Reference Modulation-Based H_∞ Control for the Hybrid Energy Storage System in DC Microgrids

Abstract

In DC microgrids, optimizing the hybrid energy storage system (HESS) current control to meet the power requirements of the load is generally a difficult and challenging task. This is because the HESS always operates under various load conditions, which are influenced by measurement disturbances and parameter uncertainties. Therefore, in this paper, we propose the H_∞ state feedback control based on the reference modulation to improve the current tracking errors of the battery (Bat) and supercapacitor (SC) in the HESS for power tracking performance. Without altering the system control signal, the reference modulation technique combines the feedforward channel and output feedback signal directly to modulate the required currents of the Bat and SC derived from the required load power. The H_∞ state feedback control based on the required Bat and SC currents modulated by the reference modulation technique is proposed to improve the current tracking errors under the influence of measurement disturbances and parameter uncertainties without a disturbance observer. The ability of the reference modulation technique to attenuate the disturbance without the use of a disturbance observer is one advantage for improving transient performance. The improvement of the HESS’s power tracking performance in DC microgrids is confirmed by study results presented under the influence of measurement disturbances for nominal parameters and parameter uncertainties.

1. Introduction

Global warming, climate change, the depletion of fossil fuel resources, and rising carbon emissions are all consequences of the growing demand for electric energy in recent years [1]. Eco-friendly energy sources are desperately needed to address these issues. As a result, electric vehicles and DC microgrids are now using renewable energy sources like wind and solar [2,3,4,5]. However, the primary reason for power fluctuations and instability is the erratic and sporadic nature of these renewable energy sources [6]. In order to address these problems, DC microgrids and electric vehicles employ energy storage systems made up of batteries (Bats) and supercapacitors (SCs) to balance the load’s power requirements [7,8,9]. However, Bats are unable to supply the required power of the load in transient response because of their high energy density and sluggish charging and discharging rates [10,11,12,13]. Because of their high power density and quick charging and discharging, SCs were developed to address Bats’ shortcomings [14,15,16,17,18,19]. However, SCs are unable to meet the high load requirement [20,21]. Thus, to solve all of the aforementioned problems in DC microgrids, a HESS comprising the high power and high energy density of SCs and Bats is needed, as illustrated in Figure 1. Controlling the HESS to meet the required power of the constantly fluctuating loads is extremely difficult because the HESS incorporates energy storage systems, such as slow dynamic in Bats and fast dynamic in SCs.

To tackle the above difficulty, PI- and PID-based control strategies for the HESS have been investigated in order to enhance the power performance of Bats and SCs [22,23,24]. The PI controller was created using frequency domain analysis [22], but stability is challenging because the method relies on small signal modeling. The conventional PID controller was created in [23,24] to be only locally stable, which lowers control quality when the load fluctuates constantly while the system is operating. The composite backstepping controllers in [25,26] and the model predictive controllers in [27,28] are two examples of the controller design studies for the HESS under load fluctuations that have been proposed to address problems [22,23,24]. However, it is challenging to guarantee control performance with these control methods because the HESS always operates with varying loads and parameter uncertainties. Multiple robust controllers, such as sliding mode controllers, have been developed to address these challenges [29,30]. However, in practice, SMCs also show discontinuity, and the chattering phenomenon in these SMCs is irreversible and challenging to minimize. Additionally, reinforcement learning control to estimate the HESS dynamics and iterative learning control to achieve smooth charging and discharging have been proposed [31,32,33]. Nevertheless, iterative computation leads to costly and sluggish transient response.

On the other hand, the stability and control quality of the HESS are significantly impacted by external disturbances, including measurement disturbances. To solve this problem, several extended state observer (ESO)-based control strategies have been proposed [34,35,36]. These include the cascade ESO in [34], the cascade–parallel ESO in [35], and the higher-order ESO in [36] to achieve disturbance rejection performance. However, these methods become more complex in terms of algorithmic implementation and computational time, and smooth transients cannot be guaranteed. Investigating DOB-based nonlinear control algorithms has also been suggested as a means of achieving robustness in control quality [37,38,39,40]. However, the HESS is a complicated system that consists of two energy storage systems with opposing features: Bats, which have a slow dynamic, and SCs, which have a fast dynamic. As a result, using a controller and a disturbance observer at the same time makes calculations more difficult. In practice, designing a disturbance observer can be particularly challenging due to the limited resources of microprocessors. Furthermore, the system parameters determine the disturbance observer’s design, and using erroneous model parameters will result in poorer estimation performance. Therefore, control methods without disturbance observers are required to attenuate the disturbances in the HESS.

To overcome the above-mentioned problems, this paper proposes the H_∞ state feedback control with reference modulation without a disturbance observer in order to improve the power tracking performance in the HESS. The following is a summary of the main ideas of the suggested approach:

First, the two conventional bidirectional DC/DC converters’ control inputs for charging and discharging are recommended in order to meet the required load power.

Second, the reference modulation technique is developed to modulate the battery and supercapacitor’s required currents from the required load power. Without altering the system control signal, the reference modulation technique directly combines the feedforward channel and output feedback signal. We demonstrate that the suggested modulation relies on a current tracking error tuning module that is model-free, offering greater customization options for the intended disturbance attenuation performance. Hence, the benefit of the suggested reference modulation technique is that it can reduce the disturbance without the need for a disturbance observer. Furthermore, the feedforward channel improves transient response by increasing the loop gain over a desired performance.

Third, to enhance current tracking errors in the HESS for power tracking performance under the influence of measurement disturbances and parameter uncertainties without a disturbance observer, the H_∞ state feedback control, which is based on the required currents of the battery and supercapacitor derived from the required load power modulated by the reference modulation technique, is proposed. The benefits of enhancing transient performance and attenuating disturbance without the need for a disturbance observer are noted in the above-mentioned second idea of the suggested approach.

Finally, the effectiveness of the proposed method is validated through study results with nominal parameters and parameter uncertainties under the influence of measurement disturbances.

This paper is composed of six sections. The hybrid energy storage system mathematical model is described in Section 2. The current tracking error dynamics are described in Section 3. The H_∞ state feedback control based on the reference modulation technique is described in Section 4. The study results for the proposed control method are given in Section 5. Finally, Section 6 presents the conclusions.

2. Mathematical Model of a Hybrid Energy Storage System

The HESS in DC microgrids consists of a load, two conventional bidirectional DC/DC converters, a battery pack, and a supercapacitor pack, as illustrated in Figure 2. The battery serves as the primary power source, and the output power is applied directly to the load while also preserving the load side’s balanced DC voltage. The SC recovers braking energy and gives the load its instantaneous peak power. Furthermore, the fully operational HESS minimizes battery damage from voltage and peak current fluctuations and enables the SC to function across a wider voltage range. The switches of the bidirectional DC/DC converters are controlled to satisfy both the energy demand strategy and the required load power. The mathematical model of the HESS during the switching phase in Figure 2 can be created using Kirchhoff’s law and the equivalent circuit model [12] is as follows:

$$\begin{array}{c}{\dot{\mathrm{i}}}_1={\displaystyle \frac{{\mathrm{V}}_{\mathrm{Bat}}}{{\mathrm{L}}_1}}-{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}{\mathrm{i}}_1-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}{\mathrm{d}}_1\\ {\dot{\mathrm{i}}}_2={\displaystyle \frac{{\mathrm{V}}_{\mathrm{SC}}}{{\mathrm{L}}_2}}-{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}{\mathrm{i}}_2-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}{\mathrm{d}}_2\\ {\dot{\mathrm{V}}}_{\mathrm{c}}=\left(1 -{ \mathrm{d}}_1\right){\displaystyle \frac{{\mathrm{i}}_1}{\mathrm{C}}}+\left(1 -{ \mathrm{d}}_2\right){\displaystyle \frac{{\mathrm{i}}_2}{\mathrm{C}}}-{\displaystyle \frac{{\mathrm{i}}_{\mathrm{load}}}{\mathrm{C}}}\end{array}$$

(1)

where ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are the battery side and supercapacitor side inductors; ${\mathrm{i}}_1$ and ${\mathrm{i}}_2$ are the battery side and supercapacitor side inductor currents; ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are the series resistances of ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$; ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are the battery side and supercapacitor side filter capacitors; C is the DC bus capacitor; V_Bat is the battery terminal voltage; ${\mathrm{V}}_{\mathrm{SC}}$ is the supercapacitor terminal voltage; ${\mathrm{V}}_{\mathrm{c}}$ is the relevant DC bus capacitor voltage; and ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ are the duty cycle of switches.

Remark 1. 

The load side is modeled as a current source instead of a resistive load because the load’s power requirements are always changing. A voltage source is used in place of the SC, internal characteristics are ignored, and the effects of uncertainty and disturbance are taken into consideration because the only goal of this study is to design the optimal current controller to satisfy the load power requirements for the HESS.

3. Current Tracking Error Dynamics

This section proposes the control inputs ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ in (1) of the switches of the bidirectional DC/DC converters in order to achieve the required load power, which varies continuously in practical applications, and stabilize the tracking current error dynamics.

Let us define the current tracking errors as follows:

$$\begin{array}{l}{\mathrm{e}}_1{=\mathrm{i}}_1^{\mathrm{d}}-{ \mathrm{i}}_1\\ {\mathrm{e}}_2{=\mathrm{i}}_2^{\mathrm{d}}-{ \mathrm{i}}_2\end{array}$$

(2)

where ${\mathrm{i}}_1^{\mathrm{d}}$ and ${\mathrm{i}}_2^{\mathrm{d}}$ represent the battery and supercapacitor’s required currents derived from the required load power. Based on (1) and (2), the following is a representation of the current tracking error dynamics:

$$\begin{array}{l}{\dot{\mathrm{e}}}_1={\dot{\mathrm{i}}}_1^{\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{Bat}}}{{\mathrm{L}}_1}}+{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}{\mathrm{i}}_1+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}{\mathrm{d}}_1\\ {\dot{\mathrm{e}}}_2={\dot{\mathrm{i}}}_2^{\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{SC}}}{{\mathrm{L}}_2}}+{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}{\mathrm{i}}_2+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}{\mathrm{d}}_2\end{array}$$

(3)

For the load power requirements, we can now propose the control inputs ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ to stabilize the current tracking error dynamics in (3) as follows:

$$\begin{array}{l}{\mathrm{d}}_1={\displaystyle \frac{{\mathrm{L}}_1}{{\mathrm{V}}_{\mathrm{c}}}}\left({\dot{\mathrm{i}}}_1^{\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{Bat}}}{{\mathrm{L}}_1}}+{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}{\mathrm{i}}_1^{\mathrm{d}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}{+\mathrm{v}}_1\right)\\ {\mathrm{d}}_2={\displaystyle \frac{{\mathrm{L}}_2}{{\mathrm{V}}_{\mathrm{c}}}}\left({\dot{\mathrm{i}}}_2^{\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{SC}}}{{\mathrm{L}}_2}}+{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}{\mathrm{i}}_2^{\mathrm{d}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}{+\mathrm{v}}_2\right)\end{array}$$

(4)

where ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ are state feedback control inputs that are designed in the next section using H∞ optimization with reference modulation.

By substituting (4) into (3), the current tracking error dynamics can be rewritten as follows:

$$\begin{array}{l}{\dot{\mathrm{e}}}_1=-{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}{\mathrm{e}}_1-{ \mathrm{v}}_1\\ {\dot{\mathrm{e}}}_2=-{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}{\mathrm{e}}_2-{ \mathrm{v}}_2\end{array}$$

(5)

Remark 2. 

In this study, we only concentrate on designing the control inputs ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ to meet the required load power by stabilizing the dynamics of the current tracking errors. As a result, we assume that earlier research has balanced the DC bus voltage in the HESS [12,39]. However, we should also note that the DC bus voltage balancing in the HESS is guaranteed when the current tracking error dynamics (3) are stabilized, as the DC bus voltage control law is dependent on the battery and supercapacitor currents [12,39].

4. H_∞ State Feedback Control Based on the Reference Modulation Technique

The reference modulation technique for the required currents of the battery and supercapacitor derived from the required load power and the optimal control design based on the reference modulation technique for ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ mentioned in Section 3 are proposed in this section. The H_∞ norm is used to guarantee the current tracking error performance under the influence of parameter uncertainties and measurement disturbances.

4.1. Reference Modulation Technique

The reference modulation technique modulates the required currents of the battery and supercapacitor derived from the required load power without directly altering the system control signal. The reference modulation technique directly combines the feedforward channel and the actual output feedback signal as follows:

$$\begin{array}{c}{\mathrm{i}}_1^{*\mathrm{d}}{=\mathrm{i}}_1^{\mathrm{d}}{+\mathrm{M}}_1\\ {\mathrm{i}}_2^{*\mathrm{d}}{=\mathrm{i}}_2^{\mathrm{d}}{+\mathrm{M}}_2\end{array}$$

(6)

where ${\mathrm{i}}_1^{\mathrm{d}}$ and ${\mathrm{i}}_2^{\mathrm{d}}$ are direct feedforward channels, and ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ are modulation modules of the actual output feedback signal as follows:

$$\begin{array}{c}{\mathrm{M}}_1{=\mathit{\alpha }}_1{\mathrm{e}}_1{+\mathit{\alpha }}_2{\displaystyle \int {\mathrm{e}}_1}\mathrm{dt}\\ {\mathrm{M}}_2{=\mathit{\alpha }}_1{\mathrm{e}}_2{+\mathit{\alpha }}_2{\displaystyle \int {\mathrm{e}}_2}\mathrm{dt}\end{array}$$

where ${\mathit{\alpha }}_1$ and ${\mathit{\alpha }}_2$ are modulation gains, and these modulation gains are determined using the study’s methodology [41].

Remark 3. 

The advantage of the suggested reference modulation technique for the required currents of the battery and supercapacitor derived from the required load power can attenuate the desired disturbance performance and improve the transient performance for reference tracking. Since the suggested modulation relies on a current tracking error tuning module in (6) that is model-free, it provides more customization options for the intended disturbance attenuation performance. Additionally, the feedforward channel raises the loop gain over a desired performance, which enhances transient response [41].

4.2. H_∞ State Feedback Control Based on the Reference Modulation Technique

Enhancing current tracking error performance in the presence of measurement disturbances and parameter uncertainties is the goal of the H_∞ state feedback control, which is based on the required currents of the battery and supercapacitor that were modulated in the previous subsection. The control inputs ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ in (4) are rewritten as follows:

$$\begin{array}{l}{\mathrm{d}}_1={\displaystyle \frac{{\mathrm{L}}_1}{{\mathrm{V}}_{\mathrm{c}}}}\left({\dot{\mathrm{i}}}_1^{*\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{Bat}}}{{\mathrm{L}}_1}}+{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}{\mathrm{i}}_1^{*\mathrm{d}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_1}}{+\mathrm{v}}_1\right)\\ {\mathrm{d}}_2={\displaystyle \frac{{\mathrm{L}}_2}{{\mathrm{V}}_{\mathrm{c}}}}\left({\dot{\mathrm{i}}}_2^{*\mathrm{d}}-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{SC}}}{{\mathrm{L}}_2}}+{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}{\mathrm{i}}_2^{*\mathrm{d}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{L}}_2}}{+\mathrm{v}}_2\right)\end{array}$$

(7)

Then, the current tracking error dynamics in (5) can be represented in the form of

$$\begin{array}{c}{\dot{\mathrm{e}}}^{*}{=\mathrm{A}}_{\mathrm{e}}{\mathrm{e}}^{*}{+\mathrm{B}}_{\mathrm{e}}{\mathrm{v}+\mathrm{B}}_1\mathrm{w}\\ {\mathrm{z}=\mathrm{C}}_1{\mathrm{e}}^{*}{+\mathrm{D}}_{12}\mathrm{v}\end{array}$$

(8)

where

$$\begin{array}{c}{\mathrm{e}}_1^{*}{=\mathrm{i}}_1^{*\mathrm{d}}-{ \mathrm{i}}_1{, \mathrm{e}}_2^{*}{=\mathrm{i}}_2^{*\mathrm{d}}-{ \mathrm{i}}_2\\ {\mathrm{e}}^{*}=\left[\begin{array}{cc}{\mathrm{e}}_1^{*}& {\mathrm{e}}_2^{*}\end{array}\right], \mathrm{v}={\left[\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_2\end{array}\right]}^{\mathrm{T}}\\ {\mathrm{A}}_{\mathrm{e}}=\left[\begin{array}{cc}-{\displaystyle \frac{{\mathrm{R}}_1}{{\mathrm{L}}_1}}& 0\\ 0& -{\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{L}}_2}}\end{array}\right]{, \mathrm{B}}_{\mathrm{e}}=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\end{array}$$

where $\mathrm{w}\in {\mathrm{R}}^{2\times 1}$ is the signal for the measurement disturbance, $\mathrm{z}\in {\mathrm{R}}^2$ is the signal of the objective function, ${\mathrm{B}}_1^{\mathrm{T}}{\mathrm{B}}_1\in {\mathrm{R}}^{2\times 2}$ is the matrix of covariance, and ${\mathrm{C}}_1\in {\mathrm{R}}^{2\times 2}$ and ${\mathrm{D}}_{12}\in {\mathrm{R}}^{2\times 2}$ are the weighting matrices of the tracking errors and control inputs, respectively.

Under the influence of measurement disturbance and parameter uncertainties, the H_∞ state feedback control, which is based on the required currents of the battery and supercapacitor that are modulated in (6), improves the current tracking error performance as follows:

$${\mathrm{v}=\mathrm{K}}_{\infty }{\mathrm{e}}^{*}$$

(9)

where ${\mathrm{K}}_{\infty }\in {\mathrm{R}}^{2\times 2}$ is the control gain.

In the optimal control design, we employed the H_∞ control in the linear matrix inequality (LMI) technique [42] to ascertain the control gain in (9). The following is a definition of the performance specification in the H_∞ norm:

$${‖{\mathrm{T}}_{\mathrm{zw}}‖}_{\infty }=\sup {\displaystyle \frac{{‖\mathrm{z}‖}_2}{{‖\mathrm{w}‖}_2}} < \mathsf{\lambda }, \forall \mathrm{w}\ne 0$$

(10)

The LMI condition is used to determine the bound $\mathsf{\lambda }$ on the H_∞ norm by the H_∞ state feedback control. The following statements are equivalent.

• There exists ${\mathrm{v}=\mathrm{K}}_{\infty }{\mathrm{e}}^{*}$ such that ${‖{\mathrm{T}}_{\mathrm{zw}}‖}_{\infty }< \mathsf{\lambda }$
• There exists ${\mathsf{\Gamma }}_{\infty } > 0$ such that

$$\left[\begin{array}{ccc}\left({\mathrm{A}}_{\mathrm{e}}{+\mathrm{B}}_{\mathrm{e}}{\mathrm{K}}_{\infty }\right){\mathsf{\Gamma }}_{\infty }{+\mathsf{\Gamma }}_{\infty }{\left({\mathrm{A}}_{\mathrm{e}}{+\mathrm{B}}_{\mathrm{e}}{\mathrm{K}}_{\infty }\right)}^{\mathrm{T}}& {\mathrm{B}}_1& {\mathsf{\Gamma }}_{\infty }{\left({\mathrm{C}}_1{+\mathrm{D}}_{12}{\mathrm{K}}_{\infty }\right)}^{\mathrm{T}}\\ {\mathrm{B}}_1^{\mathrm{T}}& -\mathrm{I}& 0\\ \left({\mathrm{C}}_1{+\mathrm{D}}_{12}{\mathrm{K}}_{\infty }\right){\mathsf{\Gamma }}_{\infty }& 0& -{\mathsf{\lambda }}^2\mathrm{I}\end{array}\right] < 0$$

We then define a new variable ${\mathsf{\Omega }}_{\infty }{=\mathrm{K}}_{\infty }{\mathsf{\Gamma }}_{\infty }$, such that the control gain can be recovered by ${\mathrm{K}}_{\infty }{=\mathsf{\Omega }}_{\infty }{\mathsf{\Gamma }}_{\infty }^{-1}$ with the following LMI condition:

$$\left[\begin{array}{ccc}{\mathrm{A}}_{\mathrm{e}}{\mathsf{\Gamma }}_{\infty }{+\mathsf{\Gamma }}_{\infty }{\mathrm{A}}_{\mathrm{e}}^{\mathrm{T}}{+\mathrm{B}}_{\mathrm{e}}{\mathsf{\Omega }}_{\infty }{+\mathsf{\Omega }}_{\infty }^{\mathrm{T}}{\mathrm{B}}_{\mathrm{e}}^{\mathrm{T}}& {\mathrm{B}}_1& {\mathsf{\Gamma }}_{\infty }{\mathrm{C}}_1^{\mathrm{T}}{+\mathsf{\Omega }}_{\infty }^{\mathrm{T}}{\mathrm{D}}_{12}^{\mathrm{T}}\\ {\mathrm{B}}_1^{\mathrm{T}}& -\mathrm{I}& 0\\ {\mathrm{C}}_1{\mathsf{\Gamma }}_{\infty }{+\mathrm{D}}_{12}{\Omega }_{\infty }& 0& -{\mathsf{\lambda }}^2\mathrm{I}\end{array}\right] < 0$$

Figure 3 displays the proposed control method’s block diagram in the HESS.

5. Study Results

This section evaluates the efficacy of the proposed approach in the HESS by performing a study using the MATLAB(R2021b)/Simulink platform.

Table 1. HESS parameters [43].

Parameter	Symbol	Value
Inductor on the Bat side	${\mathrm{L}}_1$	$2.6\times {10}^{-3} \mathrm{H}$
$\mathrm{Series} \mathrm{resistance} \mathrm{of} \mathrm{inductor} {\mathrm{L}}_1$	${\mathrm{R}}_1$	$0.2 \mathsf{\Omega }$
Inductor on the SC side	${\mathrm{L}}_2$	$1.8\times {10}^{-3} \mathrm{H}$
$\mathrm{Series} \mathrm{resistance} \mathrm{of} \mathrm{inductor} {\mathrm{L}}_2$	${\mathrm{R}}_2$	$0.15 \mathsf{\Omega }$
Filter capacitor on the Bat side	${\mathrm{C}}_1$	$0.7\times {10}^{-2} \mathrm{F}$
Filter capacitor on the SC side	${\mathrm{C}}_2$	$0.5\times {10}^{-2} \mathrm{F}$
DC bus capacitor	$\mathrm{C}$	$1.5\times {10}^{-3} \mathrm{F}$
DC bus voltage	${\mathrm{V}}_{\mathrm{c}}$	$200 \mathrm{V}$
Bat terminal voltage	${\mathrm{V}}_{\mathrm{Bat}}$	$180 \mathrm{V}$
SC terminal voltage	${\mathrm{V}}_{\mathrm{SC}}$	$150 \mathrm{V}$

and

Table 2. Control and modulation gains of the proposed approach.

Parameter	Symbol	Value
Modulation gain 1	${\mathit{\alpha }}_1$	$1\times {10}^{-2}$
Modulation gain 2	${\mathit{\alpha }}_2$	$9.8\times {10}^3$
Control gain	$\mathsf{\lambda }$	$1\times {10}^3$
PI control gains	${\mathrm{K}}_{\mathrm{P}}$	$20$
	${\mathrm{K}}_{\mathrm{I}}$	$1\times {10}^3$

display the HESS parameters and the control and modulation gains. Figure 4, Figure 5 and Figure 6 illustrate the required power profiles for the load, battery, and supercapacitor:

where

$$\mathrm{Required} \mathrm{load} \mathrm{power}=\mathrm{Required} \mathrm{Bat} \mathrm{power}+\mathrm{Required} \mathrm{SC} \mathrm{power}$$

and the battery and supercapacitor’s required currents are determined as follows, respectively:

$$\begin{array}{c}{\mathrm{i}}_1^{\mathrm{d}}={\displaystyle \frac{\mathrm{Required} \mathrm{Bat} \mathrm{power}}{{\mathrm{V}}_{\mathrm{Bat}}}}\\ {\mathrm{i}}_2^{\mathrm{d}}={\displaystyle \frac{\mathrm{Required} \mathrm{SC} \mathrm{power}}{{\mathrm{V}}_{\mathrm{SC}}}}\end{array}$$

5.1. Study Results with Nominal Parameters

In this subsection, we perform simulations with measurement disturbances injected into the inductor currents of the battery ${\mathrm{i}}_1$ and the supercapacitor ${\mathrm{i}}_2$ in Figure 7 and Figure 8 to assess how well the proposed method improves the current tracking error performance to meet the required power of the variable load in practical applications for the HESS. The following is an analysis and comparison of the simulation results in three scenarios.

• Case 1: Classical PI control.
• Case 2: Traditional H_∞ control.
• Case 3: Proposed method.

We start by examining the performance of the ${\mathrm{i}}_1$ battery’s current tracking. As seen by the battery’s required load power profile in Figure 5, the battery only takes part in the discharge process, and between 3 s and 5 s, the discharge power progressively rises to the maximum requirement of 1000 W. After 5 s of each cycle, the battery will begin to take part in the discharge process, which will lower its power from 1000 W to 0. Figure 9 and Figure 10 show the ${\mathrm{i}}_1$ battery’s current tracking performance. The effects of the measurement disturbance in Figure 7 can be lessened by the current tracking errors of cases 1, 2, and 3. However, as Figure 11a for cycle 1, Figure 11b for cycle 2, and Figure 11c for cycle 3 demonstrate, the current tracking errors of cases 1 and 2 never converge to zero during the time intervals from 3.1 s to 3.2 s, 13.1 s to 13.2 s, and 23.1 s to 23.2 s for three cycles. The current tracking errors of the battery discharge process in case 3 converge faster than those in cases 1 and 2 after being affected by the measurement disturbance because the H_∞ norm in (10) always guarantees that the efficiency of H_∞ control and the advantage of the suggested reference modulation technique of the required current, derived from the required Bat power, are applied. Furthermore, the large fixed controller gains of the classical PI control strategy used in case 1 make it evident that the transient response of case 1 is inferior to that of cases 2 and 3.

Second, we analyze the performance of the ${\mathrm{i}}_2$ supercapacitor’s current tracking. The supercapacitor participates in both the charging and discharging processes, as evidenced by the required power profile in Figure 6. The load’s primary power source is the battery, which has a 1000 W limit. However, based on Figure 4’s continuously fluctuating power requirement, the required load power before 3 s is higher than 1000 W and begins at 2000 W per cycle. The supercapacitor energy supplies an instantaneous peak power to prevent the battery from being overloaded for 3 s and then the HESS reaches the required load power. This is made possible by the supercapacitor’s rapid discharge. Additionally, the supercapacitor’s fast charging capability allows it to recover braking energy from the required load power from 6 s to 10 s per cycle. This quick energy storage guarantees that the battery will be supported during transient responses and that the required load power will be satisfied. Consequently, the battery life of the HESS is extended. The ${\mathrm{i}}_2$ supercapacitor’s current tracking performance is displayed in Figure 12 and Figure 13. In cases 1, 2, and 3, the current tracking errors during the discharge and charge of the supercapacitor can mitigate the effects of the measurement disturbance in Figure 8 on the time range of 1.1 s to 1.2 s, 11.1 s to 11.2 s, and 21.1 s to 21.2 s, respectively, for three cycles. However, under the influence of the measurement disturbance, the current tracking errors of cases 1 and 2 do not converge to zero, while those of case 3 do, as shown in Figure 14a for cycle 1, Figure 14b for cycle 2, and Figure 14c for cycle 3. Additionally, after being impacted by the measurement disturbance, the current tracking errors of the supercapacitor discharge process in case 3 converge faster than those in cases 1 and 2. Figure 15a–c demonstrate that case 3’s convergence time of the current tracking errors to zero at the start times of 0 s, 10 s, and 20 s of each cycle is quicker than cases 1 and 2. Similar to the battery, case 1’s transient response is subpar compared to cases 2 and 3. This is because of the classical PI controller’s previously stated drawbacks. Thus, we confirm that the ${\mathrm{i}}_2$ supercapacitor’s current tracking performance in case 3 is better than in cases 1 and 2. Since the H_∞ norm in (10) also guarantees the efficacy of H_∞ control, the advantage of the suggested reference modulation technique of the required current derived from the required SC power is also applied.

Finally, to assess how well the suggested approach controls the battery and supercapacitor current to provide the required load power, Figure 16 and Figure 17 display the power tracking performance results. Under the measurement disturbance shown in Figure 7 and Figure 8, the power tracking performance during the charge and discharge processes is enhanced in both scenarios 1, 2, and 3. However, as Figure 18a–c demonstrate, the power tracking errors of cases 1 and 2 under the influence of measurement disturbances on the battery and supercapacitor currents during all three cycles are unable to converge to zero, whereas the current tracking errors of case 3 rapidly do so. More specifically, in case 2, under the influence of measurement disturbances on battery and supercapacitor currents, the power tracking errors from 3.1 s to 3.2 s, from 13.1 s to 13.2 s, and from 23.1 s to 23.2 s are 32.5 W, and from 1.1 s to 1.2 s, from 11.1 s to 11.2 s, and from 21.1 s to 21.2 s are 35.3 W in all three cycles. Meanwhile, in case 1, the power tracking errors are greater than 32.5 W and 35.3 W during the disturbance-affected periods. Furthermore, at the beginning of each cycle and during times when measurement disturbances affect the battery and supercapacitor currents, the convergence time performance in case 3 is likewise faster than that in cases 1 and 2, as shown in Figure 18a–c and Figure 19a–c. The rate of change (ROC) for cases 1 and 3 is −98.57% after 0 s and −49.93% after 1.2 s and 3.2 s, according to

Table 3. Performance comparison of convergence time in the three cases.

Convergence Time Performance (s)		Case 1	Case 2	Case 3	ROC for Cases 1 and 3	ROC for Cases 2 and 3
Cycle 1 after:	0 s	0.35	0.025	0.005	−98.57%	−80%
	1.2 s	0.4	0.22	0.2003	−49.93%	−8.95%
	3.2 s	0.4	0.22	0.2003	−49.93%	−8.95%
Cycle 2 after:	10 s	0.35	0.025	0.005	−98.57%	−80%
	11.2 s	0.4	0.22	0.2003	−49.93%	−8.95%
	13.2 s	0.4	0.22	0.2003	−49.93%	−8.95%
Cycle 3 after:	20 s	0.35	0.025	0.005	−98.57%	−80%
	21.2 s	0.4	0.22	0.2003	−49.93%	−8.95%
	23.2 s	0.4	0.22	0.2003	−49.93%	−8.95%

ROC = ((Case 3 − Cases 1 or 2)/Cases 1 or 2) × 100.

for cycle 1 for further analysis. In contrast, the ROC for cases 2 and 3 is −80% after 0 s and −8.95% after 1.2 s and 3.2 s. According to these ROC results, the proposed strategy in case 3 converges more quickly than the classical PI control method in case 1 and the H_∞ control in case 2. Furthermore, it is evident that case 2’s rate of convergence is faster than case 1’s when comparing the ROCs of cases 1 and 2 with case 3. A similar methodology is used to analyze the rate of convergence in cycles 2 and 3. We verify that the proposed control approach has effectively improved power tracking performance to satisfy the required load power for the variable load while being affected by measurement disturbance in practical applications; this is because, in case 3, the effectiveness of H_∞ control is always guaranteed by the H_∞ norm in (10). In particular, the advantage of the proposed reference modulation technique in case 3 is that it attenuates disturbances without requiring the design of a disturbance observer. The traditional approach, which only employs traditional H_∞ control in case 2 and classical PI control in case 1, differs significantly from case 3’s disturbance attenuation without a disturbance observer.

5.2. Study Results with Parameter Uncertainties

In this subsection, we present more research results to evaluate the robustness of the proposed method in the HESS to parameter uncertainties. The electrical structure and prolonged charging and discharging operations of the two conventional bidirectional DC/DC converters lead to manufacturing tolerance, which implies that their parameters are not fixed and will change in response to uncertainties in the system parameters. Depending on the temperature and frequency of the environment, the electrical parameters can be more easily changed to function like the series resistances (R₁ and R₂) and inductors (L₁ and L₂). In order to assess the robustness of the suggested method, we compared the study results with uncertainties in the series resistances and inductors up to 20% of the nominal parameters as follows:

Figure 20 and Figure 21 display the current tracking performance of the battery ${\mathrm{i}}_1$ and the supercapacitor ${\mathrm{i}}_2$ results compared to the nominal parameter results. In case 2 of the suggested method, we observe that the amplitude and setting time of the current tracking errors in Figure 22 and Figure 23 are similar to the nominal parameter results. Thus, in the case of parameter uncertainties, the current tracking performance of the battery ${\mathrm{i}}_1$ and the supercapacitor ${\mathrm{i}}_2$ in Figure 20 and Figure 22 is guaranteed. This results in the outcomes illustrated in Figure 24 and Figure 25, which indicate that the power tracking performance of the HESS is also guaranteed and satisfies the power requirement of the continuously varying load in practical applications. Because the H_∞ norm in (10) always guarantees the H_∞ control based on the recommended reference modulation technique for required currents, we conclude that the suggested method is robust against measurement disturbances and parameter uncertainties.

6. Conclusions

This paper proposes a control strategy to enhance the power tracking performance of the HESS in DC microgrids, incorporating reference modulation and H_∞ state feedback control. In order to modulate the required currents of the battery and supercapacitor derived from the required load power, the reference modulation technique combined the feedforward channel with the actual output feedback signal without directly altering the system control signal. The proposed H_∞ state feedback control, which was based on the required currents modulated by the reference modulation technique, was designed to improve the current tracking errors in the battery and supercapacitor. The HESS power tracking error and battery and supercapacitor current tracking errors converged to zero in the steady state region more quickly with the suggested control strategy. The proposed control strategy showed a similar transient response to the results of the nominal parameters when parameter uncertainties were considered. The study results validated the enhancement of power tracking performance and robustness of the HESS in DC microgrids under the influence of measurement disturbances and parameter uncertainties without a disturbance observer.

The design of a reference modulation optimization technique utilizing the mixed H₂/H_∞ with multiple control objectives in power tracking performance in the HESS is one potential avenue for future research. Examining the calculation burden in hard real-time systems for the experimental environment is another potential area of future research.