufficient attenuation for these low-amplitude ripples. This can lead to substantial cost savings by allowing the use of smaller, less expensive DC-link capacitors or simplified filter topologies.
2. Identification and Utilization of Regime II: For systems where harmonics occasionally exceed Regime I, it is crucial to characterize the specific li ion battery chemistry to estimate its critical threshold $I_{2\omega,m0}$. Operating intentionally within Regime II could, in theory, offer a longevity benefit. However, this requires sophisticated battery management system (BMS) algorithms capable of real-time harmonic analysis and dynamic current control to ensure the li ion battery remains within this narrow window. The increased voltage ripple must also be accounted for in the voltage monitoring and protection circuits of the BMS.
3. Strict Enforcement of Regime III Limits: The upper bound of safe operation is defined by $I_{2\omega,m0}$. System designers must ensure, through proper inverter control, grid monitoring, and passive/active filtering, that the harmonic current injected into the li ion battery pack under worst-case grid imbalance conditions does not approach this threshold. Exceeding it poses a direct threat to the warranty period, safety, and total cost of ownership of the li ion battery energy storage system.
The safe operating envelope for a li ion battery can thus be visualized and defined by the following inequality, which must be upheld by the power conditioning system:
$$I_{2\omega,m}^{system} \le \min( I_{dc}^{max}, \, I_{2\omega,m0} )$$
where $I_{2\omega,m}^{system}$ is the worst-case second harmonic amplitude presented to the li ion battery, and $I_{dc}^{max}$ is the maximum DC current. The threshold $I_{2\omega,m0}$ is a property of the li ion battery cell and can be determined through characterization tests or derived from the aging model parameters $\gamma_1$ and $\gamma_2$.
Conclusion
The interaction between grid-induced harmonics and li ion battery longevity is nuanced and governed by the impact on phase-average currents rather than RMS values. Through electrochemical modeling, this analysis demonstrates that the terminal voltage and temperature of a li ion battery increase monotonically with the second harmonic amplitude. More importantly, the long-term aging trajectory divides into three clear regimes based on the ratio of harmonic amplitude to DC current. When the harmonic amplitude is less than or equal to the DC current (Regime I), its impact on li ion battery life is negligible. In a specific window where the harmonic amplitude moderately exceeds the DC current (Regime II), the resulting reduction in average cycle currents can potentially slow down the aging of the li ion battery. Beyond a critical threshold (Regime III), harmonic stress accelerates degradation rapidly.
These findings provide a practical, physics-based framework for optimizing the design and operation of PV-storage systems. By defining these safety and longevity regimes, engineers can make informed decisions on DC-side harmonic filtering. The insight allows for potentially reducing filter size and cost when harmonics are within Regime I, while providing a clear specification for the maximum allowable harmonic (the lesser of $I_{dc}$ and $I_{2\omega,m0}$) to prevent accelerated degradation of the valuable li ion battery asset. Ultimately, integrating this understanding of the li ion battery’s harmonic response into system-level control and protection strategies is key to achieving both economic and reliable renewable energy integration.