# Numerically Stable Implementation of Active Power Decoupling Control Strategies

Esther Kesse Ntow, Harminderjit Singh Toor, Lakshmi Varaha Iyer, Narayan

C Kar, Caniggia Viana

► To cite this version:

Esther Kesse Ntow, Harminderjit Singh Toor, Lakshmi Varaha Iyer, Narayan C Kar, Caniggia Viana. Numerically Stable Implementation of Active Power Decoupling Control Strategies. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0143. hal-05105924

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Distributed under a Creative Commons Attribution 4.0 International License
# Numerically Stable Implementation of Active Power Decoupling Control Strategies

Esther Kesse Ntow

University of Windsor
ntowe@uwindsor.ca

Harminderjit Singh Toor

University of Windsor
toor7@uwindsor.ca

Lakshmi Varaha Iyer

Magna International Inc.
LakshmiVaraha.Iyer@ieee.org

Narayan C. Kar

University of Windsor
nkar@uwindsor.ca

Caniggia Viana

University of Windsor

caniggia.viana@uwindsor.ca

*Index Terms*—Active power decoupling, control strategy, electric vehicle, power converters for EV, second harmonic ripple.

*Abstract*—In active power decoupling (APD), image ambiguity in functions like “sqrt” and “atan2” can destabilize capacitor voltage control, leading to glitches. This paper proposes a phase unwrapping technique (PUT) that stabilizes angle computation, eliminating glitches from atan2-induced image ambiguity. Simulations results recreate and eliminate atan2-induced glitches using the proposed PUT.

## I. INTRODUCTION

Instantaneous electrical power emanating from the AC end in single-phase converters in electric vehicles (EVs) and several other applications has a pulsing element at double the line's alternating current frequency in addition to a DC component. This pulsing element is the second harmonic ripple. In instances where power factor is less than one, the ripple current waveform crosses zero, causing the battery to undergo extra charge and discharge cycles, which can reduce its lifespan [1]. Various ripple elimination techniques have been proposed to eliminate this issue.

Passive solutions are bulky, owing to the high energy storage required to divert pulsating power. Current literature has redirected the focus of ripple elimination to active power decoupling (APD) strategy [2]. Control strategies are presently employed in the majority of APD techniques as seen in [1] - [9] to minimize the maximum energy rating, thus, optimize the asset utilization ratio, reduce the need for extra active electronic components, lower cost, as well as utilize closed-loop feedback control technique to optimize the overall APD efficiency [8].

A direct ripple power cancellation method based on an energy-storage inductor was implemented in

[3] to improve the power-decoupling performance and to accurately track current by using a multi-resonant controller with feed-forward control [3]. To eliminate ripple, various buffer circuitry and control strategies have been developed, categorized into three methods. The first involves APD control on the DC side, requiring a high-bandwidth controller to track high-frequency power fluctuations. However, due to the limited switch count in the DC configuration, it is suitable only for low-power applications [4]. The second method uses a separate dedicated energy port with an external circuit to control high-frequency ripples on the AC side [5]. While effective for high-power applications, this method increases weight and cost, contradicting the goals of achieving lower cost and high-power density in EVs. The third method involves compensation on the AC side, employing advanced control techniques to manage ripples within the AC circuitry [6], [7]. The latter increases converter efficiency and optimizes the utilization of storage devices, making it a suitable APD method for EV applications. In [8], the dq0 coordinate transformation is used for ripple mitigation by transforming AC quantities into DC components, allowing for independent control of these components.

Although [8] and [9] successfully eliminated ripple with an efficient strategy that meets the desired requirements in EVs, the implementation of the APD technique's control may encounter numerical instability due to the use of the “atan2” function for angle computation, especially when multiplied by $\frac{1}{2}$. The “atan2” function introduces a branch cut at $\pm\pi$, selecting one of the two possible angle solutions for the complex number. While either solution is valid, switching between them causes a sudden phase shift. This discontinuity introduces large step transients in control reference, leading to potentially destructive current glitches.

To address the numerical instability caused by the
image discontinuity in the “atan2” function, this paper proposes a numerically stable phase unwrapping technique (PUT). State-of-the-art solutions, such as the CORDIC algorithm, offer improved stability to these discontinuities, but still exhibit residual glitches that can compromise system reliability. eliminates these discontinuities, ensuring smooth angle computation and enhancing the stability of the decoupling capacitor voltage control. PUT is validated with simulations using the APD strategy presented in [8], showing how the proposed solution makes existing APD strategies, such as ones based on power phasors, implementable.

## II. SECOND ORDER RIPPLE POWER

This section analyses the second order ripple power using the power phasor as in [8]. In generality, the power equation (1) will be extensively utilized throughout this paper to underpin the various analyses and control strategies discussed.

$$p(t) = i(t) \times v(t) \quad (1)$$

where p is the power, i is current and v is voltage.

### A. Instantaneous Power from the Grid

The grid voltage and current can be defined as

$$v_g(t) = \hat{v}_g \cos(\omega t) \quad (2)$$

$$i_g(t) = \hat{i}_g \cos(\omega t + \theta) \quad (3)$$

where $\hat{i}_g$ and $\hat{v}_g$ are the peak magnitudes of current and voltage and $\theta$ is the phase difference between the two phases. The phasor representation of (2) and (3) are as follows

$$\vec{v}_g = \hat{v}_g e^{j\omega t} \quad (4)$$

$$\vec{i}_g = \hat{i}_g e^{j\omega t + \theta} = \hat{i}_g \angle \theta e^{j\omega t} \quad (5)$$

The instantaneous ac power from the grid using (1) is

$$P_g(t) = \underbrace{\frac{1}{2} \hat{v}_g \hat{i}_g \cos(2\omega t + \theta)}_{\text{2nd order ripple power}} + \frac{1}{2} \hat{v}_g \hat{i}_g \cos(\theta) \quad (6)$$

The power phasor representation of (6) is as shown below

$$\vec{P}_g = \underbrace{\frac{1}{2} \hat{v}_g (\hat{i}_g \angle \theta) e^{j2\omega t}}_{\vec{p}_r} + \underbrace{\frac{1}{2} \hat{v}_g (\hat{i}_g \angle -\theta) e^{j0\omega t}}_{\vec{p}_r} \quad (7)$$

The phasor representation of the second order ripple power is denoted as $\vec{p}_r$. Part of the oscillating ripple flows to the inductor between the inverter and the grid.

The inductance consumes and releases part of the ripple power and the phasor representation of the power is $\vec{p}_L$.

$$\vec{p}_L = \frac{1}{2} \vec{i}_L \vec{v}_L = \frac{1}{2} j \omega L (\vec{i}_g)^2 \quad (8)$$

The remaining instantaneous ripple power flowing through the converter to the output $p_{out}$ becomes

$$\vec{p}_{out}(t) = P_g + \vec{p}_r - \vec{p}_L \quad (9)$$

The unwanted power phasor i.e the 2nd order ripple power flowing to the output of the converter is

$$\vec{p}_{out} = \vec{p}_r - \vec{p}_L \quad (10)$$

Computing (10) and assuming $\theta = 0$ results in

$$\vec{p}_{out} = \left( \frac{1}{2} \hat{v}_g \hat{i}_g - \frac{1}{2} j \omega L \vec{i}_g^2 \right) e^{j2\omega t} \quad (11)$$

Using an $\alpha\beta$-to-dq transformation, the output power is transformed to obtain the rotating reference frame (RRF) representation.

$$\vec{p}_{out,dq} = p_{out,d} + j p_{out,q} = \vec{p}_{out} e^{j2\omega t} L(\vec{i}_g)^2 \quad (12)$$

$$\vec{p}_{out,dq} = \frac{1}{2} \hat{v}_g \hat{i}_g - \frac{1}{2} j \omega L \vec{i}_g^2 \quad (13)$$

### B. Active Power Decoupling

To completely eliminate ripple power, an auxiliary capacitor is utilized to decouple the output power $\vec{p}_{out}$ flowing to the batteries. The auxiliary capacitor's power, $\vec{p}_{cap}$, is a function of the capacitor voltage [8], using the following expression

$$\vec{p}_{cap} = \frac{1}{2} j \omega C_{cap} (\hat{v}_{cap} e^{j\theta_{cap}})^2 \quad (14)$$

where $\hat{v}_{cap}$ is the magnitude of the voltage applied to the capacitor and $\theta_{cap}$ is the phase angle of the capacitor voltage. Obtaining the RRF of $\vec{p}_{cap}$ as in (12) yields

$$\vec{p}_{cap,dq} = \frac{1}{2} j \omega C_{cap} (\vec{v}_{cap,dq})^2 \quad (15)$$

The output power phasor flowing to the batteries then becomes

$$\vec{p}_{out} = \vec{p}_r - \vec{p}_L - \vec{p}_{cap} \quad (16)$$

To achieve the objective of APD of making $\vec{p}_{out}$ zero, the voltage applied to the capacitor $\vec{v}_{cap}$ is regulated to achieve a particular value of $\vec{p}_{cap}$ that nullifies $\vec{p}_{out}$. This is the control strategy presented in [8].
<figure><img src="image_2.png" /><figcaption>Fig. 1: Unit circle.</figcaption></figure>

### III. APD CAPACITOR VOLTAGE ANGLE AND THE “ATAN2” FUNCTION

#### A. Decoupling Capacitor Voltage

The capacitor voltage from (15) is

$$ (\vec{v}_{cap,dq})^2 = \frac{2j\vec{p}_{cap,dq}}{\omega C_{cap}} \quad (17) $$

The phasor representation of (17) is

$$ (\vec{v}_{cap,dq})^2 = (\hat{v}_{cap,dq} \angle \theta_{cap,dq})^2 \quad (18) $$

The magnitude of the capacitor voltage $\hat{v}_{cap,dq}$ and its angle $\theta_{cap,dq}$ are as follows

$$ (\vec{v}_{cap,dq}) = d + qj \quad (19) $$

$$ \hat{v}_{cap,dq} = \sqrt{d^2 + q^2} \quad (20) $$

$$ \theta_{cap} = atan2(q, d) \quad (21) $$

$$ \theta_{cap,dq} = \frac{1}{2}(\theta_{cap}) \quad (22) $$

#### B. The “atan2” Function Problem

Considering the unit circle with four quadrants in Fig. 1, the “atan2” function as used in MATLAB extends the range of arctan from $[-\frac{\pi}{2}, \frac{\pi}{2}]$ to $[-\pi, \pi]$ by adding $\pi$ and $-\pi$ respectively.

$$ atan2 = \begin{cases} \arctan(\frac{q}{d}), & \text{if } q > 0, d > 0 \text{ (I)} \\ \arctan(\frac{q}{d}) + \pi, & \text{if } q < 0, d > 0 \text{ (II)} \\ \arctan(\frac{q}{d}) - \pi, & \text{if } q < 0, d < 0 \text{ (III)} \\ \arctan(\frac{q}{d}), & \text{if } q > 0, d < 0 \text{ (IV)} \end{cases} $$

The complex $v_{cap,dq}$ is presented in phasor as

$$ \vec{p}_{cap,dq} = |\vec{p}_{cap,dq}| e^{j \cdot atan2(\frac{q}{d})} \quad (23) $$

from (17),

$$ \vec{v}_{cap,dq} \propto \sqrt{\vec{p}_{cap,dq}} \quad (24) $$

therefore,

$$ (\vec{p}_{cap,dq})^{\frac{1}{2}} = |\vec{p}_{cap,dq}|^{\frac{1}{2}} e^{j \cdot \frac{1}{2} \cdot atan2(\frac{q}{d})} \quad (25) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2: Domain coloring phase plot of RRF capacitor power. (a) $\vec{p}_{cap,dq}$ (b) $(\vec{p}_{cap,dq})^{\frac{1}{2}}$.</figcaption></figure>

The domain coloring plots in Fig. 2 visualizes complex functions by mapping complex numbers to colors. Hue represents the argument (angle) of the complex number, while brightness indicates magnitude. Concentric circles show constant magnitudes. These plots help identify zeros, poles, and critical points, offering a clear view of the function's behavior across the complex plane in this case discontinuities [10]. In (25), the expression is not a function because it produces two outputs for a single input. To ensure a single output for each input, a line of discontinuity, known as a "branch cut", is introduced to establish boundaries. In "atan2", this branch cut occurs at $\pm\pi$ [11]. While the branch cut allows $(\vec{p}_{cap,dq})^{\frac{1}{2}}$ to function correctly by providing a single output, it also introduces discontinuity, as shown in the domain plots.

The complex number $\vec{p}_{cap,dq}$ is characterized by both its magnitude and phase, as depicted (23). In computing the square root of a complex number, the angle is halved as shown in (25). The corresponding domain coloring plot for $(\vec{p}_{cap,dq})^{\frac{1}{2}}$ is shown in Fig. 2b. The hue in Fig. 2b changes half as rapidly as in Fig. 2a in both clockwise and anticlockwise direction. As a result, disappearance of green and blue hue is observed indicating that output of complex square root function has phase range $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Observed sudden change in color at $\pi$ means output abruptly changes from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

In APD control strategies in [8] and [9], the computation of angles using the “atan2” may encounter numerical instability due to the use of the “atan2” function for angle
computation, especially when multiplied by $\frac{1}{2}$. The “atan2” function introduces a branch cut at $\pm\pi$, selecting one of the two possible angle solutions for the complex number. While either solution is valid, switching between them causes a sudden phase shift. This discontinuity disrupts smooth angle computation, particularly in APD control strategies, leading to potential glitches.

These discontinuities in the complex square root function in (24) cause corresponding discontinuities in control parameters when applied in these APD control strategies hence the need for the implementation of the proposed technique in Section IV.

## IV. PROPOSED NUMERICALLY STABLE APPROACH

The proposed PUT in Fig. 3c converts discontinuous angle $\theta_{cap,disc}$ to a continuous angle $\theta_{cap,cont}$ by continuously monitoring the output angle $\theta_{cap}$ from the “atan2” function, detecting potential discontinuities, and applying the technique. The initial angle $\theta_{cap}$ susceptible to abrupt changes at boundaries of $\pm\pi$ is calculated using the conventional “atan2” function. The output angle from “atan2” $\theta_{cap}$ is delayed by one time step and subtracted from current angle $\theta_{cap}$ to obtain the difference $\theta_{cap,diff}$ which reveals any potential phase discontinuity.

$$ \theta_{cap,diff} = \theta_{cap,delayed} - \theta_{cap}. \quad (26) $$

Discontinuity detection:

$$ \text{If } |\theta_{cap, diff}| = \begin{cases} > \pi, & \text{discontinuous} \\ < \pi, & \text{continuous} \end{cases} $$

$$ \theta = \begin{cases} \theta_{cap,diff}, & \text{if } |\theta_{cap,diff}| > \pi \\ 0, & \text{if } |\theta_{cap,diff}| \le \pi \end{cases} $$

Upon detecting a discontinuity, the PUT adjusts the delayed output angle $\theta_{cap, delayed}$ by subtracting $\theta_{cap, diff}$ which is equal to the resulting angle $\theta$, effectively smoothing the transition by “unwrapping” the phase

$$ \theta_{cap} = \theta_{cap,delayed} - \theta_{cap,diff} \quad (27) $$

This adjustment ensures that the final output angle $\theta_{cap, cont}$ is continuous and devoid of any abrupt jumps, which enhances the stability and reliability of the control system. The rectified angle $\theta_{cap}$ is then halved to obtain $\theta_{cap,dq}$ which is then passed on as the final output as shown in Fig. 3b.

## V. PERFORMANCE ANALYSIS WITH CORDIC ALGORITHM

CORDIC is an acronym for Coordinate Rotation Digital Computer. The fundamental and traditional concepts of two-dimensional geometry form the foundation of the CORDIC algorithm. It can be used to calculate trigonometric functions including “atan2” [12]. The CORDIC algorithm improves on the traditional “atan2” function by avoiding explicit trigonometric evaluations and instead using iterative vector rotations to compute angles. This approach eliminates the reliance on complex mathematical functions, significantly reducing numerical instability and minimizing the glitches caused by branch cuts at $\pm\pi$. Its efficiency and hardware-friendly nature make it ideal for real-time control applications such as APD. However, despite its inherent advantages, the CORDIC algorithm still exhibits residual glitches in angle computation, as demonstrated in Fig. 5b, Fig. 5e and Fig. 5k. These glitches highlight the need for further refinement, which is effectively addressed by the proposed PUT.

## VI. POWER PHASOR APD IMPLEMENTATION AND VALIDATION OF PROPOSED PUT APPROACH

The efficiency of the proposed approach is simulated using a single-phase AC/DC converter with two full bridges for power transfer and active power decoupling. Both architectures are shown in Fig. 3a and Fig. 3b respectively. The output power flowing the battery $\vec{p}_{out}$ serves as an input to the PI controller to determine the voltage $\vec{v}_{cap,dq}$ to be applied to the capacitor. Using the measured power $p_{out}(t)$, the voltage $\vec{v}_{cap,dq}$ is computed and then transformed back to the stationary reference frame (SRF) via the reverse Park Transformation, as depicted in Fig. 3b. Once $\vec{v}_{cap,dq}$ is determined, both the capacitor voltage $\vec{v}_{cap,dq}$ and the computed angle $\theta_{cap,dq}$ are utilized to convert the dq components to the $\alpha\beta$ components.

The AC supply $v_g$ from the grid flows through the inductor $L$, which smoothens out the high-frequency components. The single-phase AC/DC converter Fig. 4 with two full bridges operate to rectify the AC voltage into a DC voltage to supply the battery. The auxiliary capacitor $C_{cap}$ absorbs the second harmonic ripple at a controlled voltage level which makes $\vec{p}_{out}$ zero. The simulation is based on the phasor concept APD control presented in [8]. Fig. 3b was implemented with the proposed PUT disabled. As anticipated, the resulting waveforms showed glitches caused by the discontinuity of the “atan2” function, as demonstrated in Section II. The conventional “atan2” approach in Fig. 3d introduces
<figure><img src="image_4.png" /><figcaption>Fig. 3: Proposed control architecture, defining the desired voltages $v_a$, $v_b$, and $v_c$, to be applied by a suitable modulator.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Converter topology used for simulation.</figcaption></figure>

glitches in the angle $\theta_{cap,dq}$ of $\vec{p}_{cap,dq}$ as shown in Fig. 5a.

These glitches affect the decoupling capacitor-controlled voltage and the resulting current waveform, as illustrated in Fig. 5d and Fig. 5g.

The glitch caused by the discontinuity is further highlighted in Fig. 5j, which shows the control reference capacitor voltage $v_{cap,ref}$ and the actual voltage $v_{cap}$. The simulation was then implemented with the proposed PUT enabled. Fig. 5f and Fig. 5l show the capacitor voltage using the proposed technique, which demonstrates no glitches. Fig. 5l presents the control reference capacitor voltage $v_{cap,ref}$ and the actual voltage $v_{cap}$, both without glitches. Whilst the CORDIC algorithm significantly
<figure><img src="image_6.png" /><figcaption>Fig. 5: Comparison of output power angle, voltage, and current waveforms of the decoupling capacitor using (a, d, g, j) “atan2” approach, (b, e, h, k) CORDIC algorithm, and (c, f, i, l) PUT implementation.</figcaption></figure>

eliminates the glichthes as seen in Fig. 5e it does not totally eliminate it. The proposed PUT approach in Fig. 3d completely eliminates glitches in the angle $\theta_{cap,dq}$ of $\vec{p}_{cap,dq}$, as shown in Fig. 5c subsequently improving the stability of the current and voltage waveforms of the decoupling capacitor in Fig.5f and 5l making control more stable, as evidenced by the results, proving the effectiveness of the proposed PUT.

<table><caption>TABLE I: Simulation Parameters</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td></tr></thead><tbody><tr><td>$v_{battt}$</td><td>Voltage across Battery</td><td>250V</td></tr><tr><td>$v_g$</td><td>Grid Voltage</td><td>120V<sub>rms</sub></td></tr><tr><td>$f_{sw}$</td><td>Switching Frequency</td><td>10kHz</td></tr><tr><td>$\dot{C}_{cap}$</td><td>Auxiliary Capacitance</td><td>1mF</td></tr><tr><td>$L$</td><td>Machine leakage Inductance</td><td>150µH</td></tr></tbody></table>

## VII. CONCLUSION

The discontinuity of the “atan2” function causes abrupt transitions between $\pi$ and $-\pi$, resulting in glitches that induce spikes in the current flowing to the decoupling capacitor. These spikes compromise system efficiency, stability, and reliability. This paper proposes a phase

unwrapping technique that stabilizes angle computations in APD control strategies. The PUT continuously monitors the output angle from the “atan2” function, detects potential discontinuities, and applies corrective measures.
Simulation results demonstrate that the effectiveness of
the proposed technique eliminating these glitches, leading
to a more stable and reliable APD control implementation.

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