Learning-Based Approaches for Voltage Regulation and Control in DC Microgrids with CPL

Abstract

This article introduces a novel approach to voltage regulation in a DC/DC boost converter. The approach leverages two advanced control techniques, including learning-based nonlinear control. By combining the backstepping (BSC) algorithm with artificial neural network (ANN)-based control techniques, the proposed approach aims to achieve accurate voltage tracking. This is accomplished by employing the nonlinear distortion observer (NDO) technique, which enables a fast dynamic response through load power estimation. The process involves training a neural network using data from the BSC controller. The trained network is subsequently utilized in the voltage regulation controller. Extensive simulations are conducted to evaluate the performance of the proposed control strategy, and the results are compared to those obtained using conventional BSC and model predictive control (MPC) controllers. The simulation results clearly demonstrate the effectiveness and superiority of the suggested control strategy over BSC and MPC.

1. Introduction

DC microgrids are increasingly prevalent in various industrial settings due to their ease of control and cost-effectiveness [1]. However, these microgrids often face stability issues caused by the presence of a constant power load (CPL) [2]. These components exhibit a negative impedance characteristic that amplifies disturbance signals, leading to system failure and an inability to maintain a steady state [3]. Moreover, it has been observed that the presence of CPLs can give rise to low-frequency oscillations in the bus voltage, which can potentially lead to the collapse of the entire DC microgrid system [4]. Therefore, addressing these challenges has become a crucial research area, focusing on the design of a reliable controller for the nonlinear DC microgrid system, particularly for the DC/DC converter that supplies power to CPLs [5,6]. The goal is to improve the dynamic characteristics of the system while ensuring the stability of the output voltage. Figure 1 illustrates an example microgrid featuring a CPL load.

In the DC microgrids, power electronics systems are extensively utilized [7]. The conventional method of control entails the process of linearizing the system in the vicinity of the equilibrium point, followed by the utilization of a linear double-closed-loop proportional–integral (PI) controller to regulate the small-signal model [8]. Nevertheless, the employment of this particular approach poses certain challenges, given that the PI controller entails a comprehensive scrutiny of the frequency characteristics of the converter’s open and closed loop in diverse controller parameters [9]. Furthermore, when the system undergoes significant signal perturbations, it may potentially result in instability [10].

To address these challenges, academic literature has explored a range of nonlinear control techniques. These encompass active anti-disturbance control [11], sliding mode control (SMC) [12], model predictive control (MPC) [13], and various other approaches. For example, in [14], a unique approach is introduced, featuring a novel linear switching sliding mode surface. The surface is based on a combination of current and voltage tracking errors, with the aim of mitigating the potential instability caused by CPLs. Furthermore, a separate investigation [15] posits a composite BSC control approach. This methodology utilizes an NDO in order to achieve the stabilization of interleaved double dual-boost converters that supply CPLs in DC microgrids. However, these robust controllers require accurate knowledge of the system model for precise tracking. To address the issue of durability redundancy prevalent in current controllers and to achieve large-signal stabilization for DC/DC converters powering CPLs, a novel adaptive control approach was introduced in [16]. This approach effectively balances dynamic performance and system stability without necessitating the real-time optimization of control law parameters. Furthermore, in [17], a model predictive control (MPC) with maximum power point tracking (MPPT) strategy is presented for a cutting-edge high step-up DC/DC converter capable of providing a voltage input of up to 10 times the original value. Notwithstanding the fact that the MPC-based strategy allows for online parameter adjustment, extensive online calculations necessitate specific hardware conditions [18]. In [19], a sliding mode observer (SMO) was employed for predictive current control, but it suffers from a chattering phenomenon. To achieve model-free predictive current control, [20] utilized the extended state observer (ESO). Nonetheless, the ESO proves to be more appropriate for systems devoid of models and, hence, poses certain difficulties in terms of parameter configuration. A nonlinear system with a known model, on the other hand, was the subject of [21]’s utilization of the NDO. Nevertheless, the estimated error in the observers used in [22] leads to imprecise tracking, and these approaches lack learning ability. The current control methodologies need enhancements in tracking accuracy, dynamic response, and adaptability to meet the requirements of DC microgrids with a high number of CPLs.

The adoption of data-driven or model-free control methodologies, particularly those based on artificial neural networks (ANNs), is steadily gaining momentum in the field of power converters [23]. In a notable study [24], an ANN-based control scheme was introduced for the direct control of a three-phase inverter equipped with an output I.C filter, resulting in reduced total harmonic distortion (THD) and enhanced steady-state, as well as dynamic performance. In a related work, detailed in [25], the authors introduced an ANN-based control strategy for a three-phase flying capacitor multilevel inverter (FCMLI). Conversely, in a separate study [26], they presented a voltage control technique for a DC-DC boost converter, employing artificial neural networks (ANNs) to regulate the voltage in a DC/DC step-down converter through neural network estimation. The ANN was expertly trained using model predictive control (MPC) data, which used them for the voltage regulation of the DC-DC boost converter. In addition, a voltage control technique for the DC-DC boost converter using an artificial neural network (ANN) is proposed in [27].

In this study, during the training phase, the NDO is employed to estimate the converter’s output power, and state variable data are collected with full state observation. These collected data are utilized to train the ANN. During the phase of testing, the ANN that has been trained is implemented online for the purpose of regulating the output voltage of the converter. This results in the replacement of the BSC control. The efficacy of the suggested regulatory approach is assessed via simulation and contrasted with outcomes obtained from both the BSC control and MPC controllers.

In summary, this approach leverages the advantages of both the BSC algorithm and ANN-based control techniques, resulting in a comprehensive control strategy for voltage regulation in a DC microgrid. The simulation results demonstrate the effectiveness and superiority of the proposed strategy when compared to both the BSC and MPC controllers.

In comparison to the existing literature, in this study, the main contribution can be outlined as follows:

• Firstly, we introduce a novel composite voltage regulation strategy that heavily relies on learning techniques. The main objective of this strategy is to achieve a notable stabilization of the converter system responsible for powering the CPLs.
• Secondly, our approach involves the offline training of the ANN algorithm. This offline training methodology enhances the practicality and efficiency of the implementation by eliminating the requirement for extensive online computations and laborious empirical parameter adjustments.
• The proposed method offers several benefits in satisfying the loads, including a rapid response, reduced oscillations, and high-reliability tolerance. These advantages are achieved through the utilization of well-trained ANNs that exhibit high accuracy in real time. Such positive effects contribute significantly to the overall improvement of the system.

2. Materials and Methods

2.1. System Description

A conventional DC embedded microgrid shown in Figure 1 has a primary DC bus that is powered from a DC source via DC/DC source converters. In such systems, there are many DC/DC converters, rectifier circuits, and motor drive systems fed through inverters. Where a DC/DC converter powers a resistive load, tight regulation of the converter’s output voltage results in a constant power draw. Figure 2 depicts its corresponding equivalent circuit. In response to growing environmental and economic concerns, there has been a global shift towards increased adoption of renewable energy sources. This includes the utilization of photovoltaics and fuel cells, both capable of generating DC power. However, the intermittent availability of solar energy poses a challenge due to fluctuations in weather conditions and geographical locations.

To overcome this challenge, static storage integration has been implemented, involving the use of DC-only components such as batteries and mechanical energy storage systems like flywheels. These systems help mitigate the issue of intermittent availability. Additionally, on the load side, the power electronics converter responsible for supplying power to the load can function as a load if it possesses a sufficiently high control bandwidth. This load can be powered by either DC or AC power, providing enhanced flexibility and efficiency within the system.

The diagram presented in Figure 2 illustrates a DC-DC boost converter that is responsible for providing power to both CPLs and resistive loads. In order to derive the dynamic equation of the system, the application of Kirchhoff’s laws is employed effectively. The effective utilization of Kirchhoff’s laws plays a crucial role in deriving the dynamic equation of the DC-DC boost converter system.

$$\frac{d_{iL}}{d_t}=\frac{V_i}{L}-\frac{v_C\left(1-\mu \right)}{L}$$

(1)

$$\frac{d_{vC}}{d_t}=\frac{i_L\left(1-\mu \right)}{C}-\frac{v_C}{RC}-\frac{P_{CPL}}{C{v}_C}$$

(2)

In the domain of boost converters, the parameters L and C hold significant importance. Additionally, the inductor current, denoted as $i_L$, and the voltage across the capacitor, $v_C$, are also crucial factors to consider. The power of the capacitively loaded circuit CPL, denoted as $P_{CPL}$ can be expressed as the product of $v_{CPL}$ and $i_{CPL}$. The input voltage, $V_i$, is another pivotal aspect of the system. Finally, the duty cycle of the control switch signal, denoted by $\mu$, plays a crucial role in determining the control input of the system.

The current passing through capacitively loaded circuit CPLs exhibits an inverse relationship with its terminal voltage. When there is a decrease in voltage, there is a corresponding increase in current, and vice versa, which gives rise to a negative incremental impedance. This phenomenon leads to a diminution in the effective damping of the system, thereby causing instability. The instability is particularly pronounced when the system is subjected to pure CPLs [28]. To address this concern, the incorporation of resistive loads is implemented in order to augment system damping. In addition, due to the nonlinear attributes of CPLs, the identification of suitable methodologies to ensure system stability is imperative. The fundamental aim of this study is to devise a control signal, designated as $\mu$, which assures a fixed-time convergence irrespective of the initial states. This control signal guarantees that the output voltage, $v_C$, adheres to its desired value, $v_{Cr}$. To accomplish this, the duty cycle utilized for the control of the boost converter through pulse width modulation (PWM) may be ascertained through the utilization of the output signal emanating from the controller. The design of such a control system is crucial for maintaining stability and achieving the desired performance of the boost converter.

2.2. Diffeomorphism Co-Ordinate Transformation

In order to design the proposed learning-based nonlinear controller, it is essential to transform the system model given in Equations (1) and (2) into its standard form. To achieve this, a diffeomorphism can be employed based on the principles of precise feedback linearization of DC/DC converters [29]. This diffeomorphism allows for the introduction of new co-ordinates, which are interpreted as the cumulative stored energy and its rate of change. This transformation facilitates the design and implementation of the learning-based nonlinear controller, enabling effective control and regulation of the system dynamics.

Considering the inherent uncertainties in the model and the possibility of load variations, it is essential to represent the dynamic model as described in Equations (3) and (4):

$$L_b\frac{d_{i_L}}{d_t}=V_i-v_C\left(1-\mu \right)$$

(3)

$$C_b\frac{d_{vC}}{d_t}=i_L\left(1-\mu \right)-\frac{v_C}{R_S}-\frac{P_{CPL}}{v_C}$$

(4)

The designations for the inductance, capacitance, and source bus voltage are represented by the symbols $L_b$, $C_b$, and $V_i$, respectively.

The quantification of the total energy contained in the dynamic system can be expressed as in Equation (5):

$$x_1=0.5\left(L_b{i}_L^2+C_b{v}_C^2\right)$$

(5)

Taking the derivative of Equation (5) allows for the computation of its instantaneous rate of change with respect to its independent variable:

$${\dot{x}}_1=L_b{\dot{𝚤}}_L{\dot{\dot{𝚤}}}_L+C_b{v}_C\dot{v_C}=V_i i_L-\frac{v_C^2}{R_s}-P_{CPL}$$

(6)

Referring to Equation (6), it is possible to define a new state denoted as $x_2$, accompanied by an indeterminate term referred to as d₁:

$$x_2=V_i{i}_L-\frac{v_C^2}{R_b}$$

(7)

$$d_1=-P_{CPL}+\frac{v_C^2}{R_b}-\frac{v_C^2}{R_s}$$

(8)

The symbol $R_b$ represents the nominal resistance of the resistive load. ${\dot{x}}_2$ can be obtained by taking the derivative of both sides of Equation (7).

$${\dot{x}}_2=\frac{V_i{}^2}{L_b}+\frac{2v_C^2}{R_b^2{C}_b}-\left(\frac{V_i{v}_C}{L_b}+\frac{2i_L{v}_C}{R_b{C}_b}\right)\left(1-\mu \right)+\frac{2}{R_b{C}_b}\left(P_{CPL}-\frac{v_C^2}{R_b}+\frac{v_C^2}{R_s}\right)$$

(9)

Let us now proceed to define the intermediate control law denoted by $n$, together with the uncertain term, $d_2$:

$$n=\frac{V_i{}^2}{L_b}+\frac{2v_C^2}{R_b^2{C}_b}-\left(\frac{V_i{v}_C}{L_b}+\frac{2i_L{v}_C}{R_b{C}_b}\right)\left(1-\mu \right)$$

(10)

$$d_2=\frac{2}{R_b{C}_b}\left(P_{CPL}-\frac{v_C^2}{R_b}+\frac{v_C^2}{R_s}\right)$$

(11)

The system described in Equations (3) and (4) can undergo a transformation procedure that leads to the attainment of a canonical form:

$${\dot{x}}_1=x_2+d_{1 }, {\dot{x}}_2=n+d_{2 }$$

(12)

In the system described by Equation (12), the state variables are represented by $x_1$ and $x_2$. The terms $d_1$ and $d_2$ denote uncertain components, and $n$ represents the intermediate control signal. It is evident that the system model has been articulated in Brunovsky’s canonical form, a representation that is deemed appropriate for the deployment of the BSC algorithm.

To control the DC/DC boost converter effectively, the main objective is to design a control regulation u that enables the DC bus voltage $v_C$ to asymptotically track its reference value $v_{Cr}$. After implementing a co-ordinate transformation, the goal of voltage tracking is transformed into developing an intermediate control mechanism $n$. This control mechanism is aimed at achieving asymptotic tracking of the state $x_1$ to its reference value $x_1^{\prime }$, as expressed by the equation.

$$x_1^{\prime }=\frac{1}{2}{L}_b{i}_{Lr}^2+\frac{1}{2}{C}_b{v}_{Cr}^2=\frac{1}{2}{L}_b{\left(\frac{P_L}{V_i}\right)}^2+\frac{1}{2}{C}_b{v}_{Cr}^2$$

(13)

The parameter $P_L$ represents the magnitude of the total load power, denoted as a numeric value.

$$P_L=P_{CPL}+\frac{v_{Cr}^2}{R_s}$$

(14)

Once the intermediate control signal $n$ in Equation (12) has been carefully crafted, the final control law $\mu$ in the original system (Equations (3) and (4)) can be deduced in accordance with Equation (10).

$$\mu =1-\frac{\frac{V_i{}^2}{L_b}+\frac{2v_C^2}{R_b^2{C}_b}-n}{\frac{V_i{v}_C}{L_b}+\frac{2i_L{v}_C}{R_b{C}_b}}$$

(15)

3. Design of the Proposed Controller

3.1. NDO for Power Estimation

The nonlinear distortion observer is an effective technique utilized to estimate uncertainties. Its primary purpose is to estimate uncertain parameters, enabling accurate tracking and facilitating a rapid dynamic response.

From Equations (13) and (14), it can be observed that the reference value $x_1^{\prime }$ depends on the load power, which is considered an uncertain quantity and can lead to tracking errors. To achieve accurate tracking and a rapid dynamic response, the NDO methodology is applied. This methodology is a proficient approach for estimating uncertainties, specifically for evaluating uncertain parameters d1, d2, and the total load power in Equation (14).

It is important to note that the uncertainties, represented by $di$ (for i = 1 and 2), are closely related to the load power, as described in Equations (8) and (11), respectively. Therefore, from a practical perspective, it is crucial to ensure that their values and derivatives are constrained within the appropriate bounds.

Additionally, the load power is considered to be a constant value in the state of equilibrium. Furthermore, it can be assumed that the variables $di$ and $\dot{d}$i, indexed with $i=1,2,$ which contribute to the uncertain nature of the system described in Equation (12), satisfy two distinct conditions:

$$di\left(t\right)\in L_{\infty } , \dot{d}i\left(t\right)\in L_{\infty }$$

(16)

$$t\stackrel{lim}{\to }\infty \dot{d}i\left(t\right)=0$$

(17)

It was determined that the uncertain term referred to as $d_1$ in the context of the NDO was estimated using a specific methodology:

$${\hat{d}}_1=l_1\left(x_1-p_1\right), {\dot{p}}_1=x_2+{\hat{d}}_1$$

(18)

In the context of the NDO, the term $p_1$ represents an auxiliary condition, while $l_1$ denotes a favorable constant commonly known as the NDO gain:

$${\hat{d}}_2=l_2\left(x_2-p_2\right), {\dot{ p}}_2=n+{\hat{d}}_2$$

(19)

Similarly, the uncertain term referred to as $d_2$ is approximated using estimation techniques.

Equations (18) and (19) outline the expression for the estimation error of the first- and second-order derivatives of di, where $i=1,2$:

$$\tilde{d}i=di-\hat{d}i$$

(20)

$$\dot{\tilde{d}}i=\dot{d}i-\dot{\widehat{d}}i=\dot{d}i-li\tilde{d}i$$

(21)

Based on Equation (8), (14), and (18), an estimation of the total load power, $P_L$, in Equation (14) can be derived:

$$P_L=\frac{v_{Cr}^2}{R_b}-{\hat{d}}_1$$

(22)

The reference value of state $x_1^{\prime }$ in Equation (13) is redefined and formulated again as in (23):

$$x_1^{\prime }=\frac{1}{2}\frac{L_b}{V_i{}^2}{\left(\frac{v_{Cr}^2}{R_b}-{\hat{d}}_1\right)}^2+\frac{1}{2}{C}_b{v}_{Cr}^2$$

(23)

3.2. Proposed Controller Based on Artificial Neural Network

The field of artificial intelligence is focused on the investigation and advancement of methodologies that facilitate the manifestation of human-like capabilities in machines. These capabilities encompass aspects such as reasoning, judgment, emotional experience, language comprehension, and problem solving. One prominent technique within the realm of artificial intelligence is the ANN, which emulates the structural characteristics of the human brain. However, the number of neurons present in an ANN is determined based on the specific requirements of a given problem, in contrast to the approximate 15 billion neurons found in the human brain [30]. ANNs possess the ability to learn from data and apply the acquired knowledge, thus making them extensively applicable in various domains, including but not limited to forecasting, classification, identification, and control.

Figure 3 shows the general structure of the proposed system based on the ANN. In the present study, a feedforward neural network was constructed for the purpose of controlling the converter’s output voltage. The ANN consists of three components: the input layer, the output layer, and the hidden layer. Weights are used to establish connections between all layers. The experimental data received from the input layer are subjected to multiplication by the weights that link the input layer and the hidden layer. Following this, the resultant product is transmitted to the hidden layer. The neurons situated within the concealed layer undertake the task of gathering and transmitting the inputs received to the output layer via a multiplication process involving the weights that serve as a link between the concealed layer and the output layer. The neurons present in the output layer are also involved in the reception and processing of these inputs. Activation functions are applied to these inputs to generate more accurate outputs. The determination of weight values for connections is an integral aspect of the learning process. The initial stage of learning can be described as activation, which determines whether the sum of signals entering a neuron is sufficient to activate the cell. If the total signal surpasses the threshold value, the cell becomes active (y = 1); otherwise, it remains inactive (y = 0). ANNs employ various activation functions, with the “multi-layer perceptron” model being one of the most used. In contemporary applications, the activation function commonly employed is the sigmoid function, which is a continuous and differentiable function that maps input values to a range spanning from 0 to 1. It can be defined as follows:

$$F\left(Net\right)=\frac{1}{1+e^{-Net}}$$

(24)

Utilizing the input and output data, the training algorithm engages in iterative adjustments of the synapse weights until the desired level of convergence is attained. The choice of the training algorithm plays a crucial role in obtaining satisfactory results. Numerous training algorithms are documented in the literature. In the algorithm chosen for training, the discrepancy between the output of the network and the anticipated output is retroactively propagated to adjust the weightage of the network until the said discrepancy is minimized. A neuron can be mathematically defined as follows:

$$o=f\left(wx+b\right)F$$

(25)

In the equation, w and x represent the weights and inputs, respectively, while b denotes the bias. The purpose of incorporating bias entries is to shift the activation function’s origin, thereby enhancing the learning process [30]. ANNs consist of interconnected nodes organized into layers, with data flowing through them, ultimately producing output. Each connection between nodes has a weight that determines its strength, and activation functions introduce nonlinearity into the network. Training ANNs involves using algorithms like backpropagation to adjust the weights and biases of the network based on the comparison between the network’s output and the desired result. The transfer function can be described as follows:

$$net={\displaystyle \sum }_{i=1}^n{w}_i·x_i+b$$

(26)

In this study, the artificial neural network (ANN) architecture comprises an input layer with five neurons, a hidden layer consisting of ten neurons, and an output layer with a single neuron. Out of the total dataset, which includes 160,000 data rows, 70% were allocated for training, 15% for validation, and the remaining 15% for testing. The training of the neural network model was carried out using the Levenberg–Marquardt algorithm, a specific variant of the backpropagation with scaled conjugate gradient (BSC) algorithm.

The control architecture of the proposed controller, as depicted in Figure 3, is based on the previously developed procedure. Initially, the system model undergoes a transformation into the canonical form at (12) using the exact feedback linearization technique. Subsequently, a nonlinear ANN-based controller can be created for the specific system under consideration by employing the NDO estimation procedure and the stepping-back procedure outlined in (18)–(23):

$$n=-k_2{z}_2-{\hat{d}}_2+{\ddot{x}}_1^{*}$$

(27)

During the training of the artificial neural network (ANN) using data obtained from the BSC controller, we introduced variations in the CPL power values. This approach allowed us to create dynamic responses for the ANN, tailored to different scenarios. Throughout the ANN training process, we worked with a dataset containing 300,000 data points within a 2 ms timeframe. Leveraging this dataset, we designed an ANN block with five input parameters and one output. During the training of the ANN, the intermediate controller formula of the BSC control (27) is employed as the output data [29]. The training process incorporates the utilization of the canonical form parameters x1 and x2, along with the data d1, d2, and $x_1^{\prime }$ acquired from the NDO. Through the integration of NDO and ANN, the composite controller has been meticulously crafted to provide a rapid dynamic response and precise bus voltage tracking, all while maintaining exceptional signal stability.

4. Simulation Results

To validate the performance of the proposed controller depicted in Figure 3 and its integration with the system model illustrated in Figure 2, extensive tests were conducted using MATLAB/Simulink. The objective was to assess its capability to ensure stable operation under large signals and precise tracking, even in the presence of uncertainties. For a comprehensive understanding of the system parameters employed in these tests, please consult

Table 1. Simulation parameters.

Symbol	Description	Value
$v_{Cr}$	Load bus voltage reference	100 V
$V_i$	Converter input voltage	50 V
$f_s$	Switching frequency	20 kHz
$L_b$	Converter inductance	648 μH
$C_b$	Converter capacitance	241 μF
$l_1, l_2$	NDO constant	5000, 2000

.

The simulation results obtained from the proposed controller’s CPL load variation are illustrated in Figure 4. Initially, a load of 100 W is connected to the CPL bus, and a resistive load of 200 W is present. Subsequently, the CPL load increases from 100 W to 160 W over a duration of 0.7 s. Then, within 0.1 s, the CPL load further rises from 160 W to 200 W. After 0.13 s, the CPL load is reset back to its initial state of 100 W. The conspicuous attainment of smooth transient performance, which exhibits a settling time of approximately 2 ms and a maximum transient voltage deviation of less than 1%, even when subjected to triple load changes, is unmistakable. Moreover, following the completion of the transitions, the load bus voltage is accurately regulated to the reference value of 100 V. This observation emphasizes the successful performance of the proposed controller in achieving smooth transitions and precise tracking during the integration of the CPL.

A comparative analysis was conducted to evaluate the stability of the suggested controller under various CPL loads, in comparison to the conventional PI controller. Figure 5 illustrates the results of this analysis. The CPL underwent step changes for both controllers, with a surge from 100 W to 200 W within 0.1 s. The simulation results demonstrated the stability of both systems. Notably, the proposed controller exhibited a rapid return to the reference value after a brief 0.1 s oscillation of approximately 1 V, whereas the PI controller required more time, with an oscillation of approximately 5 V before achieving stability. This highlights that the proposed controller experiences oscillations five times less pronounced than the PI controller. However, when comparing the settling times, it becomes evident that the proposed control system achieves stability in a mere 2.4 milliseconds, whereas the PI controller takes 10 milliseconds to stabilize. This emphasizes that the proposed control system boasts a settling time that is four times superior to that of the PI controller. Furthermore, when the constant power load (CPL) decreases from 200 W to 100 W in just 0.12 s, the system under the PI controller persistently operates with elevated emissions. In contrast, the proposed controller exhibits a voltage response with less pronounced oscillations, although the current experiences more significant oscillations. This increase in current oscillation within the brief 0.12 s timeframe can be attributed to the higher power variation observed during the artificial neural network (ANN) training. Nevertheless, the proposed control system achieves stability with notably reduced oscillations in comparison to the PI controller. Consequently, the suggested controller ensures smooth and accurate voltage monitoring with a stable operation under CPL load changes, while the traditional PI controller falls short. Therefore, the proposed controller possesses the advantage of superior signal stability, justifying its superiority.

Simulations were conducted to compare the performance of the proposed controller with a nominal MPC controller, and the results are presented in Figure 6. It is important to note that these simulations were performed with a nominal input voltage of 50 V, similar to the comparison made earlier with the conventional PI controller. In Figure 6, the swift increase in constant power load (CPL) from 100 W to 200 W within a mere 0.1 s reveals the exceptional performance of the proposed controller. It swiftly restores the DC bus voltage to the desired reference value in just 2.4 milliseconds, accompanied by a minimal overshoot of 0.99 V. In contrast, the MPC controller exhibits a slightly higher overshoot of 2.48 V and a more extended settling time of 10 milliseconds. These results unequivocally underscore the superior responsiveness of the proposed controller compared to the MPC controller. Moreover, when the CPL power decreases from 200 W to 100 W in a mere 0.12 s, the MPC controller successfully brings the DC bus voltage back to the reference value in 10 milliseconds, albeit with a larger overshoot of 2.85 V. In contrast, the proposed controller outperforms this response, reaching the desired reference voltage value within a rapid 1.1 milliseconds and displaying a smaller voltage swing of 0.76 V. This response is even better than the performance observed during the CPL power increase. Consequently, the nominal MPC controller showed a significantly inferior performance compared to the proposed controller, highlighting the advantages of the proposed controller in this scenario.

A comparison was made between the proposed controller and the nonlinear BSC controller. The results of this comparison are shown in Figure 7. The BSC controller is known for its ability to stabilize CPLs with fast dynamics and precise tracking. In the simulation, the CPL power rose rapidly from 100 W to 200 W in 0.1 s and dropped back to 100 W in 0.12 s. As seen in Figure 7, the proposed control reached the reference voltage value in a period of 2.4 ms with an overshoot of 0.99 V, as in the comparison in Figure 6. On the other hand, the BSC controller settled at the 100 V output reference voltage in about 5 ms with a 1.15 V oscillation. As soon as the CPL power is drawn from 200 W to 100 W, the recommended controller settles into the 100 V reference voltage in 1.1 ms with a 0.76 V oscillation, while the BSC settles into the reference DC bus voltage in 5.23 ms with a 1.06 V oscillation. It is clear from the results that both controllers can stabilize CPLs with fast dynamics and accurate tracking. However, the proposed controller exhibits a lower voltage drop and faster localization performance during transients.

The simulation outcomes for MPC and BSC, both of which are recommended controllers and nonlinear control methods, are presented in Figure 8. However, the outcomes of the conventional PI controller, which were found to be inferior to those of the MPC and BSC controllers, are not included in Figure 8.

To gain a deeper insight into the advantages of the proposed controller, we present the simulation results and measurements in Figure 8. Detailed information regarding voltage overshoot values and settling times of the proposed controller, as well as those of the alternative controllers, in response to CPL power variations, is outlined in

Table 2. Proposed control, and response of MPC and BSC controls to CPL.

Controllers	Settling Time (0.1 ms)	Settling Time (0.12 ms)	Overshoot (0.1 ms)	Overshoot (0.12 ms)
Proposed control	2.4 ms	1.1 ms	0.99 V	0.76 V
MPC control	10 ms	10 ms	2.48 V	2.85 V
BSC control	5 ms	5.23 ms	1.15 V	1.06 V

. Upon reviewing the data in the Table 2, it becomes evident that the proposed ANN-based controller outperforms both MPC and BSC controllers, exhibiting superior performance in terms of both voltage overshoot and settling time. Furthermore, the proposed controller demonstrates precise and uninterrupted voltage regulation, maintaining consistent operation even in the face of changes in CPL power. As can be seen in these comparisons, the signal stability offered by the proposed ANN-based controller has shown a good performance compared to other controllers.

5. Discussion

The recommended controller boasts several advantages, including robust voltage monitoring, stable operation even in the face of constant-power-load (CPL) power uncertainties, and durability. When compared to the traditional PI controller, the proposed ANN-based controller significantly outperforms it in terms of both voltage regulation and settling time, as clearly demonstrated in Figure 5. Furthermore, as depicted in Figure 6, in comparison to a standard MPC controller, the proposed controller excels in both precise voltage tracking and rapid settling time due to the integration of NDO. In a comparison with a robust controller like BSC, the proposed controller exhibits optimized performance with seamless transitions, as evident in Figure 7. This is particularly evident in the overshoot, oscillation, and settling time illustrated in Figure 4. However, it is essential to acknowledge that the proposed approach has some limitations. Similar to other nonlinear techniques, calibration relies on practical experience, involving trial and error to achieve the desired dynamic behavior. Fine-tuning parameters like l1 and l2 associated with NDO, as well as d1 and d2 control gains, is necessary to optimize performance during ANN training. In future work, we aim to expand the scope of this methodology to encompass various converter topologies, making it a more comprehensive strategy.

6. Conclusions

This document introduces a novel methodology for regulating DC/DC boost converters that supply CPLs by utilizing NDO and learning-based combined nonlinear control. Initially, the system model is transformed into the canonical form. Load power is evaluated using the NDO methodology to ensure precise voltage tracking, thereby leading to an enhanced dynamic response. Subsequently, the ANN is trained using data obtained from the NDO and the boost converter. The resulting composite controller is illustrated in a straightforward linear structure, featuring parameters meticulously designed for optimal precision. The proposed approach offers the advantage of precise voltage monitoring while simultaneously exhibiting remarkable resilience against external and system uncertainties. Moreover, this controller design procedure can also be applied to stabilize alternative DC/DC converters. To demonstrate the effectiveness of the proposed ANN-based controller, a comparative analysis is conducted with conventional PI, MPC, and BSC controllers via MATLAB/Simulink simulation. Based on the aforementioned outcomes, it is evident that the proposed control method exhibits commendable performance in terms of settling time and overshoot, thereby showcasing superior robustness and dynamics.