An Adaptive Extended Kalman Filter with Passive Control for DC-DC Converter Supplying Constant Power and Constant Voltage Loads

Abstract

This article introduces an integrated control scheme combining an Adaptive Extended Kalman Filter (AEKF) with a Passivity-Based Control (PBC) approach to stabilize a DC-DC boost converter feeding both constant voltage and constant power loads (CPLs) in DC microgrids. Unlike conventional observers, the AEKF adapts its covariance matrices in real time to accurately estimate both system states and the unknown load dynamics introduced by CPLs, thereby eliminating the need for additional sensors and enhancing estimation convergence. Coupled with the PBC, the estimated disturbances are compensated via a feedforward path, significantly improving the system’s resilience to input voltage fluctuations and load variations. Through a Lyapunov-based stability analysis, the combined strategy is proven to ensure large-signal stability while maintaining a rapid transient recovery profile, even under significant parametric uncertainties. The simulation of this algorithm was implemented using PLECS, thoroughly validating the effectiveness and robustness of the proposed method.

1. Introduction

In recent years, amid the technological evolution of power systems, DC microgrids have emerged as a promising integrated architecture incorporating renewable energy sources, DC loads, and energy storage systems. Compared to their AC counterparts, DC microgrids offer distinct advantages, including a simpler structure, relatively lower control complexity, and the absence of reactive power and phase-related issues [1,2,3]. Their distributed control schemes further contribute to enhanced response speeds. Within the DC microgrid framework, the DC-DC boost converter is a critical component, necessitating high bidirectional energy conversion efficiency [4,5,6]. Consequently, developing novel control strategies for Boost converters that account for the influence of constant power loads (CPLs), mitigate disturbances, and enable rapid voltage regulation is of paramount importance.

Power electronic loads in DC microgrids are primarily categorized into resistive loads and constant power loads. Resistive loads, due to their passive nature, introduce positive damping into the system [7,8,9]. In contrast, CPLs, which maintain constant power draw, exhibit negative impedance characteristics at their input terminals. This negative impedance interaction with other system converters can reduce overall system damping, compromise stability, and introduce oscillation risks [10,11,12,13]. The increasing penetration of renewable energy sources further exacerbates these stability challenges posed by CPLs. Power system stability analysis is typically divided into large-signal [14,15,16,17,18] and small-signal [19,20,21] stability, based on operating scenarios and disturbance magnitudes, with current research predominantly focused on the latter. Primary mitigation strategies include passive damping [20,21], which adds passive components to improve damping directly but increases cost and weight, counter to lightweight trends; and active damping [22,23], which modifies the control structure to reshape system impedance. However, active damping is often limited to linearized small-signal models around an equilibrium point, restricting its applicability.

Given the inherent nonlinearity of power electronic converters and the negative impedance of CPLs, employing nonlinear control strategies is essential to enhance the global stability of DC microgrid systems. To improve converter stability, nonlinear control methods such as Passivity-Based Control (PBC) [24], Back-stepping Control (BSC) [25], Model Predictive Control (MPC) [26], Sliding Mode Control (SMC) [27], and PWM-based nonlinear control [28] have been progressively applied. However, challenges remain in improving output voltage tracking accuracy and system efficiency, leading to extensive research on combined control strategies. The Back-stepping Control method proposed in [29] is an effective nonlinear approach for enhancing tracking precision and system efficiency, but it requires accurate estimation of system uncertainties and disturbances to minimize tracking error. Observer technology, particularly the Nonlinear Disturbance Observer (NDO) [24], offers a promising solution for estimating disturbances without additional sensors, providing effective disturbance rejection and uncertainty compensation in nonlinear systems. For instance, ref. [30] employed an NDO combined with Back-stepping Control to enhance the large-signal stability of a conventional Boost converter system. While these strategies effectively ensure large-signal stability, their high dependency on accurate system modeling makes them susceptible to inevitable steady-state deviations under random source and load disturbances. Integrating compensation loops at relevant stages can reduce or eliminate these deviations, but often at the expense of dynamic response speed.

To address the aforementioned challenges, this paper proposes a Passivity-Based Control (PBC) method for DC-DC converters based on an Adaptive Extended Kalman Filter (AEKF). This approach not only significantly enhances the large-signal stability and robustness of the system but also incorporates precise current limiting control and rapid voltage regulation capabilities, thereby effectively ensuring the stability and reliability of DC microgrids under various complex operating conditions. The proposed control methodology primarily integrates PBC with Kalman filtering techniques. Firstly, an AEKF is designed for the real-time and accurate estimation of time-varying system parameters, notably load power and input voltage. Subsequently, an adaptive PBC technique, utilizing the state variables estimated by the filter, is developed to control the DC-DC boost converter with a CPL. Furthermore, the design of the proposed nonlinear control loop circumvents the difficulties associated with obtaining nominal parameters and performing complex multi-degree-of-freedom parameter tuning inherent in traditional methods, resulting in a relatively simple and practical control structure.

2. System Modeling and Problem Formulation

The object of this study is a typical DC microgrid architecture and its equivalent circuit, depicted in Figure 1. This system features a source bus that is intricately connected to a diversified array of power generation and storage technologies through appropriate power electronic interfaces. These include intermittent renewable sources like photovoltaic arrays and wind turbines, dispatchable generators, battery energy storage systems (BESS), and a point of common coupling to an external AC grid via an AC-DC converter. This multi-faceted integration strategy empowers the source bus to dynamically and flexibly draw power from the most available or economical sources, thereby significantly enhancing the overall reliability, resilience, and operational adaptability of the microgrid. The system’s energy management system (EMS) employs sophisticated optimization algorithms to coordinate power flows, ensuring optimal utilization of renewable resources while maintaining the state of charge of the BESS for grid support services.

A pivotal component in this architecture is the DC-DC boost converter, which interfaces the source bus with the load bus. Its primary function is twofold: first, to ensure precise and stable voltage regulation on the load bus, which is fundamental for high-quality power delivery; and second, to efficiently step up the variable, often lower voltage from the source side to a higher, regulated voltage level required by the loads. The converter’s design must account for wide input voltage variations characteristic of renewable sources, necessitating advanced control techniques to maintain stability across the entire operating range. The performance of this converter is therefore critical, demanding high conversion efficiency and robust control capabilities to maintain a continuous and stable power supply under varying generation and consumption scenarios. Its dynamic response must be sufficiently fast to mitigate disturbances propagating through the system, particularly those originating from load-side perturbations.

The load bus supplies power to a mix of load types, primarily categorized into resistive loads and CPLs. Resistive loads present a linear relationship between current and voltage, contributing naturally to system damping and stable operation. In stark contrast, CPLs, such as tightly regulated power electronic converters found in electric vehicle charging stations, server farms, and variable-speed drive systems, maintain a constant power draw. This behavior manifests as a negative impedance characteristic at their input terminals, meaning an increase in input voltage results in a decrease in input current. This negative impedance introduces non-linearity and can significantly degrade system damping through positive feedback mechanisms. The interaction between this negative impedance and the source-side converters can lead to voltage oscillations, instability, and even system collapse, presenting a major challenge for DC microgrid stability. This instability phenomenon is further exacerbated in cascaded converter systems, where the impedance interaction between source and load converters can create forbidden operating regions. Consequently, the design and implementation of sophisticated, robust control strategies for the interface DC-DC converter are not just beneficial but imperative to mitigate these destabilizing effects and ensure the stable, reliable, and efficient operation of the entire DC microgrid system. Advanced control methodologies must incorporate both large-signal stability guarantees and adaptive capabilities to handle the time-varying nature of modern load profiles.

A mathematical model of the DC-DC boost converter needs to be established to serve as the foundation for subsequent controller design and stability analysis [31]. Without loss of generality, and assuming continuous conduction mode, the average dynamic model of the DC-DC boost converter can be represented by the following state-space equations:

$$\left\{\begin{array}{l}L\frac{d{i}_L}{dt}=E-(1-\mu )v_o\\ C\frac{d{v}_o}{dt}=(1-\mu )i_L-\frac{v_o}{R}-\frac{P_{CPL}}{v_o}\end{array}\right.$$

(1)

where L, C, E, v_o, i_L, μ, R, and $P_{CPL}$ represent the inductance, output capacitance, input voltage, output voltage, inductor current, switching duty cycle, resistive load, and constant power load of the DC-DC boost converter, respectively. Furthermore, the input voltage and the power of the constant power load are state variables that require estimation.

This model captures the essential nonlinear dynamics of the system. The term $\frac{P_{CPL}}{v_o}$ explicitly embodies the negative impedance characteristic of the CPL, as the current it draws decreases with increasing voltage. It is crucial to note that within the context of the proposed control strategy, the power of the constant power load $P_{CPL}$ are treated as unknown time-varying parameters or state variables that require real-time estimation for precise control synthesis.

3. Adaptive Extended Kalman Filter (AEKF) and Stability Analysis

3.1. Adaptive Extended Kalman Filter (AEKF)

To accurately estimate the state variables using an Extended Kalman Filter, the dynamic model (1) is rewritten and can be expressed in the kinetic form of state-space representation as [32]:

$$\left\{\begin{array}{l}\dot{x}=Sx+d\rho +BE\\ y=hx\end{array}\right.$$

(2)

In Equation (2), the state variable matrix is $x={\left[\begin{array}{cc}{i}_L& v_0\end{array}\right]}^T$.

E is the input voltage, S is the system matrix, B is the input matrix, d is the nonlinear term matrix, h is the output matrix, y is the output quantity, and $\rho$ is the influence factor based on the capacitor voltage. Then:

$$S=\left[\begin{array}{cc}0& -{\displaystyle \frac{1-\mu }{L}}\\ {\displaystyle \frac{1-\mu }{C}}& -{\displaystyle \frac{1}{RC}}\end{array}\right],B=\left[\begin{array}{cc}0& {\displaystyle \frac{1}{L}}\\ 0& 0\end{array}\right],d=\left[\begin{array}{c}0\\ -{\displaystyle \frac{P_{CPL}}{C}}\end{array}\right],h=\left[\begin{array}{cc}1& 1\end{array}\right]$$

(3)

$$\rho ={\displaystyle \frac{1}{v_0}}$$

(4)

Based on the proposed Extended Kalman Filter method, the power of the constant power load needs to be considered as one of the state variables and added to the state-space expression, resulting in a new state variable matrix:

$$x_{a\mathrm{e}kf}=\left[{\displaystyle \frac{x}{P}}\right]$$

(5)

Due to the power characteristics of the constant power load, it follows that P equals 0. At this point, the state variable matrix becomes:

$$x_{a\mathrm{e}kf}=\left[{\displaystyle \frac{x}{0}}\right]=f\left(x_{a\mathrm{e}kf},u\left(t\right)\right)$$

(6)

where $u\left(t\right)$ is the time-varying system input, and $f$ is a function of $u\left(t\right)$ and $x_{a\mathrm{e}kf}$, representing the Euler equation of the dynamic model after incorporating the output power into the state variable matrix. Furthermore, the output of the EKF algorithm is updated based on changes in the state variables:

$$y=\left[h|0\right]\left[\begin{array}{c}x\\ P\end{array}\right]$$

(7)

After adding noise, the new form of the equation set is obtained, $\omega$ and $\nu$ can be considered as Gaussian white noise with zero mean. Their covariance matrices are denoted as $Q$ and $R$, respectively, and their specific values need to be specified by the user during application. The state noise $Q$ can be determined through adaptive training, while the observation noise is generally related to sensor accuracy. The following equations are obtained:

$$\left\{\begin{array}{l}{\dot{x}}_{a\mathrm{e}kf}=f\left(x_{aekf},u\left(t\right)\right)+\omega \\ y=h{x}_{aekf}+\nu \end{array}\right.$$

(8)

Discretization is performed using the Euler method:

$$\left\{\begin{array}{l}{x}_{a\mathrm{e}kf,k+1}=x_{a\mathrm{e}kf,k}+T_{s}f\left(x_{a\mathrm{e}kf,k},u{\left(t\right)}_k\right)+{\omega }_k\\ y_k=\left[h|0\right]x_{a\mathrm{e}kf,k}+{\nu }_k\end{array}\right.$$

(9)

where $f\left(x_{aukf,k},u{\left(t\right)}_k\right)$ represents the system dynamic model at time k, and $T_s$ is the sampling time. The nonlinear function $f\left(x_{aukf,k},u{\left(t\right)}_k\right)$ is incompatible with the EKF. Instead, its Jacobian, given in (10), is employed within the EKF algorithm.

$$F_k={\displaystyle \frac{\partial f}{\partial x_{\mathrm{aekf}}}}{\mid }_{({\widehat{x}}_{\mathrm{aekf},k},u_k,k)}$$

(10)

Here, ${\widehat{x}}_{\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{f},k}$ denotes the parameter estimated by the EKF at the k-th step. The term $f_{{\widehat{x}}_{\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{f},k},u_k,k}$ represents the dynamic model utilizing these estimated values. As a result, for the initial values of ${\widehat{x}}_0$, $p_0$, $R_0$, and $Q_0$, the EKF algorithm consists of two stages: the time update and the measurement update, as given below.

$$\left\{\begin{array}{l}{\widehat{x}}_{\mathrm{aekf},k}^{-}=T_{s}f\left({\widehat{x}}_{\mathrm{aekf},k-1},u_{k-1}\right)+{\widehat{x}}_{aekf,k-1}\hfill \\ P_{\mathrm{aekf},k}^{-}=F_{k-1}{p}_{\mathrm{aekf},k-1}{F}_{k-1}^T+Q_{k-1}\hfill \\ Q_k=\lambda Q_{k-1}+(1-\lambda )\left(K_{\mathrm{aekf},k}{d}_{\mathrm{aekf},k}{d}_{\mathrm{aekf},k}^T{K}_{\mathrm{aekf},k}^T\right)\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{l}{K}_{\mathrm{aekf},k}=p_{\mathrm{aekf},k}^{-}{H}_k^T{\left(H_k{p}_{\mathrm{aekf},k}^{-}{H}_k^T+R_K\right)}^{-1}\hfill \\ {\widehat{x}}_{\mathrm{aekf},k}=K_{k,aekf}\left(y_k-H{\widehat{x}}_{\mathrm{aekf},k}^{-}\right)+{\widehat{x}}_{\mathrm{aekf},k}^{-}\hfill \\ p_{\mathrm{aekf},k}=(I-K_{k,\mathrm{aekf}}{H}_k)p_{aekf,k}^{-}\hfill \\ R_k=\lambda R_{k-1}+(1-\lambda )\left({\epsilon }_{\mathrm{aekf},k}{\epsilon }_{aekf,k}^T+H_k{p}_{\mathrm{aekf},k}^{-}{H}_k^T\right)\hfill \end{array}\right.$$

(12)

The calculations for ${\epsilon }_{\mathrm{a}\mathrm{e}\mathrm{k}\mathrm{f},k}$ and $d_{\mathrm{a}\mathrm{e}\mathrm{k}\mathrm{f},k}$ are as follows:

$$\left\{\begin{array}{l}{d}_{\mathrm{ekf},k}=[y_k-H{\widehat{x}}_{\mathrm{ekf},k}^{-}]\hfill \\ {\widehat{x}}_{\mathrm{ekf},k}^{+}={\widehat{x}}_{\mathrm{ekf},k}^{-}+K_{\mathrm{ekf},k}{d}_{\mathrm{ekf},k}\hfill \\ {\epsilon }_{\mathrm{ekf},k}=[y_k-H{\widehat{x}}_{\mathrm{ekf},k}^{+}]\hfill \end{array}\right.$$

(13)

In the proposed algorithm, the prediction step prior to measurement utilization yields the predicted state ${\widehat{x}}_{ackf,k}^{-}$ and its covariance $P_{ackf,k}^{-}$, computed via the dynamic model $f\left({\widehat{x}}_{ackf,k-1},u_{k-1}\right)$. Following the measurement update, these are refined to the estimated state ${\widehat{x}}_{ackf,k}$ and covariance $P_{ackf,k}$. The update magnitude is governed by the EKF gain $K_{ackf,k}$. A forgetting factor $\lambda \left(0 <\lambda \le 1\right)$ is introduced to weight previous estimates, where a larger $\lambda$ increases this influence. This factor directly modulates $R_k$, inducing adaptive time delays to compensate for changes. The linearized system is subsequently derived as follows:

$$F_k=I+T_s{\displaystyle \frac{\partial f\left(x_{\mathrm{aekf},k},u_k\right)}{\partial x_{\mathrm{aekf},k}}}$$

(14)

The Jacobian matrix $F_k$ is computed as ${\displaystyle \frac{\partial f\left(x_{ackf,k},u_k\right)}{\partial x_{ackf,k}}}$ for all state variables, where $f\left(x_{ackf,k},u_k\right)$ is the dynamic model and $I$ is the identity matrix.

Ultimately, the AEKF algorithm processes the measured current $i_L$ and voltage $v_C$ to estimate the Constant Power Load (CPL) power $\widehat{P}$ from the state vector ${\widehat{x}}_{aekf,k}$. This estimated power is then fed directly into the PBC controller. A key advantage of this method is its ability to accurately estimate CPL power, thereby eliminating the need for dedicated power sensors. This reduction in hardware not only lowers system cost but also inherently increases robustness against measurement noise. Furthermore, the proposed AEKF eliminates the need for trial-and-error in tuning the covariance matrices.

3.2. Stability Analyzes

The stability of the Adaptive Extended Kalman Filter (AEKF) estimation error is analyzed as follows [23]. Beginning with the state prediction and update equations omitting Euler expansion:

$$\begin{array}{l}{\widehat{x}}_{aekf,k}^{-}=f({\widehat{x}}_{aekf,k-1},u_{k-1})\\ {\widehat{x}}_{aekf,k}={\widehat{x}}_{aekf,k}^{-}+K_{k}H(x_{aekf,k}-{\widehat{x}}_{aekf,k}^{-})\end{array}$$

(15)

The dynamic error is defined as ${\epsilon }_k=x_{aekf,k}-{\widehat{x}}_{aekf,k}^{-}.$ Applying Taylor’s expansion and grouping high-order terms into ${\phi }_k$, the error dynamics are linearized as:

$${\epsilon }_k=(A_{k-1}-K_{k}H{A}_{k-1}){\epsilon }_{k-1}+{\phi }_k$$

(16)

To assess exponential behavior, consider a Lyapunov function $v\left({\epsilon }_k\right)$ bounded by $C_1\parallel {\epsilon }_k{\parallel }^2\le$ $v\left({\epsilon }_k\right)\le C_2\parallel {\epsilon }_k{\parallel }^2$. If its difference $\mathsf{\Delta }v\left({\epsilon }_k\right)\le -C_3\parallel {\epsilon }_k{\parallel }^2$ for positive constants $C_1,C_2,C_3$, and the inequality ${\tilde{F}}_k^{-T}{P}_k^{-1}{\tilde{F}}_k^{-}\le \left(1-\gamma \right)P_{k-1}^{-1}$ holds, then the system is exponentially stable, satisfying $\parallel {\epsilon }_k\parallel \le \beta \parallel {\epsilon }_0\parallel {\gamma }^k$ for $\beta >0,0 <\gamma <1.$

4. Passivity-Based Control (PBC) and Stability Analysis

4.1. Adaptive PBC

Adaptive Passivity-Based Control Voltage Loop: To facilitate the design of the passivity-based controller, the state-space model (1) is expressed in the following matrix form:

$$H\dot{X}+[G+R(x)]X=\Gamma$$

(17)

In the equation, $H$ is the transient quantity in the passivity-based controller, $X$ is the state variable matrix in the passivity-based controller, $\Gamma$ is the input voltage matrix in the passivity-based controller, $G$ is the voltage transformation coefficient matrix in the passivity-based controller and $R(x)$ is the system impedance.

$$\begin{array}{c}X=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}{i}_L\\ v_o\end{array}\right),H=\left(\begin{array}{cc}L& 0\\ 0& C\end{array}\right),\Gamma =\left(\begin{array}{c}E\\ 0\end{array}\right)\\ G=\left(\begin{array}{cc}0& (1-\mu )\\ -(1-\mu )& 0\end{array}\right), R(x)=\left(\begin{array}{cc}0& 0\\ 0& (\frac{1}{R}+\frac{P_{CPL}}{x_2^2})\end{array}\right)\end{array}$$

(18)

Obviously, G = −G^T, R = R^T. Equation (17) satisfies the characteristics of the Euler–Lagrange model, providing a model basis for the design of the passivity-based controller. The design of the passivity-based control voltage loop is divided into two stages:

(1) Energy Shaping Stage: Using $X=\tilde{X}+X_d$ for transformation, (17) can be reshaped into

$$H\dot{\tilde{X}}+[G+R(x_m)]\tilde{X}=\Gamma -\left(H{\dot{X}}_d+[G+R(x_d)]X_d\right)$$

(19)

In the above equation, $X_d$ and $\tilde{X}$ represent the reference value and the disturbance value of X, respectively, where $R(x_m)$ is the actual system impedance matrix, and $R(x_d)$ is the reference value of the system impedance at the equilibrium point.

(2) Damping Injection Stage: This stage modifies the system dissipation function by injecting the virtual damping matrix $R_{V}X$ into both sides of (3), which can be expressed as

$$H\dot{\tilde{X}}+[G+R_p]\tilde{X}=\Gamma -\left(H{\dot{X}}_d+[G+R(x_d)]X_d-R_V\tilde{X}\right)$$

(20)

$$R_V=\left(\begin{array}{cc}{R}_{1V}& 0\\ 0& \frac{1}{R_{\mathrm{V}}}\end{array}\right), R(x_d)=\left(\begin{array}{cc}{R}_{1V}& 0\\ 0& \frac{1}{R}+\frac{P_{CPL}}{x_{2d}^2}\end{array}\right), R_p=R(x_m)+R_V=\left(\begin{array}{cc}0& 0\\ 0& (\frac{1}{R_{\mathrm{V}}}+\frac{1}{R}-\frac{P_{CPL}}{x_2\cdot x_{2d}})\end{array}\right)$$

(21)

To ensure that the system with the injected virtual damping matrix satisfies Lyapunov stability, thus possessing a globally stable equilibrium point for the system ($X=0$). The system’s energy function is selected as follows:

$$W(\tilde{X})=0.5{\tilde{X}}^{T}H\tilde{X}$$

(22)

Then, the time derivative of the energy function (22) can be expressed as

$$\begin{array}{ll}\dot{W}(\tilde{X})& ={\tilde{X}}^{T}H\dot{\tilde{X}}\\ & ={\tilde{X}}^T\left(\Gamma -\left(H{\dot{X}}_d+[G+R(x_d)]X_d-R_V\tilde{X}\right)-[G+R_p]\tilde{X}\right)\\ & ={\tilde{X}}^T\left(\Gamma -\left(H{\dot{X}}_d+[G+R(x_d)]X_d-R_V\tilde{X}\right)\right)-{\tilde{X}}^T{R}_p\tilde{X}\end{array}$$

(23)

To ensure global stable convergence of the system, the injected virtual damping coefficient R_v should make all elements of the matrix R_p positive. Under this premise, the control law for the passivity-based control voltage loop can be designed as

$$\Gamma -\left(H{\dot{X}}_d+[G+R(x_d)]X_d-R_V\tilde{X}\right)=0$$

(24)

Although the control law for the passivity-based control voltage loop obtained from Equation (24) can operate with multiple rated parameters, the randomness and time-varying nature of source and load side parameters can lead to steady-state deviations from the control objective. Therefore, substituting the estimated values of output power from AEKF into (24) can be expressed as

$$i_{L,ref}={\displaystyle \frac{{\widehat{P}}_{CPL}}{E}}-{\displaystyle \frac{v_{ref}}{R_{\mathrm{V}}E}}(v_o-v_{ref})+{\displaystyle \frac{{v^2}_{ref}}{RE}}$$

(25)

Based on the inherent properties of the converter and Equation (24), the control result for the switching duty cycle μ can be obtained.

$$\mu =1-\left[{\displaystyle \frac{E+R_{1v}\left(i_L-i_{L,ref}\right)}{v_{ref}}}\right]$$

(26)

Furthermore, by applying a current limiting block after the current reference, the peak current during startup and disturbances can be suppressed, thereby reducing the current stress on the equipment. Combining the passivity-based control voltage loop and the Extended Kalman Filter forms a nonlinear controller with a simple structure, high robustness, and low computational burden. Integrating the above controller design, the final proposed adaptive Extended Kalman Filter passivity-based control method is shown in Figure 2. It should be noted that while the stability condition requires R_p to be positive definite, the practical selection of R_1v is also constrained by the PWM switching frequency and modulator dynamics, as discussed in [33].

4.2. Stability Analyzes

The stability analysis of the proposed passivity-based control (PBC) algorithm is conducted within the formal framework of Lyapunov stability theory. Following the completion of energy reshaping and damping injection phases, the closed-loop system dynamics are characterized by the following differential equation:

$$H\dot{\tilde{X}}+[G+R_i(x)]\tilde{X}=0$$

(27)

The system energy function is selected as the Lyapunov function candidate:

$$\mathit{V}={\displaystyle \frac{1}{2}}{\tilde{\mathit{X}}}^{\mathit{T}}\mathit{H}\tilde{\mathit{X}}>0\hspace{1em}(\forall \tilde{\mathit{X}}\ne 0)$$

(28)

Since the inertia matrix H is positive definite and symmetric, V is consequently positive definite. To examine system stability, we compute the time derivative of the Lyapunov function along the trajectories of the system:

$$\dot{\mathit{V}}={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}{\tilde{\mathit{X}}}^{\mathit{T}}\mathit{H}\tilde{\mathit{X}}\right)={\tilde{\mathit{X}}}^{\mathit{T}}\mathit{H}\dot{\tilde{\mathit{X}}}$$

(29)

Substituting the system equation yields:

$$\dot{\tilde{\mathit{X}}}=-{\mathit{H}}^{-1}[\mathit{G}+{\mathit{R}}_i(x)]\tilde{\mathit{X}}$$

(30)

Expanding this expression:

$$\dot{\mathit{V}}=-{\tilde{\mathit{X}}}^{\mathit{T}}\mathit{G}\tilde{\mathit{X}}-{\tilde{\mathit{X}}}^{\mathit{T}}{R}_i(x)\tilde{\mathit{X}}$$

(31)

Given that matrix G is skew-symmetric (G = −G^T), it follows that:

$${\tilde{\mathit{X}}}^{\mathit{T}}\mathit{G}\tilde{\mathit{X}}=0$$

(32)

$$\dot{\mathit{V}}=-{\tilde{\mathit{X}}}^{\mathit{T}}{R}_i(x)\tilde{\mathit{X}}<0$$

(33)

According to Lyapunov stability theory, the system is globally asymptotically stable at the equilibrium point $\tilde{X}=0$. Regardless of initial conditions, the system state error $\tilde{X}$ converges to zero over time.

The AEKF is designed to estimate the unknown, time-varying CPL power in real time by treating it as an augmented state variable (Equations (4) and (5)). This accurate estimate $\widehat{\mathrm{P}}$ is then fed forward into the PBC law (Equations (25) and (26)). The PBC controller, through its energy shaping and damping injection stages, inherently compensates for the negative impedance effect of the CPL. The injected virtual damping restores system passivity, ensuring large-signal stability despite the nonlinear load characteristic.

This analysis demonstrates that the passivity-based control algorithm ensures system stability under all operating conditions, thereby providing theoretical assurance for practical engineering applications.

5. Simulation Results

To comprehensively assess both the steady-state and transient performance of the proposed Adaptive Extended Kalman Filter-based Passivity-Based Control strategy for DC microgrid DC-DC converters, a detailed simulation model was implemented within the PLECS simulation environment. The system configuration comprises a single Boost converter supplying power to parallel-connected constant power load and resistive load, where the CPL is practically realized through a current source combined with resistive components.

The converter control system was rigorously evaluated under multiple training scenarios designed to simulate realistic operational challenges. These included step changes in load conditions, abrupt variations in reference voltage commands, and sudden disturbances in the Boost converter’s input voltage. Through these comprehensive tests, the dynamic performance and robustness of the proposed control strategy were systematically evaluated under diverse operating conditions, with control parameters being progressively refined through this process. The simulation parameters employed for this validation study are detailed in

Table 1. System configuration parameters.

Symbol	Representation	Value
Boost converter
C	Capacitance	940 μF
L	Inductance	2 mH
E	Input Voltage	100 V
R	Load Resistance	100 Ω
vref	Reference Output Voltage	200 V
P	Initial CPL Power	500 W
fsw	Switching Frequency	20 kHz
Proposed Strategy
TS	Discretizing Time	5 × 10−5
RVi	Virtual Resistor	1.5
Qi,m,1, Qi,m,2, Qi,m,3	Initial Covariance of ProcessNoise	10−3, 10−4, 0.6
Pi,m,1, Pi,m,2, Pi,m,3	Initial Covariance of States	0.05, 0.05, 0.05
${\widehat{x}}_0$	Initial Value of States	0.1, 0.1, 0.1

.

5.1. Case 1: Constant Power and Constant Impedance Load Changes

The proposed control algorithm demonstrates exceptional dynamic performance and robustness when subjected to sudden changes in constant power load (CPL). As illustrated in Figure 3, when the system experiences a step change in CPL power from 500 W to 900 W at t = 0.2 s, both the output voltage and inductor current exhibit minor, brief transients. Critically, the system recovers to its steady-state values rapidly. A subsequent step reduction in CPL power from 900 W back to 500 W occurs at t = 0.6 s, followed by a step decrease in the resistive load R to 50 Ω at t = 1.0 s. In both cases, the output voltage and inductor current revert to their steady-state values promptly, with a remarkably short settling time of 8 ms and a minimal voltage overshoot constrained within 2 V. The output voltage profile maintains a smooth trajectory during these transients, devoid of significant oscillations, which verifies the algorithm’s efficacy in suppressing power oscillations induced by load disturbances. The swift response of the inductor current further underscores the controller’s superior capability for rapid power balance regulation.

5.2. Case 2: Constant Power and Input Voltage Changes

The proposed control algorithm also exhibits outstanding dynamic regulation and robustness in response to input voltage disturbances. As shown in Figure 4, the system was tested under a sequence of step changes. Initially, a CPL power step from 500 W to 900 W at t = 0.2 s causes brief deviations in the output voltage and inductor current, which are quickly stabilized within a settling time of 10 ms and a voltage overshoot below 1 V. When the CPL power steps back down from 900 W to 500 W at t = 0.6 s, the system demonstrates a faster recovery, with a settling time of 8 ms and an overshoot controlled within 1 V. Furthermore, when subjected to an input voltage step change from 100 V to 80 V at t = 1.0 s, the system again manages the transient effectively, stabilizing within 9 ms. Throughout these tests, the voltage waveform transitions smoothly, and all fluctuations in load voltage and power remain within 10%. These results collectively highlight the controller’s strong capability for rapid compensation of input-side disturbances.

5.3. Case 3: Comparative Analysis with Conventional Methods

The proposed control method demonstrates rapid bus voltage stabilization, achieving a mere 1 V overshoot and an 8 ms settling time. In contrast, the PBC + NDO control method yields a larger overshoot of 1.7 V and a longer settling time of 20 ms. As shown in Figure 5, these comparative results highlight the exceptional robustness of the proposed approach to load disturbances.

6. Experimental Results

To validate the superior performance of the proposed converter, an experimental setup was constructed for testing. The setup consists of an oscilloscope (Yokogawa DL850), a DC power supply (ITECH IT-M3433,150 V/12 A), an electronic load (Chroma 63804, 350 V/45 A/4500 W), a sampling circuit, a DC-DC boost converter, and an RT-Box1. The Chroma electronic load can be configured in either resistive mode or constant power load (CPL) mode to accommodate different load types. The control algorithm is developed in PLECS on a host computer, which generates code for the RT-Box. This code enables the RT-Box to acquire voltage and current signals via its analog-to-digital converters (ADCs) and execute the control algorithm, outputting a 20 kHz PWM signal to the DC-DC boost converter. Variations in source-side and load-side operating conditions are implemented using the programmable DC power supply and the electronic load. Furthermore, to faithfully reflect the control performance, the parameters of the experimental platform are kept consistent with those used in the simulation studies. where v_o denotes the output voltage, i_L represents the inductor current, and i_o is the output current.

6.1. Case 1: Constant Power Load Changes

The system was initialized with an input voltage of 100 V and a constant power load of 200 W. The experimental waveforms are shown in Figure 6. During the test, the constant power load was stepped up to 400 W and, after a period of time, stepped back down to 200 W. Accordingly, the inductor current increased from 2 A to 4 A and then returned to 2 A, while the output current rose from 1 A to 2 A and then recovered to 1 A. The settling time was measured as 23 ms, and the overshoot was limited to 2.1 V. The experimental results demonstrate that the proposed control method can rapidly stabilize the bus voltage under abrupt load changes.

6.2. Case 2: Input Voltage Changes

Next, experimental studies were conducted on the proposed control method under input voltage disturbances to evaluate its robustness and stability against source-side variations. The input voltage of the system was varied in the following sequence: 100 V → 80 V → 100 V. Figure 7 illustrates the comparative experimental waveforms corresponding to the input voltage change from 100 V to 80 V and back to 100 V. It can be observed that the proposed control method is able to rapidly stabilize the bus voltage, with a voltage overshoot of 4 V and a settling time of 62 ms.

6.3. Case 3: Output Voltage Changes

Finally, the dynamic tracking performance of the proposed control method is evaluated by studying its response to changes in the output voltage reference. In this experiment, the output voltage reference of the converter is stepped from 200 V to 190 V while the load is 200 Ω. As shown in Figure 8, the output voltage rapidly tracks the new reference within 30 ms without any voltage overshoot. The peak inductor current during the transient is 1.7 A. These results demonstrate that the proposed control method provides excellent voltage regulation capability.

7. Conclusions

This paper has proposed a novel control methodology for DC-DC converters that integrates an Extended Kalman Filter (EKF) with Passivity-Based Control (PBC), effectively enhancing system stability under Constant Power Load (CPL) conditions. Initially, the designed Extended Kalman Filter controller accurately estimates the time-varying load power by leveraging real-time converter measurements. This predicted information is subsequently fed into the proposed passivity-based controller to generate PWM signals for driving the power switches, thereby regulating the boost converter supplying the CPLs.

The presented strategy significantly improves large-signal stability and system robustness, while also providing precise current limiting and rapid voltage regulation. These capabilities collectively ensure reliable and stable operation of DC microgrids under various complex operating scenarios. Furthermore, the nonlinear control loop eliminates the conventional requirements for detailed nominal parameters and complex multi-degree-of-freedom tuning, resulting in a relatively simple and practical control structure. Simulation studies validate the superiority of the proposed approach, demonstrating exceptional control performance across various operating conditions.