Capacitor-Energy-Based Super-Twisting Sliding Mode Control for Flywheel Energy Storage System DC-Bus Voltage in DC Microgrid

Abstract

To address the DC-link voltage control issue in flywheel energy storage systems (FESSs), a DC-link voltage control strategy using a capacitor-energy-based super-twisting sliding mode controller (CE-STSMC), integrated with a disturbance observer, is proposed in this article. First, an exponential term is incorporated into the STSMC algorithm to enhance its convergence rate. Then, the improved STSMC is employed as the voltage-loop controller to mitigate the insufficient anti-disturbance capability of conventional control methods. To improve the system robustness, a nonlinear disturbance observer (NDOB) is developed to estimate the load power. The estimated disturbance is further feedforward-compensated into the improved STSMC controller. Finally, experiments are carried out on a 2.2 kW FESS prototype under DC-link voltage step and sudden load-change conditions, which demonstrates the effectiveness and superiority of the proposed control strategy.

1. Introduction

In recent years, flywheel energy storage systems (FESSs) have drawn growing interest for both academia and industry, which largely attributes to the progress in composite material technology and the distinctive benefits of FESSs, including quick transient response, minimal maintenance cost, and extended operational lifespan [1,2,3]. When FESSs are applied in DC microgrids, it is necessary to maintain the DC-link voltage to meet operational requirements in [4,5]. Otherwise, the proper functioning of critical loads and even the overall system stability could be severely compromised. A flywheel energy storage-based DC microgrid is shown in Figure 1, where permanent magnet synchronous generator (PMSG) is applied as the flywheel energy storage system.

Figure 1. DC microgrid equipped with FESSs.

When the flywheel energy storage system operates within a DC microgrid, the flywheel speed continuously adjusts to meet the energy demands of the load in [6,7]. Furthermore, the equivalent load on the DC side will appear abrupt variations in [8,9]. The DC-link capacitance gradually declines over time due to aging or temporarily rise when additional equipment is connected to the DC bus. Such capacitance variation is an important issue because the value of capacitor directly determines the DC-link energy buffer and voltage dynamics. A reduced capacitance increases the DC-link voltage ripple and disturbance sensitivity. Consequently, the DC-link voltage may exceed its allowable range, which can deteriorate power quality and affect compliance with grid-code requirements in grid-connected FESS applications. Therefore, a DC-link voltage controller should ensure robust voltage tracking under speed, load, and capacitance variations.

To improve the DC-link voltage control performance, numerous methods for voltage control have been studied, such as voltage control methods equipped with proportional integral (PI) in [10,11], sliding model control (SMC) [12], extended state observer (ESO) [13] and model predictive control (MPC) [14]. In [11], a PI controller is employed in voltage loop to generate reference current for the inner current loop. In [13], a nonlinear ESO is used to realize global linearization and for the estimation of the total disturbance. Simultaneously, to mitigate the impact of speed variations on bus voltage control, a speed-adaptive feedback rate has been devised. However, the aforementioned method achieves DC-link voltage control through localised linearisation, which is applicable only near the steady-state operating point. The motor speed significantly fluctuates for the operation of the flywheel energy storage system, which deteriorates the control performance across the entire speed range.

SMC is widely recognized for robust control of nonlinear systems under uncertainties and disturbances, hence the SMC are widely employed as DC-link voltage controllers [15]. Nevertheless, the robustness of SMC is usually obtained at the expense of increased control gains, and excessive gains tend to induce pronounced chattering, potentially stimulating high-frequency dynamics [16]. To deal with the problems, many improved SMC strategies have been proposed, such as sliding mode control based on improved convergence rate, observer-based sliding mode control, and higher-order sliding mode control.

In [17], an outer-loop SMC with an integral switching surface and an improved exponential reaching law is proposed for voltage regulation in PMSG system. The proposed method can attenuate chattering while improving load-step transients and eliminating steady-state DC-link voltage error, which has been validated by experiments. For the purposes of improve the dynamic performance of the DC-link voltage for the FESSs, an nonlinear function is employed to replace the sign or sat function of the SMC controller in [18]. Although increasing the convergence rate can mitigate sliding mode oscillations, this comes at the cost of increased system complexity, thereby necessitating the design of additional parameters. Moreover, the high-order sliding model methods are regarded as an alternatively strategy to suppress the chatting phenomenon in [19,20,21,22].

Owing to the simple structure and robust performance, the super-twisting sliding mode control (STSMC) has received significant attention within both industry and academia. However, the super-twisting sliding mode algorithm requires that the disturbance be bounded. To address this limitation, an observer was introduced into the STSMC [23]. In [24], a reduced extended state observer (RESO) is incorporated into the sliding model speed controller. System uncertainties including parameter mismatches and unmodeled dynamics are lumped into a total disturbance, which is estimated by the RESO and then compensated. Based on the same idea, in [25], a load observer based on sliding mode is proposed to further improve the anti-load disturbance capability for the SMC method. In [26], a super-twisting nonlinear extended state observer incorporated with robust integral of the sign of the error is developed to estimate the load disturbance for the servo system. However, when embedded within sliding mode controllers, the aforementioned observers require complex parameter design and feature relatively intricate structures.

Based on the above analysis, a capacitor-energy-based super-twisting sliding mode controller (CE-STSMC) with a simple nonlinear disturbance observer (NDOB) is proposed in this article to enhance the disturbance-rejection capability of flywheel energy storage systems (FESSs), where the dominant disturbances mainly arise from sudden load application and the wide-range speed variation during the discharge process. First, the mathematical model based on the capacitive energy is established and analyzed. Based on the capacitive energy model, the CE-STSMC is designed to enhance the system robustness. Moreover, an adaptive continuous function is introduced to replace the discontinuous signum function, thereby alleviating chattering. In addition, considering the influence of total disturbance in the control system of FESSs, especially load and speed disturbances, an NDOB is introduced. The proposed CE-STSMC leverages high-order sliding-mode features to mitigate observer-induced chattering while enhancing disturbance estimation accuracy. The system robustness is further enhanced by feeding the estimated total disturbance into the CE-STSMC. Finally, the superiority of the proposed method are verified by the experiments.

The rest of this article is organized as follows. Section 2 establishes the mathematical model based on capacitive energy. Section 3 presents a NDOB integrated into the STCEC for improving the system robustness. Section 4 provides the simulation and experimenta results that demonstrted the effectiveness of the proposed method, followed by the conclusion.

2. Capacitor-Energy-Based FESSs Model

Figure 2 shows the topology of a flywheel energy storage system based on a permanent magnet synchronous motor (PMSM). The primary structure of FESSs is composed of PMSM coupled with a coaxial flywheel for kinetic energy storage, a three-phase bidirectional converter, capacitors on the DC link, and an equivalent DC load. In Figure 2, C represents the DC-link capacitor, $R_L$ stands for equivalent load, encompassing both linear and nonlinear components. $i_{dc}$ is the DC-link current, and $i_{Load}$ represents the load current. Based on the synchronous rotating coordinate transformation, the mathematical model of FESSs can be expressed as [27]

$$\begin{array}{cc}\hfill L_{d}p{i}_d& =-R_s{i}_d+{\omega }_e{L}_q{i}_q+u_d\hfill \\ \hfill L_{q}p{i}_q& =-R_s{i}_q-{\omega }_e{L}_d{i}_d-{\omega }_e{\psi }_f+u_q\hfill \\ \hfill T_e=& 1.5P_n{i}_q[(L_d-L_q)i_d+{\psi }_f]\hfill \\ \hfill C{\displaystyle \frac{d{u}_{dc}}{dt}}& =i_{dc}-i_{Load}\hfill \end{array}$$

(1)

where $i_d$, $i_q$, $u_d$, $u_q$, $L_d$ and $L_q$ are the d- and q-axes components of current, voltage and inductance, respectively; p is the differential operator, i.e., $p=d/dt$; ${\psi }_f$, $P_n$ and $T_e$ are flux linkage, number of pole pairs and electromagnetic torque, and $u_{dc}$ represents DC-link voltage, respectively.

Figure 2. The main circuit of the FESSs.

The power balance equation between the two sides of the converter in the FESS can be expressed as

$$u_{dc}{i}_{dc}=-1.5(u_d{i}_d+u_q{i}_q)-P_{switch}$$

(2)

where $P_{switch}$ is the switching loss and the negative sign indicates that the permanent magnet synchronous motor is operating in generator mode.

Substituting (1) into (2), then

$$\begin{array}{c}\hfill C{u}_{dc}{\displaystyle \frac{d{u}_{dc}}{dt}}+u_{dc}{i}_{Load}+P_{switch}=-T_e{\omega }_m-1.5\left\{R_s(i_d^2+i_q^2)+L_q{i}_q{\displaystyle \frac{d{i}_q}{dt}}+L_d{i}_d{\displaystyle \frac{d{i}_d}{dt}}\right\}\end{array}$$

(3)

It can be observed that the DC-link voltage loop is nonlinear; therefore, the traditional PI controller cannot achieve satisfactory performance. In order to linearize (3), the process of energy transfer is subsequently taken into account, and (3) can be rewritten as

$${\displaystyle \frac{d{E}_c}{dt}}={\displaystyle \frac{d}{dt}}\left(0.5C{u}_{dc}^2\right)=-T_e{\omega }_m-P_{load}-P_{cu}-P_L-P_{switch}$$

(4)

where $P_{load}=u_{dc}{i}_{load}$ is the load power, $P_{cu}=1.5R_s(i_d^2+i_q^2)$ is the copper power of the PMSM, $P_L=1.5p(0.5L_d{i}_d^2+0.5L_q{i}_q^2)$ is the inductive energy storage power.

When the power loss is treated as total consumed power, the (4) is derived as

$${\displaystyle \frac{d{E}_c}{dt}}=b{i}_q+P_{total}$$

(5)

where $b=-1.5{\psi }_f{\omega }_e$, $P_{total}=-(P_{load}+P_{cu}+P_L+P_{switch})$.

3. Proposed Robust Voltage Control Method for FESSs

As previously discussed, the FESS is sensitive to the load disturbance when controlled by a conventional proportional–integral (PI) regulator, making it challenging to maintain satisfactory dynamic performance. Sliding mode control (SMC) is recognized for its strong robustness against system uncertainties and external disturbances in [28,29]; nevertheless, the inherent chattering effect significantly restricts its practical application. To address this issue, a super-twisting sliding mode voltage controller is developed to enhance disturbance rejection while reducing chattering behavior. Moreover, a nonlinear disturbance observer is designed to further strengthen the robustness of the system, particularly against load variations. The observer continuously estimates load disturbances in real time, and the estimated value is employed as a feedforward term to compensate for the controller’s output.

3.1. Capacitor-Energy-Based Super-Twisting Sliding Mode Controller Design

Considering the control objectives of the FESSs and the design principle of the STSMC, the variable of the system is defined as

$$e=E_c^{*}-E_c$$

(6)

where $E_c^{*}$ denotes the reference value of the DC-link capacitor energy, given by $E_c^{*}=0.5C{\left(u_{dc}^{*}\right)}^2$, with $u_{dc}^{*}$ being reference DC-link voltage, respectively.

Then, the sliding surface is defined as

$$s=e=E_c^{*}-E_c$$

(7)

By differentiating (7) and combining it with the capacitor-energy dynamics in (5), the derivative of the sliding variable can be obtained as

$$\dot{s}={\dot{E}}_c^{*}-{\dot{E}}_c={\dot{E}}_c^{*}-b{i}_q-P_{\mathrm{total}}$$

(8)

Since $E_c^{*}$ is constant within each control interval, ${\dot{E}}_c^{*}=0$, and thus

$$\dot{s}=-b{i}_q-P_{\mathrm{total}}$$

(9)

In this paper, the STSMC output u is regarded as the desired reaching law of the sliding variable, namely

$$\dot{s}=u$$

(10)

where the improved STSMC out u is designed as

$$\left\{\begin{array}{c}u=-k_1{\left|s\right|}^{{\alpha }_1}\mathrm{sgn}\left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}-\lambda s+\vartheta \hfill \\ \dot{\vartheta }=-\epsilon \mathrm{sgn}\left(s\right)\hfill \end{array}\right.$$

(11)

where u and $\vartheta$ are control laws of STSMC, $k_1$, $k_2$, $\lambda$ and $\epsilon$ are the positive switching gains, respectively. ${\alpha }_1$ and ${\alpha }_2$ are the exponent gains.

Compared to traditional STSMC, the enhanced STSMC incorporates an additional nonlinear term $k_2{\left|s\right|}^{{\alpha }_2}$ and a linear term $\lambda s$. The added term can increase the convergence rate of the sliding mode controller when $\parallel s\parallel$ > 1. Therefore, the dynamic performance of the STSMC can be improved.

Therefore, the reference q-axis current can be obtained from

$$b{i}_q^{*}+P_{\mathrm{total}}=u$$

(12)

Substituting the STSMC law in (9) into the above equation yields

$$i_q^{*}={\displaystyle \frac{1}{b}}\{-k_1{\left|s\right|}^{{\alpha }_1}\mathrm{sgn}\left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}-\lambda s-\int \epsilon \mathrm{sgn}\left(s\right)-P_{total}\}$$

(13)

To alleviate the chattering effect caused by the discontinuous signum function, a continuous switching function is introduced to replace $\mathrm{sgn}\left(s\right)$ in the STSMC law. In this paper, $\theta \left(s\right)$ is constructed as

$$\theta \left(s\right)={\displaystyle \frac{s}{\left|s\right|+{\sigma }_0{e}^{-\gamma \left|s\right|}}}$$

(14)

where ${\sigma }_0$ is the initial boundary layer width, $\gamma$ represents exponential decay factor. It can be found that the proposed function $\theta \left(s\right)$ possesses the following properties: When $\left|s\right|$ approaches zero, the function $\theta \left(s\right)$ approaches $s/{\sigma }_0$, which forms a local linear region around the equilibrium, reducing steady-state chattering. And when $\left|s\right|$ approaches infinity, the function $\theta \left(s\right)$ approaches $sign\left(s\right)$, which guarantees fast convergence in the transient stage.

Finally, the control law is derived as

$$i_q^{*}={\displaystyle \frac{1}{b}}\{-k_1{\left|s\right|}^{{\alpha }_1}\theta \left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}-\lambda s-\int \epsilon \theta \left(s\right)-P_{total}\}$$

(15)

The influence of measurement noise and parameter uncertainties can be regarded as bounded lumped disturbances in the sliding-mode dynamics. Since the proposed super-twisting sliding mode controller is designed under bounded disturbance conditions, the closed-loop system can still maintain finite-time convergence when the control gains are properly selected. Moreover, the disturbance observer compensates the low-frequency load disturbance and parameter mismatch, while the continuous control law of the super-twisting algorithm helps reduce the chattering caused by measurement noise. Therefore, the proposed method is expected to be robust against measurement noise and parameter uncertainties.

The parameters of the proposed STSMC are determined by balancing the convergence speed and chattering suppression. Specifically, $k_1$ and $k_2$ are used to regulate the nonlinear reaching rate, $\lambda$ is adopted to improve the response speed for large sliding errors, and $\epsilon$ determines the updating speed of the auxiliary variable $\vartheta$. Meanwhile, ${\sigma }_0$ and $\gamma$ are used to adjust the variation rate of the continuous switching function, respectively. In this paper, the parameters are set as Table 1.

Table 1. Parameter selection of the proposed STSMC.

Parameters	Value	Parameters	Value
${\alpha }_1$	0.5	$k_1$	800
${\alpha }_2$	0.5	$k_2$	40
$\lambda$	10	$\epsilon$	200
${\sigma }_0$	100	$\gamma$	300

Stability Analyses

Consider the Lyapunov candidate function

$$V={\displaystyle \frac{s^2}{2}}$$

(16)

Then combining (10) and (15), the derivative of Lyapunov candidate function is

$$\begin{array}{cc}\hfill \dot{V}& =s\dot{s}\hfill \\ \hfill & =s(-k_1|s{|}^{{\alpha }_1}\theta \left(s\right)-k_2|s{|}^{{\alpha }_2}-\lambda s-\int \epsilon \theta \left(s\right))\hfill \\ \hfill & =-k_1{\left|s\right|}^{{\alpha }_1+1}-k_2{\left|s\right|}^{{\alpha }_2+1}-\lambda s^2-\epsilon \Omega \hfill \end{array}$$

(17)

where $\Omega$ is a positive constants. With an adequately chosen $\epsilon$, the inequality holds without difficulty. According to Lyapunov stability analysis, the voltage loop fulfills the reaching condition, guaranteeing finite-time convergence of the sliding motion and overall system stability under the CE-STSMC control law. The configuration of the voltage outer-loop controller employing the CE-STSMC algorithm is depicted in Figure 3.

Figure 3. Block diagram of the STSMC.

3.2. Design of the Nonlinear Disturbance Observer

As can be seen in (15), the control law includes unknown total loss power, which will affect the selection of controller parameters. Although large control gain can overcome the effect of the disturbance, it will lead large chattering. However, observing disturbances through an observer and compensating for them can reduce controller gain, thereby minimizing chattering. Thereby, an NDOB is designed to estimate the total loss power online. Then the estimated disturbance can be feedforward into the CE-STSMC.

According to (5), the NDOB can be designed as

$$\left\{\begin{array}{c}\dot{z_p}=-m{z}_p-m[p\left(x\right)+b{i}_q]\hfill \\ {\widehat{P}}_{total}=z_p+p\left(x\right)\hfill \end{array}\right.$$

(18)

where $z_p$, m and ${\widehat{P}}_{total}$ are the estimated values for the internal variables of the detector, the gain coefficient, and the estimated total loss power; the $p\left(x\right)$ is the function to be designed, where $p\left(x\right)=m{E}_c$ can be taken here.

The estimated error in defining the total power loss is defined as

$$e_p=P_{total}-{\widehat{P}}_{total}$$

(19)

Then the derivative of the error is

$${\dot{e}}_p={\dot{P}}_{total}-{\dot{\widehat{P}}}_{total}$$

(20)

Substituting (5) and (18) into above equation, then

$${\dot{e}}_p={\dot{P}}_{total}-{\dot{z}}_p-m{\dot{E}}_c={\dot{P}}_{total}-m{e}_p$$

(21)

For the real flywheel energy storage systems, the total power loss is bounded, and the derivative approaches zero. Hence, when the gain m is positive, the estimated error will converge to zero as time approaches infinity, which means the observer can estimate the total power loss. Moreover, the convergence rate of the observer is determined by m; the faster convergence rate can be achieved with a larger value of m. In practical implementation, m is selected by considering both the convergence rate and the noise sensitivity of the observer. In this paper, $m=25$ is adopted in experiments, which provides a satisfactory compromise between fast convergence and smooth estimation.

By applying the Laplace transform to (18), the transfer function of the observer can be obtained as

$$\begin{array}{cc}\hfill {\widehat{P}}_{total}& ={\displaystyle \frac{ms}{s+m}}{E}_c\left(s\right)-{\displaystyle \frac{m}{s+m}}b{i}_q\left(s\right)\hfill \\ \hfill & ={\displaystyle \frac{s}{s/m+1}}{E}_c\left(s\right)-{\displaystyle \frac{1}{s/m+1}}b{i}_q\left(s\right)\hfill \end{array}$$

(22)

It can be seen in (22) that the NDOB possesses low-pass filtering characteristics. Consequently, the output values require no further filtering and can effectively reduce the ripple of observed disturbances. Simultaneously, since the estimated total power loss of the observation contains the approximate differential of $E_c$, an excessively large m value will increase measurement noise, adversely breaking system stability.

After integrating the disturbance estimation mechanism, the final composite control law for the voltage control is derived by feeding the observed total power loss into the STSMC, as follows:

$$i_q^{*}={\displaystyle \frac{1}{b}}\{-k_1{\left|s\right|}^{{\alpha }_1}\theta \left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}-\lambda s-\int \epsilon \theta \left(s\right)-{\widehat{P}}_{total}\}$$

(23)

The overall control structure of the proposed method is presented in Figure 4. The super-twisting sliding-mode controller (STSMC) is employed in the outer loop to regulate the DC-link voltage and generate q-axis reference current for inner current loop. Meanwhile, a nonlinear disturbance observer (NDOB) estimates the lumped disturbance online and provides feedforward compensation to enhance robustness. The inner current loop is implemented in the synchronous $dq$ frame with PI regulators, whose gains are designed via the pole-zero cancellation method [30]. In addition, an MTPA block is incorporated to improve efficiency by allocating the d-axis current reference.

Figure 4. Control block diagram of the proposed method.

3.3. Stability Analysis of the Proposed Method

To demonstrate the stability of the proposed method, the candidate Lyapunov function is

$$V_2={\displaystyle \frac{1}{2}}{s}^2+{\displaystyle \frac{1}{2}}{e}_p^2$$

(24)

Combined with the (14) formula, the derivative of the candidate Lyapunov function is

$$\begin{array}{cc}\hfill {\dot{V}}_2& =s\dot{s}+e_p{\dot{e}}_p\hfill \\ \hfill & =s(-k_1{\left|s\right|}^{{\alpha }_1}-k_2{\left|s\right|}^{{\alpha }_2}-\lambda s-\epsilon \Omega )+e_p{\dot{e}}_p\hfill \\ \hfill & \le -(\lambda +{\displaystyle \frac{\epsilon \Omega }{s^2}})s^2+e_p{\dot{e}}_p\hfill \\ \hfill & \le \left|\lambda +{\displaystyle \frac{\epsilon \Omega }{s^2}}\right|s^2+\left|e_p{\dot{e}}_p\right|\hfill \end{array}$$

(25)

Both $e_p$ and ${\dot{e}}_p$ are bounded. This means there must exist constants $L_1$ and $L_2>0$ such that

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le L_1{s}^2+\left|e_p{\dot{e}}_p\right|\hfill \\ \hfill & =L_1(2V_1-e_p^2)+\left|e_p{\dot{e}}_p\right|\le 2L_1{V}_2+L_2\hfill \end{array}$$

(26)

where $L_1>\left|\lambda +{\displaystyle \frac{\epsilon \Omega }{s^2}}\right|$, $L_2>-L_1{e}_p^2+\left|e_p{\dot{e}}_p\right|$

Based on (26), the boundedness of $V_2$ is ensured, and consequently the trajectories of s and $e_p$ are uniformly bounded throughout the convergence process. Thus, prior to the finite-time bounded convergence of the observer errors, neither s nor $e_p$ can exhibit unbounded growth.

Next, the convergence property of the sliding variable is further analyzed. Considering the observer estimation error, the closed-loop sliding-mode dynamics can be written as

$$\dot{s}=-k_1{\left|s\right|}^{{\alpha }_1}\theta \left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}\theta \left(s\right)-\lambda s-\epsilon \Omega +e_p,$$

(27)

According to the stability analysis of the observer, $e_p$ is bounded in finite time. Thus, there exists a positive constant ${\overline{e}}_p$ such that $|e_p|\le {\overline{e}}_p$.

The Lyapunov candidate function is selected as

$$V_s={\displaystyle \frac{1}{2}}{s}^2$$

(28)

Taking the derivative of $V_s$ gives

$$\begin{array}{cc}\hfill {\dot{V}}_s=& s\dot{s}\hfill \\ \hfill =& -k_1{\left|s\right|}^{{\alpha }_1}s\theta \left(s\right)-k_2{\left|s\right|}^{{\alpha }_2}s\theta \left(s\right)-\lambda s^2-\epsilon s\Omega +s{e}_p.\hfill \end{array}$$

(29)

Since

$$s\theta \left(s\right)={\displaystyle \frac{s^2}{\left|s\right|+{\sigma }_0{e}^{-\gamma \left|s\right|}}}\ge 0$$

(30)

and $\epsilon s\Omega \ge 0$, one obtains

$${\dot{V}}_s\le -k_1{\left|s\right|}^{{\alpha }_1}s\theta \left(s\right)-\lambda s^2+{\overline{e}}_p\left|s\right|$$

(31)

For the case of $\left|s\right|>{\delta }_s$, where ${\delta }_s>0$ is a prescribed small boundary layer, the derivative of $V_s$ can be expressed as

$${\dot{V}}_s\le -k_1{\kappa }_{\delta }{\left|s\right|}^{{\alpha }_1+1}-\lambda s^2+{\overline{e}}_p\left|s\right|$$

(32)

As the control gains are designed to satisfy

$$k_1{\kappa }_{\delta }{\delta }_s^{{\alpha }_1}+\lambda {\delta }_s>{\overline{e}}_p$$

(33)

then ${\dot{V}}_s<0$ for $\left|s\right|>{\delta }_s$. Therefore, the sliding variable can converge to the boundary layer $\left|s\right|\le {\delta }_s$.

For the case of $\left|s\right|\le {\delta }_s$, the continuous function $\theta \left(s\right)$ behaves as a local linear function around the origin. Therefore,

$${\dot{V}}_s\le -{\displaystyle \frac{k_1}{{\delta }_s+{\sigma }_0}}{\left|s\right|}^{{\alpha }_1+2}-\lambda s^2+{\overline{e}}_p\left|s\right|$$

(34)

When the observer estimation error converges to zero, ${\overline{e}}_p=0$, and thus ${\dot{V}}_s\le -\lambda s^2\le 0$. Hence, the sliding variable asymptotically converges to zero inside the boundary layer. When the observer estimation error is bounded, the sliding variable is ultimately bounded by a small neighborhood around the origin.

Consequently, the proposed controller-observer structure is practically finite-time stable. The sliding variable reaches a small boundary layer in finite time and further converges to zero if the observer estimation error vanishes.

4. Experimental Result

To verify the effectiveness of the proposed method in this article, the experiments have been carried out. The parameters of the FESSs are listed in Table 2. The experimental platform is shown in Figure 5. A DC power supply is employed to accelerate the permanent magnet synchronous motor within the flywheel energy storage system to a predetermined rotational speed, thereby storing energy within the high-speed rotating flywheel. Subsequently, the DC bus is disconnected. The three-phase voltage-source inverter utilises intelligent power module (IPM). The proposed algorithm is implemented on a Texas Instruments TMS320F28335 processor (Texas Instruments, Dallas, TX, USA), where both the sampling and switching frequencies are set to 10 kHz. The DC-link voltage is measured using the LV25-P voltage module (LEM International SA, Meyrin, Switzerland). The block diagram of the proposed discharge methods is presented in Figure 4. To demonstrate the superiority of the proposed algorithm, traditional PI and NDOB are employed as comparison methods.

Table 2. Parameters of the FESSs.

Parameters	Value	Unit
Rated power	3.75	kW
Rated voltage	190	V
Rated current	11.4	A
Stator resistance	0.31	$\Omega$
d-axis inductance	2.8	mH
q-axis inductance	4.47	mH
Flux linkage	0.135	Wb
Number of pole pairs	2	-

Figure 5. Experimental setup for FESSs.

4.1. Tracking Performance Analysis

As shown in Figure 6, Figure 7 and Figure 8, the DC-link voltage tracking responses under different control schemes are compared with initial speed of 4500 r/min. During the voltage tracking process, the reference voltage is stepped down from 350 V to 250 V and then restored from 250 V to 350 V. Although all methods can regulate DC-link voltage to the command value, the dynamic performances are significantly different. The responses in Figure 6 and Figure 7 suffer from evident oscillation and voltage fluctuation during the step-change process. In comparison, the responses in Figure 8 present superior dynamic behavior, characterized by smoother transients, smaller overshoot, and faster convergence. Therefore, the proposed strategy behaves higher voltage tracking and stronger transient robustness under sudden command changes.

Figure 6. The tracking performance of DC-link voltage with reference voltage step using PI.

Figure 7. The tracking performance of DC-link voltage with reference voltage step using NDOB.

Figure 8. The tracking performance of DC-link voltage with reference voltage step using proposed method.

The above experimental results show that the q-axis current increases rapidly when the flywheel speed decreases to a very low level during the final discharge stage. This phenomenon is closely related to the physical discharge limit of the flywheel energy storage system. At very low speed, the flywheel system reaches its physical discharge limit. Since the electromagnetic power maintain the same discharge power requires a larger q-axis current as the speed decreases. When the speed approaches zero, the required current will increase. Therefore, in the practical implementation, a q-axis current saturation is introduced. Once the current limit is reached, the discharge process is terminated to avoid overcurrent and excessive DC-bus voltage drop.

4.2. Load Disturbance Suppression Analysis

In order to verify the load disturbance ability, Figure 9, Figure 10 and Figure 11 represent the performance of different methods under sudden load changes. During the experiments, the bus voltage is set as 300 V, and a resistance of 950 $\Omega$ is suddenly applied. It can be seen that under the sudden load situation, under a sudden load change, the DC-link voltage drops to 28.4 V and 20.4 V for the PI and NDOB control schemes respectively, with recovery times of 0.9 s and 0.52 s, respectively. On the contrary, the lowest voltage drop and shortest recovery time can be obtained with the proposed method. The proposed method exhibits excellent dynamic performance since the improved STSMC can rapidly generate the q-axis reference current. At the same time, the NDOB can also provide feedforward compensation for sudden load disturbances, causing the phase current to increase rapidly and accelerating the conversion of kinetic energy into electrical energy. Therefore, the disturbance rejection capability of the proposed method against sudden load application can be verified.

Figure 9. The tracking performance of DC-link voltage with load step using PI.

Figure 10. The tracking performance of DC-link voltage with load step using NDOB.

Figure 11. The tracking performance of DC-link voltage with load step using proposed method.

In order to provide a comparison with the existing method in [24], the DFTSMC method is additionally included in the sudden-load experiment at 6000 r/min. It can be observed from Figure 9, Figure 10, Figure 11 and Figure 12 that the DC-link voltage drops in both the proposed method and the other schemes at the end of the discharge process. This phenomenon can be interpreted from (5). As the flywheel speed continuously declines, the required q-axis current rises sharply and theoretically tends to infinity as the speed approaches zero, which is physically unrealizable. Therefore, in the experimental implementation of all the methods, the speed used in the controller is clamped to a small constant value once it falls below a near-zero threshold. Under this condition, the reference q-axis current can no longer provide sufficient compensation to maintain the DC-link voltage constant. The same effect can also be observed in the subsequent experimental results.

Figure 12. The tracking performance of DC-link voltage with load step using DFTSMC.

Table 3 gives the quantitative performance comparison of different control methods. It can be observed that the proposed method achieves lower overshoot, shorter settling time, and smaller steady-state error than PI, NDOB, and DFTSMSC, which further demonstrates its superior dynamic response and steady-state regulation performance.

Table 3. Quantitative performance comparison of different methods.

Index	PI	NDOB	DFTSMSC	Proposed Method
Overshoot (V)	28.4	20.4	14.2	9.8
Settling time (s)	0.9	0.52	0.28	0.1
Steady-state error (V)	4.0	2.0	0.9	0.4

4.3. Robustness Comparison

To demonstrate the robustness of the proposed method, experiments were conducted under different capacitance conditions. Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 show the experimental results of the voltage tracking and sudden load tests under the condition of $C=0.75C_{real}$, where $C_{real}$ is the real value of the capacitor in FESSs. The condition of $C=0.75C_{real}$ is considered since $0.75C_{real}$ of the rated capacitance can be viewed as the allowable lower limit in practical operation, and the effect of capacitance reduction on the controller performance becomes more significant under this condition.

Figure 13. The tracking performance of DC-link voltage with reference voltage step using PI under $C=0.75C_{real}$.

Figure 14. The tracking performance of DC-link voltage with reference voltage step using NDOB under $C=0.75C_{real}$.

Figure 15. The tracking performance of DC-link voltage with reference voltage step using proposed method under $C=0.75C_{real}$.

Figure 16. Dynamic responses of PI under a sudden load step with $C=0.75C_{real}$.

Figure 17. Dynamic responses of NDOB under a sudden load step with $C=0.75C_{real}$.

Figure 18. Dynamic responses of the proposed method under a sudden load step with $C=0.75C_{real}$.

Compared with the nominal capacitance condition shown in Figure 6, Figure 7 and Figure 8 for the voltage tracking test and Figure 9, Figure 10 and Figure 11 for the sudden load test, the PI controller and the NDOB-based method exhibit significantly degraded performance when the capacitance varies. Specifically, the voltage tracking performance is severely broken, while a larger DC-link voltage drop and a longer recovery time is clear under sudden load disturbance. These results indicate that the performances of the PI controller and the DOB-based method are significantly affected by the capacitance variation. In contrast, the performance of the proposed method is almost unaffected, thereby demonstrating its strong robustness and superior control performance even under capacitance variation.

4.4. Total Disturbance Estimation Based on DOB

Figure 19 illustrates the DC-link voltage response and the estimated total disturbance under a reference step and a load step. When the reference step is applied, the DC-link voltage exhibits a short transient drop and quickly settles to the steady state, indicating an effective dynamic response. Correspondingly, the estimated disturbance shows a sharp negative peak, which reflects the rapid variation induced by the reference change. The disturbance estimate then converges quickly with small steady-state fluctuation, demonstrating good convergence and noise immunity. When the load step occurs, only a slight voltage deviation is observed, and the voltage is restored within a short duration, indicating strong disturbance rejection capability. Meanwhile, the estimated disturbance increases promptly to a new steady-state level, accurately capturing the effect of the external disturbance. It should be noted that the estimated disturbance by the NDOB in Figure 19 is a lumped disturbance rather than the pure load power, which includes the effects of load variation, system losses, and model uncertainties.

Figure 19. DC-link voltage tracking performance with reference voltage step using proposed method.

Overall, the results verify that the proposed NDOB can achieve fast and accurate observation under both reference variation and external disturbance, thereby effectively improving the dynamic and steady-state performance of the DC-link voltage control.

To more comprehensively evaluate the performance of the proposed method, it is necessary to consider the computational overhead of the algorithm during runtime, as shown in Figure 20. From Figure 20, it can be viewed that the total execution time includes sampling, control algorithm execution, and modulation calculation. Under the 10 kHz voltage-loop frequency, the available control period is 100 μs. The execution times of PI, DOB, DFTSMC, and the proposed method are 11.5 μs, 13.0 μs, 18.4 μs, and 17.5 μs, respectively. Although the proposed method introduces a disturbance observer and an improved super-twisting sliding mode controller, its execution time is still much shorter than the control period and lower than that of DFTSMC, indicating that the proposed method has acceptable computational complexity and sufficient real-time implementation margin.

Figure 20. Execution time comparison of different control methods.

5. Conclusions

This article proposes a capacitor-energy-based super-twisting sliding mode controller integrated with an NDOB for DC-link voltage control of FESSs. The proposed control framework mainly consists of two parts: (1) the design of a capacitor-energy-based DC-link voltage controller using the STSMC, and (2) the construction of an NDOB to estimate the total disturbance caused by load variations, parameter mismatches, and wide-speed-range operation. The main conclusions are summarized as follows.

• An enhanced STSMC has been developed by introducing an additional nonlinear term and an adaptive continuous function. This design improves the convergence rate and simultaneously suppresses the chattering effect.
• The proposed NDOB can estimate the total disturbance in real time with fast transient response and high estimation accuracy, thereby improving the disturbance rejection capability of the overall system.
• Experimental results confirm that the proposed method achieves satisfactory DC-link voltage control performance and exhibits strong robustness under disturbances and varying operating conditions.