In recent years, traction battery pack systems in new energy vehicles have demanded increasingly compact, lightweight, and cost-effective current sensing solutions. To address these requirements, I proposed an elliptical Hall-element array architecture tailored for the rectangular busbar conductors inside a traction battery pack. This study covers the array parameter design, full-temperature-domain calibration, and a hybrid GWO-BP error optimization algorithm that compensates eccentricity and tilt errors.
The increasing penetration of electric vehicles has placed stringent constraints on the battery management system (BMS) within a traction battery pack. The BMS relies on accurate current measurement to estimate state of charge (SOC) and state of health (SOH). Conventional current transducers based on fluxgate technology or closed-loop Hall effect often use bulky magnetic cores, which conflict with the spatial and weight limitations of a modern traction battery pack. Shunt resistors provide high accuracy but break the galvanic isolation and introduce additional power loss. Fiber-optic current sensors are expensive and difficult to calibrate. Therefore, there is a strong engineering motivation to develop a non-contact, coreless, and easily integrable current sensor that performs well when mounted around the rectangular busbars commonly found inside a traction battery pack.
Hall element arrays have been studied by many researchers because they discard ferromagnetic cores and thereby eliminate magnetic saturation, hysteresis, and nonlinear core effects. Depending on the arrangement of the sensor elements, circular arrays, rectangular arrays, and elliptical arrays can be used. For rectangular busbars, an elliptical array offers a lower profile and a smaller installation footprint than a circular array while providing a smoother magnetic field sampling path than a rectangular array. In my research, I set the design target to maintain the measurement accuracy within 3‰ over the full temperature range, while also minimizing the influence of conductor eccentricity and inclination.
Sensor Principles and Fundamental Theory
Magnetic field sensing can be realized by several physical mechanisms. The fluxgate effect relies on the nonlinear magnetization of soft magnetic materials, offering extremely high resolution but requiring complex excitation and readout circuits. In contrast, the Hall effect provides a simple and linear relationship between the magnetic induction and the output voltage. When a current flows through a rectangular conductor inside a traction battery pack, it produces a quasi-static magnetic field that obeys Ampère’s circuital law:
$$ \oint_{L} \mathbf{H} \cdot d\mathbf{l} = \sum I_{enc} $$
For an infinitely long straight conductor, the magnetic field intensity at a distance \(\rho\) can be expressed as
$$ H = \frac{I}{2\pi \rho} $$
If the conductor has a rectangular cross-section with width \(l\) and thickness \(h\), and the cross-section is uniformly filled with current density, the conductor can be decomposed into differential current filaments. I denote the cross-sectional area as \(S = l \cdot h\). Each differential element \(dI\) is:
$$ dI = \frac{I}{S} dx dy $$
Then, for an observation point at \((a,b)\) in the plane perpendicular to the conductor, the \(x\)-component and the \(y\)-component of the magnetic field intensity generated by the entire rectangular conductor are:
$$ H_{ix} = -\frac{I}{2\pi S} \int_{-h/2}^{h/2} \int_{-l/2}^{l/2} \frac{b-y}{(a-x)^2 + (b-y)^2} dx dy $$
$$ H_{iy} = \frac{I}{2\pi S} \int_{-h/2}^{h/2} \int_{-l/2}^{l/2} \frac{a-x}{(a-x)^2 + (b-y)^2} dx dy $$
These two formulas describe the two-dimensional magnetic field distribution of a rectangular conductor when the conductor axis is perpendicular to the sensor plane and passes exactly through the center of the array. However, in a real traction battery pack, the busbar may be displaced by vibration or thermal expansion. Therefore, a two-dimensional model is insufficient. I built a three-dimensional magnetic field model to describe arbitrary conductor eccentricity and tilt.
Elliptical Array Parameter Design
An elliptical array is parameterized by its semi-major axis \(a\) and semi-minor axis \(b\), and the aspect ratio is denoted as \(AR=b/a\). For a circular array, \(AR=1\). To generate the sensor element locations on the ellipse, I compare the projection method and the uniform curve segment length (UCSL) method.
With the projection method, points are first evenly placed on a circle of radius \(a\), then projected vertically onto an ellipse with the same long axis. The coordinates of the \(i\)-th element are:
$$ P_i = \left( a\cos\left( \varphi_0 + \frac{360^\circ}{N} i\right),\; b\sin\left( \varphi_0 + \frac{360^\circ}{N} i\right) \right) $$
However, the arc segments formed by projection vary significantly, making the discrete line integral less accurate.
In the UCSL method, the circumference of the ellipse is divided into segments of equal arc length. Since no closed-form arc length exists for an ellipse, I approximated the perimeter by discretizing the ellipse into a large number of points \(K\). For a point with index \(n\), its coordinate is:
$$ E_n = \left( a\cos\frac{360^\circ n}{K},\; b\sin\frac{360^\circ n}{K} \right), \quad n=1,2,…,K $$
The arc length from the positive \(x\)-axis to the \(k\)-th point is computed recursively:
$$ s(k) = \sum_{n=1}^{k} \sqrt{ \left(E_{n,x} – E_{n-1,x}\right)^2 + \left(E_{n,y} – E_{n-1,y}\right)^2 } $$
The total perimeter is \(s(K)\), and if \(N\) Hall elements are placed on the ellipse, each curve segment has length:
$$ \Delta s = \frac{s(K)}{N} $$
An offset parameter \(s_{0,r}\) is introduced to improve the robustness of the array. I used the best offset value:
$$ s_{0,r} =
\begin{cases}
\frac{3}{8}, & N \text{ is odd} \\
\frac{1}{4}, & N \text{ is even}
\end{cases} $$
The arc length from the \(x\)-axis to the first element \(P_0\) is then:
$$ s_0 = s_{0,r} \cdot s(K) $$
Thus, the arc length to the \(i\)-th Hall element \(P_{i-1}\) becomes:
$$ s_i = s_0 + (i-1)\Delta s $$
By locating the point on the ellipse corresponding to \(s_i\), I obtain the sensor coordinate \(P_{i-1}\). The sensitive axis of each Hall element is the tangent vector to the ellipse at that point:
$$ t_{i-1} = \frac{ \left( -a\sin\frac{360^\circ m}{K}, \; b\cos\frac{360^\circ m}{K} \right) }{ \sqrt{ \left(a\sin\frac{360^\circ m}{K}\right)^2 + \left(b\cos\frac{360^\circ m}{K}\right)^2 } } $$
where \(m\) is the index of the point on the discretized ellipse whose arc length matches \(s_i\).
I compared the theoretical current error for UCSL and projection methods for different numbers of Hall elements. The UCSL method consistently showed lower error than the projection method for a given \(N\). I therefore adopted the UCSL method for the elliptical array.
The magnetic field of a rectangular conductor with the classical size of \(20\,\text{mm}\times4\,\text{mm}\) was considered. I selected a semi-major axis \(a=30\,\text{mm}\) to match the installation space inside a traction battery pack. Using MATLAB, I swept the number of Hall elements \(N\) from 3 to 8 and computed the current error at \(AR=0.33\). The error decreased with larger \(N\), but diminishing returns were observed for \(N>7\). Considering cost, I focused on \(N=6\) and \(N=7\). The aspect ratio sweep in Figure 3-4 yielded zero theoretical error at several specific \((N, AR)\) combinations. For example:
- \(N=6, AR=0.33\)
- \(N=7, AR=0.31\)
- \(N=6, AR=1\)
- \(N=7, AR=1\)
I evaluated the allowable eccentric range and error behavior for these configurations. The maximum allowable eccentric distance in the \(X\)-direction differs only by 4.2 mm between elliptical and circular arrays, while the circular array allows about 10 times more eccentric distance in the \(Y\)-direction. However, since the circular array has a much larger installation footprint, it violates the compactness requirement of a traction battery pack. Furthermore, when the conductor is eccentrically displaced to the maximum allowable distance, the circular array may produce even larger current error than the elliptical array because the element remains within the circular boundary but farther from the conductor surface.
Additional comparison showed that the \(N=6, AR=0.33\) array outperformed the \(N=7, AR=0.31\) array when the conductor was displaced in the negative \(X\)-direction. This behavior was confirmed by theoretical simulations. Considering the overall sensor accuracy and component cost, I selected the final array parameters:
$$ N = 6, \quad AR = 0.33, \quad a = 30\,\text{mm}, \quad b = 10\,\text{mm} $$
Table 1 lists the element coordinates and sensitive-axis angles derived from the UCSL method for this configuration.
| Element | Position \((x,y)\) (mm) | Angle \(\omega_i\) (deg) |
|---|---|---|
| \(P_0\) | (26.94, 4.40) | 145.79 |
| \(P_1\) | (5.57, 9.83) | 176.38 |
| \(P_2\) | (–16.59, 8.33) | 192.58 |
| \(P_3\) | (–26.94, –4.40) | 325.79 |
| \(P_4\) | (–5.56, –9.83) | 356.38 |
| \(P_5\) | (16.59, –8.33) | 12.58 |
Hardware Implementation
The hardware circuit of the proposed elliptical Hall current sensor was designed around the selected array parameters. An AKM EQ-730L single-axis linear Hall element was chosen as the primary sensing element. Its key specifications are given in Table 2.
| Parameter | Symbol | Min | Typ | Max | Unit |
|---|---|---|---|---|---|
| Supply voltage | \(V_{cc}\) | 3.0 | 5.0 | 5.5 | V |
| Supply current | \(I_{cc}\) | – | 12 | – | mA |
| Sensitivity | \(V_h\) | 110 | 130 | 150 | mV/mT |
| Output voltage range | \(V_{out}\) | 10 | – | 90 | %\(V_{cc}\) |
| Operating temperature | \(T\) | –40 | 25 | 125 | °C |
For the three-dimensional magnetic field measurement used to estimate tilt parameters, I employed the Infineon TLV493DA1B6 three-axis Hall sensor. This integrated X/Y/Z Hall sensor measures magnetic induction in all three spatial directions and communicates via the I²C interface. The electrical characteristics are summarized in Table 3.
| Parameter | Symbol | Min | Typ | Max | Unit |
|---|---|---|---|---|---|
| Supply voltage | \(V_{DD}\) | 2.8 | 3.3 | 3.5 | V |
| Standby current | \(I_{DD}\) | – | 7 | 100 | nA |
| Average operating current | \(I_{opDD}\) | – | 3.7 | – | mA |
| Sensitivity | \(V_h\) | – | 10.2 | – | LSB12/mT |
| Operating temperature | \(T\) | –40 | – | 125 | °C |
The main control unit was an STM32F103C8T6 MCU, which contains a 72 MHz ARM Cortex-M3 core, two 12-bit ADCs, and a CAN controller. Stable power was supplied by a 12 V programmable DC source. An E1205SY DC-DC module converted 12 V to 5 V for the Hall sensors and the operational amplifier circuits. The 5 V rail was further converted to 3.3 V by a TLV1117 low-dropout regulator for the MCU. The analog output from each EQ-730L element was buffered by a voltage follower and then fed into a summing amplifier. The summed voltage, together with individual element voltages, was digitized by the STM32 ADC. A TJA1050 CAN transceiver provided communication to the host PC.
Three-Dimensional Magnetic Field Model and Current Inversion
To describe conductor eccentricity and tilt, I established a coordinate system with the origin at the array center. The \(X\)-axis aligns with the long semi-axis of the ellipse and the \(Y\)-axis with the short semi-axis. The \(Z\)-axis follows the right-hand rule. The infinite rectangular conductor axis intersects the \(XOY\) plane at coordinates \((x_p, y_p, 0)\). The tilt direction is described by the angle \(\alpha\), the angle between the conductor axis and the positive \(Z\)-axis, and \(\beta\), the angle between the projection of the conductor axis on the \(XOY\) plane and the positive \(X\)-axis. The unit vector along the conductor axis is \( \mathbf{e}_l = (m,n,p)\), where
$$ m = \sin\alpha\cos\beta, \quad n = \sin\alpha\sin\beta, \quad p=\cos\alpha $$
For a Hall element located at \(P(a,b,0)\), the vector from the conductor intersection point \(Q(x_p,y_p,0)\) to the Hall element is \(\overrightarrow{QP}\). The shortest distance between the conductor and the Hall element is
$$ \rho = | \mathbf{e}_l \times \overrightarrow{QP} | $$
Based on the Biot–Savart law for a long straight conductor, the magnetic field at \(P\) can be expressed as:
$$ \mathbf{H}_i = \frac{I}{2\pi\rho} \,\mathbf{e}_{\varphi} $$
where the unit azimuthal vector is
$$ \mathbf{e}_{\varphi} = \mathbf{e}_l \times \mathbf{e}_r = \frac{ \mathbf{e}_l \times \overrightarrow{QP} }{ |\mathbf{e}_l \times \overrightarrow{QP}| } $$
When the conductor is tilted, the cross-section of the conductor on the \(XOY\) plane is no longer the original rectangle. The actual integration area \(S\) is the projection of the inclined conductor cross-section onto the plane of the array. I therefore introduced a method to reconstruct the integration area. From the tilt vector \((m,n,p)\) I computed the projected boundaries of the conductor; then, using the eccentric position \((x_p,y_p)\), I shifted the projected area to its actual location. This process is illustrated by the algorithm flow:
- Input the conductor tilt vector \((m,n,p)\).
- Derive the boundary lengths and slopes of the projected conductor cross-section.
- Construct the candidate boundary line equations.
- Input the conductor center coordinates \((x_p,y_p)\).
- Offset the boundary equations by \((x_p,y_p)\).
- Obtain the integration region equation.
Inside this reconstructed integration region, the magnetic field components produced by the differential current element \(dI\) at the Hall element position can be found. The \(x\)-component is
$$ H_{ix} = \frac{I}{2\pi A} \iint_{S} \frac{ p(b-y) }{ \left[p(a-x)\right]^2 + \left[p(b-y)\right]^2 + \left[m(a-x)-n(b-y)\right]^2 } \, dx dy $$
The \(y\)-component is
$$ H_{iy} = \frac{I}{2\pi A} \iint_{S} \frac{ p(a-x) }{ \left[p(a-x)\right]^2 + \left[p(b-y)\right]^2 + \left[m(a-x)-n(b-y)\right]^2 } \, dx dy $$
The \(z\)-component is
$$ H_{iz} = \frac{I}{2\pi A} \iint_{S} \frac{ m(a-x) – n(b-y) }{ \left[p(a-x)\right]^2 + \left[p(b-y)\right]^2 + \left[m(a-x)-n(b-y)\right]^2 } \, dx dy $$
When \(m=n=0\) and \(p=1\), these equations reduce to the two-dimensional formulas, confirming consistency.
Let \(\omega_i\) denote the angle between the X-direction sensitive axis of each Hall element and the global \(X\)-axis. Then the tangential magnetic field detected by element \(i\) is:
$$ H_{MFS,i} = H_{ix}\cos\omega_i + H_{iy}\sin\omega_i + H_{iz} $$
The total current \(I_{calc}\) is then obtained by summing the tangential field along the elliptical integration path:
$$ I_{calc} = \sum_{i=1}^{6} H_{MFS,i} \, \Delta s_i $$
In the UCSL array all \(\Delta s_i\) are equal, so
$$ \Delta s_i = \Delta s = \frac{s(K)}{N} $$
The relative current error is defined as
$$ \varepsilon = \frac{I_{calc} – I_{true}}{I_{true}} \times 100\% $$
For the traction battery pack, the target current range was set to \(-1500\,\text{A}\) to \(1500\,\text{A}\).
Error Sources and Analysis
Eccentricity Error
When the rectangular conductor is eccentrically located inside the elliptical array, the magnetic field distribution around the finite number of Hall elements changes asymmetrically. The discrete sampling cannot perfectly reconstruct the continuous line integral of Ampère’s law, producing the eccentricity error. I analyzed the theoretical error by substituting the offset conductor center \((X_p,Y_p)\) into the two-dimensional field model. The updated \(x\)-component when the conductor is displaced by \(X_p\) in \(X\)-direction is:
$$ H_{ixp} = -\frac{I}{2\pi S} \int_{-h/2 + X_p}^{h/2 + X_p} \int_{-l/2 + Y_p}^{l/2 + Y_p} \frac{b-y}{(a-x)^2 + (b-y)^2} dx dy $$
and the \(y\)-component is:
$$ H_{iyp} = \frac{I}{2\pi S} \int_{-h/2 + X_p}^{h/2 + X_p} \int_{-l/2 + Y_p}^{l/2 + Y_p} \frac{a-x}{(a-x)^2 + (b-y)^2} dx dy $$
Similar formulas hold when displacement occurs independently along the \(Y\)-axis. Theoretical analysis showed that the current error increases almost linearly with eccentric distance within the allowable displacement range. For the \(N=6, AR=0.33\) elliptical array, a pure \(X\)-direction eccentricity of 10 mm produces a theoretical error of about 4.18%, and a pure \(Y\)-direction eccentricity of 2.4 mm reaches 0.94%. These errors exceed the desired accuracy for a traction battery pack, so an optimization algorithm is necessary.
Tilt Error
Conductor tilt was analyzed using the three-dimensional field model. The tilt angles \(\alpha\) and \(\beta\) influence the projected integration region shape, causing the current inversion integral to be evaluated over an incorrect area. As \(\alpha\) increases, the projected conductor cross-section expands, causing the local field around each Hall element to deviate from the ideal vertical-conductor case. The theoretical analysis indicated that a tilt angle of \( \alpha=45^\circ\) leads to a tilt error of nearly 12.37% in the uncompensated sensor. Therefore, a robust conductor state estimation method is needed to compensate for both eccentric and tilt errors.
Full-Temperature-Domain Calibration
Initial errors in a traction battery pack current sensor arise from component tolerances, soldering offsets, and temperature drift of the Hall elements and analog circuits. I designed a cold, room-temperature, and high-temperature calibration algorithm. During calibration, the sensor output is converted from the raw ADC value to an acquired current value.
The raw current message is processed by extracting the high and low 8-bit substrings from the CAN data. Let \(ADC_{dif}\) be the difference between the high-byte decimal value and the low-byte decimal value. The analog voltage can be recovered using the known ADC reference voltage \(V_{min}=0.001\,\text{V}\), the gain \(A\), and the number of sampling points \(K=128\):
$$ V_{ADC} = ADC_{dif} \cdot V_{min} \cdot \frac{1}{K} \cdot \frac{1}{A} $$
For a sampling resistor \(R=0.47\,\Omega\), the measured current is:
$$ I_{output} = \frac{V_{ADC}}{R} $$
In the calibration process, I bias the measurement with a reference current \(I_{input}\). The current deviation is:
$$ \Delta I = I_{output} – I_{input} $$
I split the current range into three segments: small current (0–90 A), medium current (91–870 A), and large current (871–1500 A). Different amplification gains \(A\) are used for these segments. For each segment, I performed a first-order least-squares fit of \(\Delta I\) versus \(I_{output}\):
$$ \Delta I_{fit,i} = k_i I_{output} + b_i $$
A quadratic fit was also investigated, but the first-order fit gave comparable or better accuracy over most measured points and involved lower computational complexity. Therefore, the first-order linear fit was selected.
After fitting, the compensated current becomes:
$$ I_{mea} = I_{output} – \Delta I_{fit} $$
Table 4 gives the fitted slope and intercept values at room temperature for the three current segments.
| Current segment | Direction | Slope | Intercept |
|---|---|---|---|
| Small current | Positive | \(k_0=-0.00505\) | \(b_0=0.04657\) |
| Small current | Negative | \(k_1=-0.00508\) | \(b_1=-0.04342\) |
| Medium current | Positive | \(k_2=-0.00367\) | \(b_2=0.06019\) |
| Medium current | Negative | \(k_3=-0.00395\) | \(b_3=-0.10566\) |
| Large current | Positive | \(k_4=-0.00284\) | \(b_4=-1.12625\) |
| Large current | Negative | \(k_5=-0.00316\) | \(b_5=1.12059\) |
After room temperature calibration, the maximum residual relative error was reduced within 3‰ across the entire current range.
For high and low temperature calibrations, I controlled the ambient temperature at \(-40\,^\circ\text{C}\), \(25\,^\circ\text{C}\), and \(125\,^\circ\text{C}\). For each temperature point, I fitted the deviation curves. Let the low-temperature fit slope be \(l_i\), the room-temperature slope \(k_i\), and the high-temperature slope \(h_i\). The low-temperature drift coefficient is:
$$ k_{low,i} = \frac{l_i – k_i}{65} $$
since the temperature difference between \(-40\,^\circ\text{C}\) and \(25\,^\circ\text{C}\) is 65°C. The high-temperature drift coefficient is:
$$ k_{high,i} = \frac{h_i – k_i}{100} $$
where the temperature difference between \(125\,^\circ\text{C}\) and \(25\,^\circ\text{C}\) is 100°C. Thus the average unit-temperature drift coefficient is:
$$ k_i^* = \frac{k_{low,i} + k_{high,i}}{2} $$
The temperature-compensated fitted deviation at an environment temperature \(T\) is:
$$ \Delta I_{fit,tem} = \left( k_i + k_i^* T \right) I_{output} + b_i $$
Finally, the temperature-compensated current is:
$$ I_{mea,tem} = I_{output} – \Delta I_{fit,tem} $$
The experimental calibration results showed that the relative current error in the low-temperature environment was reduced by 62.59%, and in the high-temperature environment by 42.99%. After full-temperature-domain calibration, the current error remained within 3‰ at all tested temperatures. Table 5 gives the slope and intercept values at different temperature points for the medium-current segment as an example.
| Temperature | Direction | Slope | Intercept |
|---|---|---|---|
| –40°C | Positive | \(l_2=-0.00380\) | \(c_2=-0.23163\) |
| –40°C | Negative | \(l_3=-0.00392\) | \(c_3=0.22826\) |
| 25°C | Positive | \(k_2=-0.00367\) | \(b_2=0.06019\) |
| 25°C | Negative | \(k_3=-0.00395\) | \(b_3=-0.10566\) |
| 125°C | Positive | \(h_2=-0.00421\) | \(a_2=-0.65411\) |
| 125°C | Negative | \(h_3=-0.00387\) | \(a_3=0.31021\) |
Conductor State Estimation Using GWO-BP Neural Network
To accurately compensate the eccentricity and tilt errors, I need to estimate the conductor state parameters \((X_p,Y_p,m,n,p)\). Because these parameters have a highly nonlinear relation with the measured magnetic field components, I developed a hybrid neural network based on the Grey Wolf Optimizer (GWO) and a backpropagation (BP) neural network.
BP Neural Network
The BP neural network is a multilayer feedforward network with an input layer, one or more hidden layers, and an output layer. Each neuron in the hidden layer computes a weighted sum of the input data and applies a nonlinear activation function. The final output is compared with the target value, and the error is propagated backward through the network. The gradient descent method adjusts each weight and bias. A standard BP network can approximate arbitrary nonlinear functions, but its performance depends heavily on the initial weights, thresholds, and the choice of learning rate. If the initial weights fall into a poor region of the loss surface, the network may converge slowly or become trapped in a local minimum. Therefore, I adopted a global search technique to optimize these initial parameters.
Grey Wolf Optimizer
Grey Wolf Optimizer is an effective metaheuristic algorithm that mimics the social hierarchy and hunting behavior of grey wolves. The wolf population is divided into four levels: \(\alpha\), \(\beta\), \(\delta\), and \(\omega\), where \(\alpha\) is the best solution. During the hunt, the wolves surround the victim. The encircling behavior is modeled by:
$$ \mathbf{D} = |\mathbf{C}\cdot\mathbf{X}_p(t) – \mathbf{X}(t)| $$
$$ \mathbf{X}(t+1) = \mathbf{X}_p(t) – \mathbf{A}\cdot\mathbf{D} $$
where \(\mathbf{A}=2a\mathbf{r}_1 – a\) and \(\mathbf{C}=2\mathbf{r}_2\). The convergence parameter \(a\) decreases linearly from 2 to 0 over the course of iterations:
$$ a = 2 – \frac{2t}{T_{max}} $$
Since the position of the optimum is generally unknown, the GWO assumes that \(\alpha\), \(\beta\), and \(\delta\) are the first, second, and third best solutions. All other wolves update their positions according to these three leaders:
$$ \mathbf{D}_{\alpha} = |\mathbf{C}_1\cdot\mathbf{X}_{\alpha} – \mathbf{X}|, \quad \mathbf{D}_{\beta} = |\mathbf{C}_2\cdot\mathbf{X}_{\beta} – \mathbf{X}|, \quad \mathbf{D}_{\delta} = |\mathbf{C}_3\cdot\mathbf{X}_{\delta} – \mathbf{X}| $$
$$ \mathbf{X}_1 = \mathbf{X}_{\alpha} – \mathbf{A}_1\cdot\mathbf{D}_{\alpha}, \quad \mathbf{X}_2 = \mathbf{X}_{\beta} – \mathbf{A}_2\cdot\mathbf{D}_{\beta}, \quad \mathbf{X}_3 = \mathbf{X}_{\delta} – \mathbf{A}_3\cdot\mathbf{D}_{\delta} $$
$$ \mathbf{X}(t+1) = \frac{\mathbf{X}_1 + \mathbf{X}_2 + \mathbf{X}_3}{3} $$
When \(|\mathbf{A}|>1\), the wolves diverge to search globally; when \(|\mathbf{A}|<1\), they converge on the prey. This exploration–exploitation balance helps GWO avoid premature convergence and enhances global optimization ability.
GWO-BP Model Structure and Implementation
I fused the GWO algorithm with the BP neural network by using GWO to optimize the initial weights and thresholds of the BP network. The overall model structure was “18-12-5”. The input layer contained 18 nodes because the six Hall elements each provide three magnetic field components \((x,y,z)\). The hidden layer contained 12 nodes determined by experimental tuning. The output layer consisted of the five conductor state parameters \((X_p,Y_p,m,n,p)\).
The steps of the GWO-BP training process are:
- Normalize the input and output data to the range [0,1].
- Set the BP neural network architecture: input nodes \(=18\), hidden nodes \(=12\), output nodes \(=5\).
- Set the GWO parameters: population size \(pop = 20\), maximum iteration number \(Max\_iteration = 50\).
- In each iteration, treat the connection weights and biases of the BP network as the position vector of each grey wolf.
- Compute the fitness for each wolf by training the BP network on the dataset and evaluating the mean squared error.
- Update the three best wolf positions and reposition the remaining wolves.
- Repeat until the maximum iteration number is reached.
- Decode the best wolf position into the optimized initial weights and thresholds of the BP network.
- Train the final BP network and use it to predict the conductor state parameters.
The fitness function for the GWO was the mean square error of the predicted conductor state values:
$$ fitness = \frac{1}{n}\sum_{j=1}^{n} \sum_{i=1}^{5} \left( y_{ij} – z_{ij} \right)^2 $$
where \(n\) is the number of training samples, \(y_{ij}\) is the actual output value, and \(z_{ij}\) is the predicted value.
Performance Evaluation of the Conductor State Estimator
To verify the performance of the proposed model, I constructed a dataset using COMSOL Multiphysics finite-element simulations. The simulation model included a rectangular copper busbar with a cross-section of \(20\,\text{mm}\times4\,\text{mm}\) and the elliptical Hall array with the parameters obtained in Table 1. With the aid of the COMSOL-MATLAB interface, I simulated more than 2000 different conductor states by varying \(X_p\) in the range –10.5 mm to 10.5 mm, \(Y_p\) in the range –2.1 mm to 2.1 mm, \(\alpha\) in the range 0° to 45°, and \(\beta\) in the range 0° to 180°. The magnetic flux density at every Hall element was recorded and converted to the dataset input.
I used three evaluation metrics: mean absolute error (MAE), mean squared error (MSE), and mean absolute percentage error (MAPE). Their formulas are:
$$ MAE = \frac{1}{n}\sum_{i=1}^{n} \left| y_i – \hat{y}_i \right| $$
$$ MSE = \frac{1}{n}\sum_{i=1}^{n} \left( y_i – \hat{y}_i \right)^2 $$
$$ MAPE = \frac{100\%}{n}\sum_{i=1}^{n} \left| \frac{y_i – \hat{y}_i}{y_i} \right| $$
Table 6 compares the BP and GWO-BP estimators for all five output parameters.
| Output | Algorithm | MAE | MSE | MAPE |
|---|---|---|---|---|
| \(X_p\) | BP | 0.2272 | 0.0771 | 0.8419% |
| \(X_p\) | GWO-BP | 0.0498 | 0.0041 | 1.0411% |
| \(Y_p\) | BP | 0.0619 | 0.0043 | 5.2646% |
| \(Y_p\) | GWO-BP | 0.0146 | 0.0004 | 1.0621% |
| \(m\) | BP | 0.0124 | 0.0003 | 10.3490% |
| \(m\) | GWO-BP | 0.0037 | 0.0001 | 2.8585% |
| \(n\) | BP | 0.0071 | 0.0002 | 3.9567% |
| \(n\) | GWO-BP | 0.0028 | 0.0001 | 0.8626% |
| \(p\) | BP | 0.0143 | 0.0005 | 2.2479% |
| \(p\) | GWO-BP | 0.0029 | 0.0001 | 0.4173% |
The GWO-BP network outperformed the standard BP network in MAE and MSE for all five parameters. For the tilt vector components \(m\) and \(p\), the MAPE was also considerably reduced. For \(X_p\), the MAPE increased slightly, but the significantly lower MAE and MSE indicated a more stable prediction with fewer large deviations. In the traction battery pack environment, conductor eccentricity often reaches several millimeters, and tilt errors are nonnegligible. The improved estimation accuracy of GWO-BP is beneficial for reducing the final current measurement errors.
Error Compensation Strategy
Once the conductor state has been estimated by the GWO-BP network, I apply the following compensation procedure:
- Use the predicted \((X_p,Y_p,m,n,p)\) to reconstruct the integration region for the inclined conductor.
- Compute the field integrals from equations above for each Hall element position.
- Calculate the new Hall-element output components by projecting the computed field onto the element sensitive axes.
- Sum the resulting weighted tangential field to obtain a corrected current value.
This approach separates the global nonlinear inversion problem into two modules: a neural-network-based parameter estimator and an analytical magnetic field integrator. The combination effectively suppresses both eccentric error and tilt error.
Experimental Setup and Test Platform
To verify the complete system in a scenario matching a traction battery pack, I built a hardware test platform. The platform components are listed in Table 7.
| Module | Equipment model | Manufacturer | Key specification |
|---|---|---|---|
| Host PC | IPC-610-L | ADVANTECH | Industrial computer |
| Current generator | Sorensen SGX | AMETEK | Programmable DC source up to 600 A |
| Current reversal | Arduino-based IGBT board | Self-developed | Bidirectional current control |
| Data acquisition | DAQ6510 | KEITHLEY | 80-channel data logger |
| Sensor power supply | TH6313 | TONGHUI | 12 V DC output |
| Temperature chamber | B-TH-120B | BOYI | –40°C to 150°C, ±0.5°C |
The host PC sent control commands to the power source and the current reversal board. The DAQ6510 scanned all analog outputs, while the temperature chamber regulated the environmental temperature. The sensor under test was placed inside the chamber with the rectangular busbar passing through the center of the elliptical array. The communication with the sensor was implemented using CAN bus, and a LabVIEW-based software platform was developed for automated testing. The test software allowed me to set multiple current points and multiple temperature points in a sequence, and record the sensor outputs automatically. The software flow begins with parameter initialization, then executes loops over current points and temperature points, verifying the stability of both sources. Real-time data were displayed and saved for post-processing.
Finite Element Simulation
I used COMSOL Multiphysics to simulate the magnetic field of the rectangular busbar and to validate the magnetic field distribution used in the theoretical formulas. The simulation model included a rectangular copper bar with conductivity \(5.998\times10^{7}\,\text{S/m}\). The surrounding domain was set to air. After building the geometry and assigning material properties, I applied adaptive mesh refinement and solved the steady-state electric current and magnetic field equations. The simulation confirmed that the magnetic flux density contours around a rectangular busbar are approximately elliptical. This observation supports the use of an elliptical Hall array for the traction battery pack current sensor. The magnetic flux density distribution in the vicinity of the array is shown for a typical current of 1000 A. The field is strongest near the surface of the busbar and decays with distance, confirming that the six Hall elements sample a useful portion of the field while remaining within a sufficiently compact region.
Experimental Results and Discussion
Full-Temperature Calibration Results
After calibrating at \(-40\,^\circ\text{C}\), \(25\,^\circ\text{C}\), and \(125\,^\circ\text{C}\), I measured the residual current error for the full current range. At room temperature the compensated current error was reduced by 85.58%, and the maximum absolute error remained below 3‰. At the low-temperature point, the compensation reduced the error by 62.59%, and at the high-temperature point by 42.99%. In all temperature cases, the relative error was no greater than 3‰. This ensures stable operation of the sensor in a traction battery pack over the specified ambient temperature region.
De-eccentricity experiments
In the de-eccentricity experiment, I moved the rectangular busbar along the \(X\)- or \(Y\)-axis while recording the current measurement error. Figure 5-21 showed the error before optimization, after BP-based optimization, and after GWO-BP-based optimization. For the \(X\)-direction displacement, the uncompensated error reached 4.18% at 10 mm. The BP optimization reduced the error, but the GWO-BP optimization brought the error down to 1.46%, a reduction of 65.07%. For the \(Y\)-direction displacement, the GWO-BP algorithm reduced the maximum error from 0.94% to 0.51%, yielding a 45.74% improvement. The remaining error after compensation is acceptable because the practical maximum displacement is limited by the mechanical constraints inside a traction battery pack.
De-tilt experiments
In the tilt experiments, I fixed the conductor center at the array origin and rotated the conductor. For \(\beta=20^\circ\), the error versus \(\alpha\) was measured. The uncompensated sensor error grew sharply for \(\alpha>15^\circ\), reaching 12.37% at \(\alpha=45^\circ\). With the GWO-BP-based compensation, the error at \(\alpha=45^\circ\) dropped to 2.95%, a decrease of 76.15%. For \(\alpha=0^\circ\), the error versus \(\beta\) showed a local minimum near \(\beta=20^\circ\), and the GWO-BP algorithm reduced the maximum error by 62.92%. The improved robustness in the presence of substantial tilt is important for the harsh vibration environment of a traction battery pack.
Final accuracy verification
The fully calibrated and optimized sensor was tested at three temperatures and over a current range of \(-1500\,\text{A}\) to \(+1500\,\text{A}\). Three different sensor samples were measured. The residual relative error at every tested point stayed within 3‰, as shown in Table 8. The results confirm the repeatability and reliability of the proposed sensor for high-precision current measurement in a traction battery pack.
| Temperature | Maximum relative error (%) | Minimum relative error (‰) |
|---|---|---|
| –40°C | 0.249 | 0.021 |
| 25°C | 0.213 | 0.012 |
| 125°C | 0.286 | 0.033 |
Comparison with Conventional Solutions
To highlight the advantages of the proposed sensor, I compared its mass and footprint with a conventional fluxgate current sensor and a Hall-element circular array sensor. The results are summarized in Table 9. The proposed elliptical array demonstrates a significant reduction in both weight and installation area while maintaining acceptable worst-case eccentric and tilt errors after compensation.
| Sensor type | Mass (g) | Installation area (mm²) | Max X-eccentric error after compensation (%) | Max Y-eccentric error after compensation (%) |
|---|---|---|---|---|
| Fluxgate current sensor | 69.5 | 3290 | — | — |
| Circular Hall array | 15.5 | 2827.4 | ≤5.96 | ≤1.31 |
| Proposed elliptical Hall array | 11.9 | 908.5 | ≤1.46 | ≤0.51 |
The proposed sensor reduces weight by 82.9% compared with the fluxgate solution and installation area by 72.4% compared with the circular array. This compactness is highly valuable for a battery pack, where every millimeter of space is critical for maximizing energy density and simplifying thermal management.
Conclusions and Future Work
In this paper, I have presented the design, calibration, error analysis, and experimental validation of an elliptical Hall-element array current sensor for traction battery pack applications. The main conclusions are summarized as follows.
First, the UCSL method outperforms the projection method for sampling a discrete elliptical integration path. By selecting the optimal aspect ratio \(AR=0.33\) and six Hall elements, the sensor achieves a compact installation area of only 908.5 mm² while maintaining a theoretical error compatible with the requirements of a traction battery pack.
Second, the full-temperature-domain calibration algorithm, based on first-order least-squares fitting and temperature-drift compensation, effectively corrects the initial gain and offset errors. After calibration, the sensor maintains a relative error of less than 3‰ over the operational temperature range from \(-40\,^\circ\text{C}\) to \(125\,^\circ\text{C}\) and over a current range from \(-1500\,\text{A}\) to \(1500\,\text{A}\).
Third, the GWO-BP-based conductor state estimation method provides reliable estimates of the eccentric position and tilt direction. Combining these estimates with the reconstructed integration area significantly reduces the sensitivity against eccentric and tilted busbar installations. Experimental results show reductions of 65.07%, 45.74%, and 76.15% in \(X\)-eccentric, \(Y\)-eccentric, and tilt-induced errors, respectively.
Finally, the proposed current sensor is lightweight, compact, and free of magnetic cores, making it attractive for automotive mass production and easy integration into the battery management system of a traction battery pack.
Future work will address the effects of neighboring conductors and curved busbars in a real traction battery pack. Magnetic crosstalk from adjacent phase busbars can introduce additional errors, and the busbar may not always remain perfectly straight. I plan to extend the current inversion model to include interference currents and to develop an online adaptive calibration method that can be performed during vehicle operation. In addition, the real-time performance of the GWO-BP algorithm can be improved by model compression or by using more lightweight extreme learning machines. Finally, I intend to perform vehicle-level road tests to evaluate the long-term stability of the sensor in actual driving and charging cycles.